# Convexification for a Coefficient Inverse Problem of Mean Field Games

Michael V. Klibanov1, Jingzhi Li2 and Zhipeng Yang3 1 Department of
Mathematics and Statistics, University of North Carolina at Charlotte,
Charlotte, NC, 28223, USA  2 Department of Mathematics & National Center for
Applied Mathematics Shenzhen & SUSTech International Center for Mathematics,
Southern University of Science and Technology, Shenzhen 518055, P. R. China  3
Department of Mathematics, Southern University of Science and Technology,
Shenzhen 518055, P. R. China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The globally convergent convexification numerical method is constructed for a
Coefficient Inverse Problem for the Mean Field Games System. A coefficient
characterizing the global interaction term is recovered from the single
measurement data. In particular, a new Carleman estimate for the Volterra
integral operator is proven, and it stronger than the previously known one.
Numerical results demonstrate accurate reconstructions from noisy data.

††: Inverse Problems

, ,

  *

Keywords: global convergence, numerical studies, mean field games

##  1 Introduction

The mean field games (MFG) theory examines the collective behavior of an
infinite number of rational agents. This theory was initially introduced in
the seminal works of Lasry and Lions [31, 32] and Huang, Caines, and Malhamé
[16, 17]. A commonly recognized serious mathematical advantage of the MFG
theory is that it is based on a universal system of coupled nonlinear
parabolic Partial Differential Equations (PDEs) known as the Mean Field Games
System (MFGS) [1]. At this point of time the MFG theory is the single
mathematical model of a broad spectrum of the societal phenomena that uses a
universal system of PDEs [8].

In addition to the nonlinearity, there are two other quite substantial
challenges in working with the MFGS. The first challenge is that those two
equations have two opposite directions of time. Therefore, the conventional
theory of parabolic PDEs is inapplicable to the MFGS. The second challenge is
due to the presence of an integral operator, the so-called “global interaction
term”, in one of equations of the MFGS, see below in this section. This
presence is very unusual in the theory of parabolic equations. On one hand,
there is no applied meaning of the MFGS without this term. On the other hand,
the presence of this integral term prevents a straightforward application of
previously developed theory of CIPs for one parabolic PDE to CIPs for the
MFGS.

With the increasing significance of social sciences in the modern society,
MFG-based mathematical modeling of social phenomena has a potential of a
substantial societal impact [8]. Indeed, this theory finds a rapidly growing
number of applications in such areas as, e.g. finance, combating corruption,
cybersecurity, interactions of electrical vehicles, election dynamics, robotic
control, etc. For a far non-exhaustive list of references describing these
applications, we refer to, e.g. [1, 10, 11, 16, 17, 30, 31, 32, 43].

Definition 1.1. _We call a problem for the MFGS the “forward problem” if it
consists in finding a solution of this system under the assumption that its
coefficients are known. And we call a problem for the MFGS the “Coefficient
Inverse Problem (CIP)” if it is required in it to find a coefficient(s) of
this system using data occurring in some measurement events._

Definition 1.2. _We call a numerical method for a CIP “globally convergent” is
there is a rigorous guarantee that it converges to the true solution of this
problem without an advanced knowledge of any point in a sufficiently small
neighborhood of this solution._

Given a broad range of applications of the MFG theory, it becomes important to
address several mathematical questions for both forward problems and CIPs for
the MFGS. We introduce here the first globally convergent numerical method for
a CIP for the MFGS. This is a version of the so-called “convexification
method”. The convexification was first originated in the theoretical work of
Klibanov [18]. Next, publication [4] has removed some obstacles for
computations, which were present in [18]. Since then a number of works were
published, which synthesize analytical and numerical parts of various versions
of the convexification for a number of CIPs, see, e.g. [22, 23, 21, 33] and
references cited therein. In particular, the most recent publication [29] of
this research group is about the convexification method for a forward problem
for the MFGS.

Conventional numerical methods for CIPs are based on the least squares
minimization, see, e.g. [2, 3, 5, 6, 9, 13, 14, 15, 39]. While quite powerful,
this technique has a drawback: it suffers from the phenomenon of multiple
local minima and ravines of cost functionals, see, e.g. [41] for a good
numerical example. This phenomenon, in turn means that convergence of
corresponding algorithms to true solutions can be rigorously guaranteed only
if the starting point of such an algorithm is located in a sufficiently small
neighborhood of that solution. The latter means local convergence. In fact,
the convexification method avoids the phenomenon of local minima and ravines.

This paper consists of two parts:

  1. 1.

Construction and global convergence analysis of the convexification method for
our CIP. As a by-product, we establish uniqueness result for our CIP.

  2. 2.

Numerical studies. As the second by-product, we present here a new procedure
of data generation for the MFGS.

Another new element of this publication, which is interesting in its own
right, is a new Carleman estimate for a Volterra-like integral operator, see
Theorem 4.1. This estimate is stronger than the previously known one of, e.g.
[19, Lemma 1.10.3] and [23, Lemma 3.1.1].

Our CIP is about the recovery of the coefficient $k\left(x\right)$ in

$k\left(x\right)\int\limits_{\Omega}K\left(x,y\right)p\left(y,t\right)dy,$
(1.1)

see section 2 for notations. The integral in (1.1) is called the “global
interaction term”. Such a term is not a part of all previous works on CIPs.
The applied meaning of this term is explained in [24, section 5]. More,
precisely, the kernel $K(x,y)$ is the influence on the agent, who occupies the
state $x,$ by the agent occupying the state $y$. The integral (1.1) is the
average influence on the agent occupying the state $x,$ by the rest of agents.

Let a small number $\delta>0$ be the level of noise in the input data for our
CIP. To specify Definition 1.2 for our case, the term “global convergence”
refers to the convergence analysis resulting in a theorem, which guarantees
that if $\delta\rightarrow 0,$ then the iterative solutions generated by our
method converge to the true solution of our CIP (if it exists) starting from
any point within a predefined convex bounded set in a Hilbert space with a
fixed but arbitrary diameter $d>0$. In simple terms, we do not need a good
first guess for the solution. An explicit estimate of the convergence rate is
also given here. Results of our computational experiments demonstrate a good
accuracy of computed solutions in the presence of a random noise in the input
data.

The authors are aware about only two previous publications on numerical
studies of CIPs for the MFGS [10, 12]. Numerical methods of these references
are significantly different from ours, and their global convergence property
is not proven.

Just as in all other previous publications of this research group about both
forward and inverse problems for the MFGS [24]-[29], we work here with the
input data resulting from a single measurement event. We refer to [27, 28] for
previous analytical works on CIPs for the MFGS, where questions of uniqueness
and stability of those problems are addressed for the single measurement case.
In publications [34, 35], uniqueness of some inverse problems for the MFGS was
proven for the case of infinitely many measurements.

The work of Klibanov and Averboukh [24] is the first one, in which Carleman
estimates were introduced in the MFG theory. We quite essentially use Carleman
estimates here, so as in all our works on the MFG theory [24]-[29]. Both the
convexification method and [27, 28] use the idea of the paper of Bukhgeim and
Klibanov [7], where the method of Carleman estimates was introduced in the
field of Inverse Problems for the first time, see, e.g. [20, 23, 19] and
references cited therein for some follow up publications.

Remark 1.1. _Minimal smoothness requirements traditionally hold little
significance in the theory of Ill-Posed and Inverse Problems, as seen in
publications like, e.g.[23, 38], [40, Theorem 4.1]. Consequently, such
requirements are not a primary concern below._

This paper is arranged as follows. In section 2 we present the MFGS and
formulate our CIP. In section 3 we present the version of the convexification
method for this CIP. In section 4 we formulate theorems of our global
convergence analysis and prove one of them. In sections 5-7 we prove three
more theorems formulated in section 4. Section 8 is dedicated to numerical
studies. Summary of results is given in section 9. All functions considered
below are real valued ones.

##  2 Problem Statement

Let $x=\left(x_{1},...,x_{n}\right)$ denotes points in $\mathbb{R}^{n}.$
Denote $\overline{x}=\left(x_{2},...,x_{n}\right).$ Let
$a,b,M,A_{i}>0,i=2,...,n$ and $T>0$ be some numbers and $a<b$. Let the number
$\gamma\in\left(0,1\right).$We set the domain $\Omega\subset\mathbb{R}^{n}$ as
a rectangular prism,

$\left.\begin{array}[]{c}\Omega=\left\\{x:a<x_{1}<b,-A_{i}<x_{i}<A_{i},i=2,...,n\right\\},\mbox{
}\\\ \Omega_{1}=\left\\{-A_{i}<x_{i}<A_{i},i=2,...,n\right\\},\mbox{
}\Gamma=\partial\Omega\cap\left\\{x_{1}=b\right\\},\\\
Q_{T}=\Omega\times\left(0,T\right),\mbox{
}S_{T}=\partial\Omega\times\left(0,T\right),\mbox{
}\Gamma_{T}=\Gamma\times\left(0,T\right),\\\ Q_{\gamma
T}=\Omega\times\left(\left(1-\gamma\right)T/2,\left(1+\gamma\right)T/2\right)\subset
Q_{T}.\end{array}\right.$ (2.1)

We now specify the kernel $K\left(x,y\right),$ $x,y\in\Omega$ of the integral
operator (1.1) of the global interaction term of the MFGS. It was noticed in
[37, section 4.2] that a good choice for $K\left(x,y\right)$ would be the
product of Gaussians. Since Gaussian approximates the $\delta-$function in the
sense of distributions, then we set

$\left.\begin{array}[]{c}K_{1}\left(x,y\right)=k\left(x\right)\delta\left(x_{1}-y_{1}\right)\overline{K}_{1}\left(x,\overline{y}\right),\\\
k\left(x\right)\in L_{\infty}\left(\Omega\right),\mbox{
}\left\|k\right\|_{L_{\infty}\left(\Omega\right)}\leq M,\\\
\overline{K}_{1}\left(x,\overline{y}\right)\in
L_{\infty}\left(\Omega\times\Omega_{1}\right),\mbox{
}\left\|\overline{K}_{1}\right\|_{L_{\infty}\left(\Omega\times\Omega_{1}\right)}\leq
M.\end{array}\right.$ (2.2)

We also consider the second form of the kernel $K\left(x,y\right)$ as the one
in [28]:

$K_{2}\left(x,y\right)=H\left(y_{1}-x_{1}\right)\overline{K}_{2}\left(x,y\right).$
(2.3)

where $H$ is the Heaviside function,

$H\left(z\right)=\left\\{\begin{array}[]{c}1,z>0,\\\ 0,z<0.\end{array}\right.$

Below $K\left(x,y\right)$ is any of functions (2.2), (2.3). Let the
coefficient

$k\left(x\right)\in C\left(\overline{\Omega}\right).$ (2.4)

Thus, below

$\left.\begin{array}[]{c}k\left(x\right)\int\limits_{\Omega}K\left(x,y\right)f\left(y,t\right)dy=k\left(x\right)\int\limits_{\Omega_{1}}\overline{K}_{1}\left(x,\overline{y}\right)f\left(x_{1},\overline{y},t\right)dy\mbox{,}\\\
\mbox{in the case }(\ref{2.2}),\\\ \mbox{and
}k\left(x\right)\int\limits_{\Omega}K\left(x,y\right)f\left(y,t\right)dy=k\left(x\right)\int\limits_{x_{1}}^{b}\left(\int\limits_{\Omega_{1}}\overline{K}_{2}\left(x,\overline{y}\right)f\left(y,t\right)d\overline{y}\mbox{
}\right)dy_{1}\mbox{ }\\\ \mbox{in the case }(\ref{2.3}),\\\ \forall f\in
L_{\infty}\left(Q_{T}\right).\end{array}\right.$ (2.5)

We consider the MFGS of the second order in the following form [1]:

$\left.\begin{array}[]{c}v_{t}(x,t)+\Delta v(x,t){-r(x)(\nabla
v(x,t))^{2}/2}-\\\
-k\left(x\right)\int\limits_{\Omega}K\left(x,y\right)p\left(y,t\right)dy-s\left(x,t\right)p\left(x,t\right)=0,\mbox{
}\left(x,t\right)\in Q_{T},\\\ p_{t}(x,t)-\Delta
p(x,t){-\mbox{div}(r(x)p(x,t)\nabla v(x,t))}=0,\mbox{ }\left(x,t\right)\in
Q_{T},\end{array}\right.$ (2.6)

where $s\left(x,t\right),p\left(x,t\right)$ is the local interaction term.
Here $v(x,t)$ is the value function and $p(x,t)$ is the density of players. We
assume that

$\left.\begin{array}[]{c}r\in C^{1}\left(\overline{\Omega}\right);\mbox{
}s,s_{t}\in L_{\infty}\left(Q_{T}\right),\\\
\left\|r\right\|_{C^{1}\left(\overline{\Omega}\right)},\left\|s\right\|_{L_{\infty}\left(Q_{T}\right)},\left\|s_{t}\right\|_{L_{\infty}\left(Q_{T}\right)}\leq
M.\end{array}\right.$ (2.7)

Coefficient Inverse Problem (CIP). _Assume that the following functions_
$v_{0}\left(x\right)$, $p_{0}\left(x\right)$, $g_{0,i}\left(x,t\right)$,
$g_{1,i}\left(x,t\right)$_are known:_

$\left.\begin{array}[]{c}v\left(x,T/2\right)=v_{0}\left(x\right),\mbox{
}p\left(x,T/2\right)=p_{0}\left(x\right),\mbox{ }x\in\Omega,\\\
v\mid_{S_{T}}=g_{0,1}\left(x,t\right),\mbox{
}p\mid_{S_{T}}=g_{0,2}\left(x,t\right),\\\
v_{x_{1}}\mid_{\Gamma_{T}}=g_{1,1}\left(x,t\right),\mbox{
}p_{x_{1}}\mid_{\Gamma_{T}}=g_{1,2}\left(x,t\right).\end{array}\right.$ (2.8)

_Determine the coefficient_ $k\left(x\right)$_in ( 2.6), assuming (2.4)._

Note that since the Neumann boundary conditions are not given at
$S_{T}\diagdown\Gamma_{T}$ then the boundary data in (2.6) are incomplete. In
(2.6)__ $T/2$ can be replaced with any fixed number $t_{0}\in\left(0,T\right)$
without any significant changes in our derivations below.

Remark 2.1. _The data ( 2.6) are generated by a single measurement event. As
to a real game, it was pointed out in [28, section 2.2 ] that these data can
be obtained via polling game participants._

##  3 Convexification

Below we work either with (2.2) or with (2.3) and keep (2.5).__ Let $c>0$ be a
number. Assume that

$\mbox{in the case }\mbox{(\ref{2.2})}\mbox{
}\left|\int\limits_{\Omega_{1}}\overline{K}_{1}\left(x,\overline{y}\right)p_{0}\left(x_{1},\overline{y}\right)d\overline{y}\right|\geq
c,\mbox{ }x\in\overline{\Omega}.$ (3.1)

Let $\eta\in\left(0,b-a\right)$ be a number. Denote

$\chi\left(x_{1}\right)=\left\\{\begin{array}[]{c}1,x_{1}\in\left[0,b-\eta\right),\\\
0,x_{1}\in\left[b-\eta,b\right].\end{array}\right\\}$

In addition to (3.1), we also assume that

$\left.\begin{array}[]{c}\mbox{In the case (\ref{2.3})}\\\
k\left(x\right)=0,\mbox{ }x_{1}\in\left(b-\eta,b\right),\\\
\chi\left(x_{1}\right)\int\limits_{x_{1}}^{a}\left(\int\limits_{\Omega_{1}}\overline{K}_{2}\left(x,\overline{y}\right)p_{0}\left(y\right)d\overline{y}\mbox{
}\right)dy_{1}\geq c,\mbox{ }x\in\overline{\Omega}.\end{array}\right.$ (3.2)

Denote

$u\left(x,t\right)=v_{t}\left(x,t\right),\mbox{
}m\left(x,t\right)=p_{t}\left(x,t\right).$ (3.3)

Using (2.8) and (3.3), we obtain

$\left.\begin{array}[]{c}v(x,t)=\int\limits_{T/2}^{t}u(x,\tau)d\tau+v_{0}\left(x\right),\mbox{
}\left(x,t\right)\in Q_{T},\\\
p(x,t)=\int\limits_{T/2}^{t}m(x,\tau)d\tau+p_{0}\left(x\right),\mbox{
}\left(x,t\right)\in Q_{T}.\end{array}\right.$ (3.4)

We have:

$u\left(x,T/2\right)=u\left(x,t\right)-\int\limits_{T/2}^{t}u_{t}\left(x,\tau\right)d\tau.$
(3.5)

Setting in the first line of (2.6) $t=T/2$ and using (3.1)-(3.5), we obtain

$\left.\begin{array}[]{c}k\left(x\right)=u\left(x,T/2\right)f\left(x\right)+F\left(x\right),\\\
\mbox{or, equivalently,
}k\left(x\right)=\left(u\left(x,t\right)-\int\limits_{T/2}^{t}u_{t}\left(x,\tau\right)d\tau\right)f\left(x\right)+F\left(x\right),\mbox{
}\\\ F\left(x\right)=\left[\left(\Delta v_{0}-r\left(\nabla
v_{0}\right)^{2}/2-s\left(x,T/2\right)p_{0}\right)\left(x\right)\right]f\left(x\right),\end{array}\right.$
(3.6)

where the function $f\left(x\right)$ is defined as:

$\left.f\left(x\right)=\left\\{\begin{array}[]{c}\left[\int\limits_{\Omega_{1}}\overline{K}_{1}\left(x,\overline{y}\right)p_{0}\left(x_{1},\overline{y}\right)d\overline{y}\right]^{-1},\\\
\mbox{in the case }\emph{(\ref{2.2}),}\\\
\chi\left(x_{1}\right)\left[\int\limits_{x_{1}}^{a}\left(\int\limits_{\Omega_{1}}\overline{K}_{2}\left(x,\overline{y}\right)p_{0}\left(y\right)d\overline{y}\mbox{
}\right)dy_{1}\right]^{-1},\\\ \mbox{in the case
}\emph{(\ref{2.3}).}\end{array}\right\\}\right.$ (3.7)

Differentiate equations (2.6) with respect to $t.$ Substituting (3.3)-(3.7) in
resulting equations and using (2.5), we obtain two nonlinear integral
differential equations with the lateral Cauchy data for the vector function
$\left(u,m\right)\left(x,t\right).$ The first equation is:

$\left.\begin{array}[]{c}L_{1}\left(u,m,v_{0},p_{0}\right)=u_{t}+\Delta
u{-r\nabla u}\left(\int\limits_{T/2}^{t}\nabla u(x,\tau)d\tau+\nabla
v_{0}\left(x\right)\right)-\\\
-f\left(x\right)\left[\left(u\left(x,t\right)-\int\limits_{T/2}^{t}u_{t}\left(x,\tau\right)d\tau\right)+F\left(x\right)\right]\int\limits_{\Omega}K\left(x,y\right)m\left(y,t\right)dy-\\\
-sm-
s_{t}\left(\int\limits_{T/2}^{t}m(x,\tau)d\tau+p_{0}\left(x\right)\right),\mbox{
}\left(x,t\right)\in Q_{T}.\end{array}\right.$ (3.8)

The second equation is:

$\left.\begin{array}[]{c}L_{2}\left(u,m,v_{0},p_{0}\right)=m_{t}-\Delta
m(x,t){-}\\\ -{\mbox{div}}\left[{r(x)m\left(\int\limits_{T/2}^{t}\nabla
u(x,\tau)d\tau+\nabla v_{0}\left(x\right)\right)}\right]-\\\
-\mbox{div}\left[{r(x)\nabla
u}\left(\int\limits_{T/2}^{t}m(x,\tau)d\tau+p_{0}\left(x\right)\right)\right]=0,\mbox{
}\left(x,t\right)\in Q_{T}.\end{array}\right.$ (3.9)

The lateral Cauchy data for functions $u\left(x,t\right)$ and
$m\left(x,t\right)$ are:

$\left.\begin{array}[]{c}u\mid_{S_{T}}=\partial_{t}g_{0,1}\left(x,t\right),\mbox{
}m\mid_{S_{T}}=\partial_{t}g_{0,2}\left(x,t\right),\\\
u_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,1}\left(x,t\right),\mbox{
}m_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,2}\left(x,t\right).\end{array}\right.$
(3.10)

Suppose that problem (3.8)-(3.10) is solved. Then the unknown coefficient
$k\left(x\right)$ can be recovered via the first line of (3.6). Therefore, we
focus below on the numerical solution of problem (3.8)-(3.10). In our
derivations below we need functions $u_{x_{i}},u_{t},m_{x_{i}},m_{t},\Delta
u,\Delta m\in C\left(\overline{Q}_{T}\right),$ see Remark 1.1. Therefore, we
assume that functions $u,m\in H^{k_{n}}\left(Q_{T}\right),$ where
$k_{n}=\left[\left(n+1\right)/2\right]+3,$ where
$\left[\left(n+1\right)/2\right]$ is the largest integer not exceeding
$\left(n+1\right)/2.$ By embedding theorem there exists a constant
$C_{0}=C_{0}\left(Q_{T}\right)>0$ depending only on the domain $Q_{T}$ such
that

$H^{k_{n}}\left(Q_{T}\right)\subset C^{2}\left(\overline{Q}_{T}\right),\mbox{
}\left\|y\right\|_{C^{2}\left(\overline{Q}_{T}\right)}\leq
C_{0}\left\|y\right\|_{H^{k_{n}}\left(Q_{T}\right)},\mbox{ }\forall y\in
H^{k_{n}}\left(Q_{T}\right).$ (3.11)

Introduce four Hilbert spaces:

$\left.\begin{array}[]{c}H^{2,1}\left(Q_{T}\right)=\left\\{u:\left\|u\right\|_{H^{2,1}\left(Q_{T}\right)}^{2}=\sum\limits_{\left|s\right|+2k\leq
2}\left\|D_{x}^{s}D_{t}^{k}u\right\|_{H^{2,1}\left(Q_{T}\right)}^{2}<\infty\right\\},\\\
H_{0}^{2,1}\left(Q_{T}\right)=\left\\{u\in
H^{2,1}\left(Q_{T}\right):u\mid_{S_{T}}=u_{x_{1}}\mid_{\Gamma_{T}}=0\right\\},\\\
H=\left\\{\left(u,m\right):\left\|\left(u,m\right)\right\|_{H}^{2}=\left\|u\right\|_{H^{k_{n}}\left(Q_{T}\right)}^{2}+\left\|m\right\|_{H^{k_{n}}\left(Q_{T}\right)}^{2}<\infty\right\\},\\\
H_{0}=\left\\{\begin{array}[]{c}\left(u,m\right)\in H:\\\
u\mid_{S_{T}}=u_{x_{1}}\mid_{\Gamma_{T}}=m\mid_{S_{T}}=m_{x_{1}}\mid_{\Gamma_{T}}=0.\end{array}\right\\}\end{array}\right.$
(3.12)

Remark 3.1. _Below_ $\left[,\right]$_is the scalar product in the space_ $H.$

Let $R>0$ be an arbitrary number. Consider two sets $B\left(R\right)$ and
$B_{0}\left(R\right)$ defined as:

$B\left(R\right)=\left\\{\begin{array}[]{c}\left(u,m\right)\in H:\\\
u\mid_{S_{T}}=\partial_{t}g_{0,1}\left(x,t\right),\mbox{
}m\mid_{S_{T}}=\partial_{t}g_{0,2}\left(x,t\right),\\\
u_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,1}\left(x,t\right),\mbox{
}m_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,2}\left(x,t\right),\\\
\left\|\left(u,m\right)\right\|_{H}<R,\end{array}\right\\}$ (3.13)
$B_{0}\left(R\right)=\left\\{\left(u,m\right)\in
H_{0}:\left\|\left(u,m\right)\right\|_{H}<R\right\\}.$ (3.14)

Let $\alpha$ be a rational number represented as

$\alpha=\frac{n_{1}}{n_{2}}\in\left(0,\frac{1}{3}\right),$ (3.15)

where $n_{1}>0$ and $n_{2}>0$ are two odd integers. Let $\lambda\geq 1$ be a
large parameter which will be chosen later. We now consider a new Carleman
Weight Function (CWF) $\varphi_{\lambda}\left(x,t\right),$

$\varphi_{\lambda}\left(x,t\right)=\exp\left[2\lambda\left(x_{1}^{2}-\left(t-T/2\right)^{1+\alpha}\right)\right].$
(3.16)

Note that since $1+\alpha=\left(n_{1}+n_{2}\right)/n_{2}$ and since the number
$n_{1}+n_{2}$ is even, then $\left(t-T/2\right)^{1+\alpha}$ is defined for
both $t>T/2$ and $t<T/2,$ and the function
$\varphi_{\lambda}\left(x,t+T/2\right)$ is even with respect to $t$. The
novelty of the CWF (3.16) is due to the fact that the case $1+\alpha=2$ in the
conventional one, see, e.g. [21, formula (3.12)] and [23, formula (9.20)].
Below we need the form (3.16) with $1+\alpha\in\left(1,4/3\right)$ in order to
obtain proper estimates for the Volterra integrals in (3.8), (3.9) containing
$u_{t}$ and $\Delta u.$ Clearly

$\max_{\overline{Q}_{T}}\varphi_{\lambda}\left(x,t\right)=e^{2\lambda b^{2}}.$
(3.17)

Consider four functionals mapping the set $\overline{B\left(R\right)}$ in
$\mathbb{R},$

$\left.\begin{array}[]{c}J_{1,\lambda}\left(u,m\right)=\int\limits_{Q_{T}}\left(L_{1}\left(u,m\right)\right)^{2}\varphi_{\lambda}dxdt,\\\
J_{2,\lambda}\left(u,m\right)=\int\limits_{Q_{T}}\left(L_{2}\left(u,m\right)\right)^{2}\varphi_{\lambda}dxdt,\\\
J_{3,\beta}\left(u,m\right)=\beta\left\|\left(u,m\right)\right\|_{H}^{2},\\\
J_{\lambda,\beta}\left(u,m\right)=e^{-2\lambda
b^{2}}\lambda^{3/2}J_{1,\lambda}\left(u,m\right)+e^{-2\lambda
b^{2}}J_{2,\lambda}\left(u,m\right)+J_{3,\beta}\left(u,m\right),\end{array}\right.$
(3.18)

where the operators $L_{1}\left(u,p\right)$ and $L_{2}\left(u,p\right)$ are
defined in (3.8) and (3.9), and $\beta\in\left(0,1\right)$ is the
regularization parameter. The multiplier $e^{-2\lambda b^{2}}$ is included in
$J_{\lambda,\beta}$ to balance the terms in the first two lines of (3.18) with
the term in the third line, see (3.17). We solve problem (3.8)-(3.10) via
solving the following Minimization Problem:

Minimization Problem. _Minimize the functional_
$J_{\lambda,\beta}\left(u,m\right)$_in ( 3.18) on the set
_$\overline{B\left(R\right)}$_defined in ( 3.13)._

##  4 Convergence Analysis

###  4.1 Carleman estimates

Theorem 4.1 (a new Carleman estimate for a Volterra-like integral). _Let_
$d>0$_be a number and let the number_ $\alpha$ _be as in ( 3.15), where
_$n_{1},n_{2}$_are two odd numbers. Then the following Carleman estimate of
the Volterra-like integral holds for all functions_ $f\in
L_{2}\left(-d,d\right)$ _and for all_ $\lambda>0:$ __

$\int\limits_{-d}^{d}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\leq\frac{1}{\lambda^{3/2}}\cdot\frac{d^{\left(1-3\alpha\right)/2}}{\sqrt{2}\left(1+\alpha\right)^{3/2}}\int\limits_{-d}^{d}f^{2}e^{-2\lambda
t^{1+\alpha}}dt.$ (4.1)

Remark 4.1. _As stated in section 3,_ $1+\alpha=2$ _in the conventional case.
However, the corresponding conventional analog of estimate ( 4.1) is weaker
than (4.1) since _$\lambda^{-3/2}$_is replaced then with_
$\lambda^{-1}>>\lambda^{-3/2}$_for large values of_ $\lambda,$ _see[19, Lemma
3.1.1], [23, Lemma 3.1.1] for the conventional case. We also refer to the
proof of Theorem 4.3 for a more detailed explanation._

Theorem 4.2._Let_ $\varphi_{\lambda}\left(x,t\right)$_be the Carleman Weight
Function defined in ( 3.16). Then there exist a sufficiently large number
_$\lambda_{0}=\lambda_{0}\left(\Omega,\alpha,T\right)\geq 1$_and a number_
$C_{0}=C_{0}\left(\Omega,\alpha,T\right)>0,$_both numbers depending only on
listed parameters, such that the following two Carleman estimates are valid:_

$\left.\begin{array}[]{c}\int\limits_{Q_{T}}\left(u_{t}\pm\Delta
u\right)^{2}\varphi_{\lambda}dxdt\geq\left(C_{0}/\lambda\right)\int\limits_{Q_{T}}\left(u_{t}^{2}+\sum\limits_{i,j=1}^{n}u_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+\\\
+C_{0}\int\limits_{Q_{T}}\left(\lambda\left(\nabla
u\right)^{2}+\lambda^{3}u^{2}\right)\varphi_{\lambda}dxdt-\\\
-C_{0}\left(\left\|u\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|u\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\exp\left[-2\lambda\left(\left(T/2\right)^{1+\alpha}-b^{2}\right)\right],\mbox{
}\\\ \forall\lambda\geq\lambda_{0},\mbox{ }\forall u\in
H_{0}^{2,1}\left(Q_{T}\right).\end{array}\right.$ (4.2)

An analog of Theorem 4.2 was proven in [21] and [23, Theorem 9.4.1] for the
case $1+\alpha=2$ and for the operator $\partial_{t}-\Delta.$ The proof for
our case is quite similar with some insignificant deviations. Therefore, we
omit the proof of this theorem.

###  4.2 Global strong convexity and uniqueness

Theorem 4.3 (the central result). _Assume that_

$\left\|v_{0}\right\|_{C^{2}\left(\overline{\Omega}\right)},\mbox{
}\left\|p_{0}\right\|_{C^{1}\left(\overline{\Omega}\right)}<R$ (4.3)

_and conditions ( 2.1)-(2.5), (3.1) and (3.2) are satisfied. Then:_

_1\. The functional_ $J_{\lambda,\beta}\left(u,m\right)$_has Fréchet
derivative_ $J_{\lambda,\beta}^{\prime}\left(u,m\right)\in H_{0}$_at every
point_ $\left(u,m\right)\in\overline{B\left(R\right)},$_and this derivative is
Lipschitz continuous on_ $\overline{B\left(R\right)}$_, i.e. there exists a
number_ $D=D\left(\Omega,T,M,c,\alpha,R,\lambda\right)>0$_depending only on
listed parameters such that for all_
$\left(u_{1},m_{1}\right),\left(u_{2},m_{2}\right)\in\overline{B\left(R\right)}$__

$\left\|J_{\lambda,\beta}^{\prime}\left(u_{2},m_{2}\right)-J_{\lambda,\beta}^{\prime}\left(u_{1},m_{1}\right)\right\|_{H}\leq
D\left\|\left(u_{1},m_{1}\right)-\left(u_{2},m_{2}\right)\right\|_{H}.$ (4.4)

_2\. Let_ $\lambda_{0}=\lambda_{0}\left(\Omega,\alpha,T\right)\geq 1$_be the
number of Theorem 4.2. There exist a sufficiently large number_
$\lambda_{1}=\lambda_{1}\left(\Omega,\alpha,T,M,c,R\right)\geq\lambda_{0}$_and
a number_ $C_{1}=C_{1}\left(\Omega,\alpha,T,M,c,R\right)>0,$_both numbers
depending only on listed parameters, such that if_
$\lambda\geq\lambda_{1}$_and_
$\beta\in\left[2e^{-\lambda\left(T/2\right)^{1+\alpha}},1\right],$_then the
functional_ $J_{\lambda,\beta}$_is strongly convex on_
$\overline{B\left(R\right)},$_i.e. the following inequality holds:_

$\left.\begin{array}[]{c}J_{\lambda,\beta}\left(u_{2},m_{2}\right)-J_{\lambda,\beta}\left(u_{1},m_{1}\right)-\left[J_{\lambda,\beta}^{\prime}\left(u_{1},m_{1}\right),\left(u_{2},m_{2}\right)-\left(u_{1},m_{1}\right)\right]\geq\\\
\geq C_{1}\exp\left[-2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]\left\|\left(u_{2},m_{2}\right)-\left(u_{1},m_{1}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2}+\\\
+\left(\beta/2\right)\left\|\left(u_{2},m_{2}\right)-\left(u_{1},m_{1}\right)\right\|_{H}^{2},\mbox{
}\\\
\forall\left(u_{1},m_{1}\right),\left(u_{2},m_{2}\right)\in\overline{B\left(R\right)},\mbox{
}\forall\geq\lambda_{1},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (4.5)

_3\. For_ $\lambda$_and_ $\beta$_as in item 2, the functional_
$J_{\lambda,\beta}$_has unique minimizer_
$\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right)$_on the set_
$\overline{B\left(R\right)},$_and the following inequality holds:_

$\left[J_{\lambda,\gamma}^{\prime}\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right),\left(u_{\min,\lambda,\beta}-u,m_{\min,\lambda,\beta}-m\right)\right]\leq
0,\mbox{ }\forall\left(u,m\right)\in\overline{B\left(R\right)}.$ (4.6)

Remarks 4.2:

  1. 1.

_Below_ $C_{1}>0$_denotes different numbers depending only on the above listed
parameters._

  2. 2.

_Even though this theorem requires that_ $\lambda$_should be sufficiently
large, our extensive computational experience with the convexification method
tells us that reasonable values of_ $\lambda\in\left[1,5\right]$_can always be
selected to obtain good accuracy of numerical results, see, e.g.[4, 21, 23,
22, 29, 33] and references cited therein as well as subsection 6.2 below. This
is basically because, like in any asymptotic theory, only numerical studies
can indicate which specific ranges of parameters are reasonable._

###  4.3 The accuracy of the minimizer and uniqueness

Suppose that there exists a 2d vector function $G=\left(G_{1},G_{2}\right)\in
B\left(R\right).$ By (3.13) this means that this vector function is an
extension of boundary conditions of (3.10) inside of the domain $Q_{T},$

$\left.\begin{array}[]{c}G_{1}\mid_{S_{T}}=\partial_{t}g_{0,1}\left(x,t\right),\mbox{
}G_{2}\mid_{S_{T}}=\partial_{t}g_{0,2}\left(x,t\right),\\\
\partial_{x_{1}}G_{1}\mid_{\Gamma_{T}}=\partial_{t}g_{1,1}\left(x,t\right),\mbox{
}\partial_{x_{1}}G_{2}\mid_{\Gamma_{T}}=\partial_{t}g_{1,2}\left(x,t\right).\end{array}\right.$
(4.7)

By one of principles of the theory of Ill-Posed Problems [42], we assume that
there exists an exact solution $\left(k^{\ast},u^{\ast},m^{\ast}\right)\in
C\left(\overline{\Omega}\right)\times B^{\ast}\left(R-\delta\right)$ of CIP
(2.4), (2.8) with the exact, noiseless data

$v_{0}^{\ast},p_{0}^{\ast},\partial_{t}g_{0,1}^{\ast},\partial_{t}g_{0,2}^{\ast},\partial_{t}g_{1,1}^{\ast},\partial_{t}g_{1,2}^{\ast},$
(4.8)

where

$B^{\ast}\left(R-\delta\right)=\left\\{\begin{array}[]{c}\left(u,m\right)\in
H:\\\ u\mid_{S_{T}}=\partial_{t}g_{0,1}^{\ast},\mbox{
}m\mid_{S_{T}}=\partial_{t}g_{0,2}^{\ast},\\\
u_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,1}^{\ast},\mbox{
}m_{x_{1}}\mid_{\Gamma_{T}}=\partial_{t}g_{1,2}^{\ast},\\\
\left\|\left(u,m\right)\right\|_{H}<R-\delta.\end{array}\right\\},$ (4.9)

where $\delta>0$ is a sufficiently small number characterizing the level of
noise in the input data. Hence, there exists a vector function
$G^{\ast}=\left(G_{1}^{\ast},G_{2}^{\ast}\right)\in B^{\ast}\left(R\right)$
satisfying direct analogs of boundary conditions (4.7), in which the right
hand sides are replaced with functions listed in (4.8). Thus,

$\left(u^{\ast},m^{\ast}\right),G^{\ast}=\left(G_{1}^{\ast},G_{2}^{\ast}\right)\in
B^{\ast}\left(R-\delta\right).$ (4.10)

Furthermore, by (3.6) the exact coefficient $k^{\ast}\left(x\right)$ is

$k^{\ast}\left(x\right)=u^{\ast}\left(x,T/2\right)f^{\ast}\left(x\right)+F^{\ast}\left(x\right),$
(4.11)

where functions $f^{\ast}\left(x\right)$ and $F^{\ast}\left(x\right)$ are
obtained from functions $f\left(x\right)$ and $F\left(x\right)$ in (3.6),
(3.7) via replacing the pair $\left(v_{0},p_{0}\right)$ with the pair
$\left(v_{0}^{\ast},p_{0}^{\ast}\right).$

We assume that

$\left.\begin{array}[]{c}\left\|v_{0}^{\ast}\right\|_{C^{2}\left(\overline{\Omega}\right)},\left\|p_{0}^{\ast}\right\|_{C^{1}\left(\overline{\Omega}\right)}<R,\\\
\left\|G-G^{\ast}\right\|_{H}<\delta,\\\
\left\|v_{0}-v_{0}^{\ast}\right\|_{C^{2}\left(\overline{\Omega}\right)},\left\|p_{0}-p_{0}^{\ast}\right\|_{C^{1}\left(\overline{\Omega}\right)}<\delta.\end{array}\right.$
(4.12)

Let $\left(u,m\right)\in B\left(R\right)$ be an arbitrary pair of functions.
Denote

$\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)=\left(u^{\ast},m^{\ast}\right)-G^{\ast},\mbox{
}\left(\widetilde{u},\widetilde{m}\right)=\left(u,m\right)-G.$ (4.13)

Note that by (4.9), (4.10) and (4.12)

$\left\|G\right\|_{H}<R.$ (4.14)

By (3.12)-(3.14) and (4.12)-(4.14)

$\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right),\left(\widetilde{u},\widetilde{m}\right)\in\overline{B_{0}\left(2R\right)}.$
(4.15)

Also, let $L_{i}\left(u,m,v_{0},p_{0}\right),$ $i=1,2$ be operators defined in
(3.8), (3.9). Then

$L_{i}\left(u^{\ast},m^{\ast},v_{0}^{\ast},p_{0}^{\ast}\right)=L_{i}\left(\widetilde{u}^{\ast}+G_{1}^{\ast},\widetilde{m}^{\ast}+G_{2}^{\ast},v_{0}^{\ast},p_{0}^{\ast}\right)=0,\mbox{
}i=1,2.$ (4.16)

Introduce a new functional $I_{\lambda,\beta},$

$I_{\lambda,\beta}:\overline{B_{0}\left(2R\right)}\rightarrow\mathbb{R},\mbox{
}I_{\lambda,\beta}\left(\widetilde{u},\widetilde{m}\right)=J_{\lambda,\beta}\left(\widetilde{u}+G_{1},\widetilde{m}+G_{2}\right).$
(4.17)

It follows from (3.13), (3.14), (4.14) and triangle inequality that

$\left(\widetilde{u}+G_{1},\widetilde{m}+G_{2}\right)\in\overline{B\left(3R\right)},\mbox{
}\forall\left(\widetilde{u},\widetilde{m}\right)\in\overline{B_{0}\left(2R\right)}.$
(4.18)

Theorem 4.4 (the accuracy of the minimizer and uniqueness of the CIP).
_Suppose that conditions of Theorem 4.3 as well as conditions ( 4.8)-( 4.17)
are met. In addition, let inequalities (3.1) and (3.2) be valid when
_$p_{0}$_is replaced with_ $p_{0}^{\ast}.$_Then:_

_1\. The functional_ $I_{\lambda,\beta}$_has the Fréchet derivative_
$I_{\lambda,\beta}^{\prime}\left(\widetilde{u},\widetilde{m}\right)\in
H_{0}$_at any point_
$\left(\widetilde{u},\widetilde{m}\right)\in\overline{B_{0}\left(2R\right)}$_and
the analog of ( 4.4) holds._

_2\. Let_ $\lambda_{1}=\lambda_{1}\left(\Omega,\alpha,T,M,c,R\right)\geq 1$_be
the number of Theorem 4.3. Consider the number_
$\lambda_{2}=\lambda_{1}\left(\Omega,\alpha,T,M,c,3R\right)\geq\lambda_{1}$_.
Then for any_ $\lambda\geq\lambda_{2}$_and for any choice of the
regularization parameter_
$\beta\in\left[2e^{-\lambda\left(T/2\right)^{1+\alpha}},1\right)$_the
functional_ $I_{\lambda,\beta}$_is strongly convex on the set_
$\overline{B_{0}\left(2R\right)}$_, has unique minimizer_

$\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)\in\overline{B_{0}\left(2R\right)}$
(4.19)

_on this set, and the analog of ( 4.6) holds._

_3\. Let_ $\rho\in\left(0,1\right)$_be an arbitrary number. Choose the number_
$\gamma=\gamma\left(\rho\right)$_as_

$\gamma=\gamma\left(\rho\right)=\left(\frac{\rho}{3-\rho}\right)^{1/\left(1+\alpha\right)}\in\left(0,\left(\frac{1}{2}\right)^{1/\left(1+\alpha\right)}\right).$
(4.20)

_Choose the number_
$\delta_{0}=\delta_{0}\left(\Omega,\alpha,\mu,T,M,c,R,\rho\right)\in\left(0,1\right)$_so
small that_

$\ln\left\\{\delta_{0}^{-2/\left[\left(T/2\right)^{1+\alpha}\left(1+\gamma^{1+\alpha}\right)\right]}\right\\}\geq\lambda_{2}>1.$
(4.21)

_For any_ $\delta\in\left(0,\delta_{0}\right)$_choose in_
$I_{\lambda,\beta}$_parameters_
$\lambda=\lambda\left(\delta,\rho,\alpha\right)\geq\lambda_{2}$_and_
$\beta=\beta\left(\delta,\rho,\alpha\right)$_as_

$\lambda\left(\delta,\rho,\alpha\right)=\ln\left\\{\delta^{-2/\left[\left(T/2\right)^{1+\alpha}\left(1+\gamma^{1+\alpha}\right)\right]}\right\\},$
(4.22)

$\beta=2e^{-\lambda\left(T/2\right)^{1+\alpha}}.$ (4.23)

_Denote_

$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)=\left(\widetilde{u}_{\min,\lambda,\beta}+G_{1},\widetilde{m}_{\min,\lambda,\beta}+G_{2}\right)\in\overline{B\left(3R\right)},$
(4.24)

_see ( 4.18). Then the following accuracy estimates hold:_

$\left\|\left(\overline{u}_{\min,\lambda,\beta}-u^{\ast},\overline{m}_{\min,\lambda,\beta}-m^{\ast}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}\leq
C_{1}\delta^{1-\rho},\mbox{ }\forall\delta\in\left(0,\delta_{0}\right),$
(4.25)
$\left\|k_{\min,\lambda,\beta}-k^{\ast}\right\|_{L_{2}\left(\Omega\right)}\leq
C_{1}\delta^{1-\rho},\mbox{ }\forall\delta\in\left(0,\delta_{0}\right).$
(4.26)

_In_ _( 4.26), the function _$k^{\ast}\left(x\right)$ _is computed via (
4.11), and the function _$k_{\min,\lambda,\beta}\left(x\right)$_is computed
via ( 3.6) and (3.7 ). _

_4\. Next, based on ( 4.14) and (4.25), it is reasonable to assume that_

$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)\in\overline{B\left(R\right)}.$
(4.27)

_Then the vector function_
$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)$
_is the unique minimizer_
$\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right)$ _of the
functional_ $J_{\lambda,\beta}$_on the set_
$\overline{B\left(R\right)},$_which is found in Theorem 4.3, i.e._

$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)=\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right),$
(4.28)

_and, therefore, estimate ( 4.25) remains valid for
_$\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right).$

_5\. (Uniqueness). There exists at most one vector function_
$\left(k^{\ast},u^{\ast},m^{\ast}\right)$_satisfying the above conditions._

Remark 4.4. _If_ $\rho\approx 0$_in ( 4.25), (4.26), then these are almost
Lipschitz stability estimates. In fact, stability estimates, similar with ones
in (4.25), (4.26), were obtained in [28] for a different CIP for MFGS (2.6)._

###  4.4 The gradient descent method

Assume now that

$\left(u^{\ast},m^{\ast}\right)\in B^{\ast}\left(R/3-\delta\right),\mbox{
}R/3-\delta>0,$ (4.29)
$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)=\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right)\in
B\left(R/3\right),$ (4.30)

where parameters $\lambda$ and $\beta$ are the ones in (4.22) and (4.23).__ It
follows from (4.25) that assumption (4.30) is reasonable, as soon as (4.29) is
true.

We construct now the gradient descent method of the minimization of the
functional $J_{\lambda,\beta}.$ Consider an arbitrary pair of functions

$\left(u_{0},m_{0}\right)\in B\left(R/3\right).$ (4.31)

Let $\mu>0$ be the step size of the gradient descent method. The iterative
sequence of this method is:

$\left(u_{n},m_{n}\right)=\left(u_{n-1},m_{n-1}\right)-\mu
J_{\lambda,\beta}^{\prime}\left(u_{n-1},m_{n-1}\right),\mbox{ }n=1,2,...$
(4.32)

Note that since $J_{\lambda,\beta}^{\prime}\in H_{0}$ by Theorem 4.3, then all
pairs $\left(u_{n},m_{n}\right)$ have the same boundary conditions, see
(3.12).

Theorem 4.5._Let_ $\lambda\geq\lambda_{2}$_, where_ $\lambda_{2}$_was chosen
in Theorem 4.4. Let_ $\rho,\gamma,\lambda$_and_ $\beta$_be the parameters
chosen in Theorem 4.4_._Assume that conditions ( 4.29)-(4.31) hold. Then there
exists a number _$\mu_{0}\in\left(0,1\right)$_such that for any_
$\mu\in\left(0,\mu_{0}\right)$ _there exists a number_
$\theta=\theta\left(\mu\right)\in\left(0,1\right)$_such that for all_ $n\geq
1$__

$\left.\begin{array}[]{c}\left(u_{n},m_{n}\right)\in B\left(R\right),\mbox{
}\\\
\left\|u_{n}-u^{\ast}\right\|_{{}_{H^{2,1}\left(Q_{\gamma,T}\right)}}+\left\|m_{n}-m^{\ast}\right\|_{{}_{H^{2,1}\left(Q_{\gamma,T}\right)}}+\left\|k_{n}-k^{\ast}\right\|_{L_{2}\left(\Omega\right)}\leq\\\
\leq
C_{1}\delta^{1-\rho}+C_{1}\theta^{n}\left(\left\|u_{\min,\lambda,\beta}-u_{0}\right\|_{H^{k_{n}}\left(Q_{T}\right)}+\left\|m_{\min,\lambda,\beta}-m_{0}\right\|_{H^{k_{n}}\left(Q_{T}\right)}\right),\end{array}\right.$
(4.33)

_where functions_ $k_{n}\left(x\right)$_are computed via direct analogs of (
3.6), (3.7) with the replacement of _$u^{\ast}$_with_ $u_{n}.$

Proof. We prove this theorem, assuming that Theorems 4.3 and 4.4 hold. The
first line of (4.33), existence of numbers $\mu_{0}$ and
$\theta\left(\mu\right)$ as well as estimate

$\left.\begin{array}[]{c}\left\|u_{n}-u_{\min,\lambda,\beta}\right\|_{{}_{H^{k_{n}}\left(Q_{T}\right)}}+\left\|m_{n}-m_{\min,\lambda,\beta}\right\|_{{}_{H^{k_{n}}\left(Q_{T}\right)}}+\left\|k_{n}-k^{\ast}\right\|_{L_{2}\left(\Omega\right)}\leq\\\
\leq
C_{1}\theta^{n}\left(\left\|u_{\min,\lambda,\beta}-u_{0}\right\|_{H^{k_{n}}\left(Q_{T}\right)}+\left\|m_{\min,\lambda,\beta}-m_{0}\right\|_{H^{k_{n}}\left(Q_{T}\right)}\right).\end{array}\right.$
(4.34)

follow immediately from (4.29)-(4.32) and [22, Theorem 6]. Next, triangle
inequality, (4.25), (4.26), (4.30) and (4.34) lead to the desired result. $\
\square$

Remark 4.4. _Since_ $R>0$_is an arbitrary number and in ( 4.31)
_$\left(u_{0},m_{0}\right)$_is an arbitrary point of the set_
$B\left(R/3\right),$_then Theorem 4.5 implies the_ global  _convergence of the
iterative procedure ( 4.32 ), see Definition 1.2. Note that the gradient
descent method for a conventional functional converges only locally._

##  5 Proof of Theorem 4.1

Note that $1+\alpha=\left(n_{1}+n_{2}\right)/n_{2}.$ Since $n_{1}+n_{2}$ is an
even number, then

$t^{1+\alpha}=\left(-t\right)^{1+\alpha},$ (5.1)

and, also, $t^{1+\alpha}$ makes sense for $t<0.$ We have:

$\left.\begin{array}[]{c}\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt=\\\
=\left(2\lambda\left(1+\alpha\right)\right)^{-1}\int\limits_{0}^{d}\partial_{t}\left(-e^{-2\lambda
t^{1+\alpha}}\right)\left(1/t^{\alpha}\right)\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt=\\\
=-\left(2\lambda\left(1+\alpha\right)\right)^{-1}e^{-2\lambda
d^{1+\alpha}}d^{-\alpha}\left(\int\limits_{0}^{d}f\left(\tau\right)d\tau\right)^{2}-\\\
-\left(2\lambda\left(1+\alpha\right)\right)^{-1}\alpha\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(1/t^{\alpha+1}\right)\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt+\\\
+\left(\lambda\left(1+\alpha\right)\right)^{-1}\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(1/t^{\alpha}\right)f\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)dt.\end{array}\right.$
(5.2)

Since terms in the third and fourth lines of (5.2) are negative, then (5.2)
implies:

$\int\limits_{0}^{a}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\leq\frac{1}{\lambda\left(1+\alpha\right)}\int\limits_{0}^{a}e^{-2\lambda
t^{1+\alpha}}\frac{1}{t^{\alpha}}f\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)dt.$
(5.3)

Hence, applying Cauchy-Schwarz inequality to the right hand side of (5.3), we
obtain

$\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\leq$
$\leq\frac{1}{\lambda\left(1+\alpha\right)}\left[\int\limits_{0}^{d}f^{2}e^{-2\lambda
t^{1+\alpha}}dt\right]^{1/2}\left[\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\frac{1}{t^{2\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\right]^{1/2}.$
(5.4)

Estimate $I,$ where

$I=\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(1/t^{2\alpha}\right)\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt.$
(5.5)

We have

$\left.\begin{array}[]{c}I=\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(1/t^{2\alpha}\right)\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\leq\\\
\leq\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}\left(\int\limits_{0}^{t}f^{2}\left(\tau\right)d\tau\right)dt=\int\limits_{0}^{d}f^{2}\left(\tau\right)\left[\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt\right]d\tau.\end{array}\right.$

Hence,

$I\leq\int\limits_{0}^{d}f^{2}\left(\tau\right)\left[\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt\right]d\tau.$ (5.6)

Estimate now the interior integral in the right hand side of (5.6),

$\left.\begin{array}[]{c}\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt=\left(2\lambda\left(1+\alpha\right)\right)^{-1}\int\limits_{\tau}^{d}\partial_{t}\left(-e^{-2\lambda
t^{1+\alpha}}\right)t^{1-3\alpha}dt=\\\
=-\left(2\lambda\left(1+\alpha\right)\right)^{-1}e^{-2\lambda
d^{1+\alpha}}d^{1-3\alpha}+\left(2\lambda\left(1+\alpha\right)\right)^{-1}e^{-2\lambda\tau^{1+\alpha}}\tau^{1-3\alpha}+\\\
+\left(1-3\alpha\right)\left(2\lambda\left(1+\alpha\right)\right)^{-1}\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{-3\alpha}dt.\end{array}\right.$

Hence,

$\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt\leq\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\tau^{1-3\alpha}+\frac{1-3\alpha}{2\lambda\left(1+\alpha\right)}\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{-3\alpha}dt.$ (5.7)

Next, since the function $e^{-2\lambda t^{1+\alpha}}$ is decreasing with
respect to $t$, then

$\frac{1-3\alpha}{2\lambda\left(1+\alpha\right)}\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{-3\alpha}dt\leq\frac{1-3\alpha}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\int\limits_{\tau}^{d}t^{-3\alpha}dt=$
$=\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\left(d^{1-3\alpha}-\tau^{1-3\alpha}\right).$

Hence, in (5.7)

$\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt\leq\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\tau^{1-3\alpha}+\frac{1-3\alpha}{2\lambda\left(1+\alpha\right)}\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{-3\alpha}dt\leq$
$\leq\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\tau^{1-3\alpha}+\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}d^{1-3\alpha}-\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}\tau^{1-3\alpha}=$
$=\frac{1}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}d^{1-3\alpha}.$

Thus, we have proven that

$\int\limits_{\tau}^{d}e^{-2\lambda
t^{1+\alpha}}t^{1-2\alpha}dt\leq\frac{d^{1-3\alpha}}{2\lambda\left(1+\alpha\right)}e^{-2\lambda\tau^{1+\alpha}}.$

Hence, (5.5) and (5.6) imply:

$\left[\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\frac{1}{t^{2\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\right]^{1/2}\leq\frac{d^{\left(1-3\alpha\right)/2}}{\lambda^{1/2}\left(2\left(1+\alpha\right)\right)^{1/2}}\left(\int\limits_{0}^{d}f^{2}\left(t\right)e^{-2\lambda
t^{1+\alpha}}dt\right)^{1/2}.$

Substituting this in (5.4), we obtain

$\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(\tau\right)d\tau\right)^{2}dt\leq\frac{d^{\left(1-3\alpha\right)/2}}{\lambda^{3/2}\sqrt{2}\left(1+\alpha\right)^{3/2}}\int\limits_{0}^{d}f^{2}e^{-2\lambda
t^{1+\alpha}}dt.$ (5.8)

Next, using (5.1) and (5.8), we obtain

$\displaystyle\int\limits_{-d}^{0}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{t}^{0}f\left(\tau\right)d\tau\right)^{2}dt=\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{0}^{t}f\left(-\tau\right)d\tau\right)^{2}dt\leq$
$\leq\frac{1}{\lambda^{3/2}}\cdot\frac{d^{\left(1-3\alpha\right)/2}}{\sqrt{2}\left(1+\alpha\right)^{3/2}}\int\limits_{0}^{d}e^{-2\lambda
t^{1+\alpha}}f^{2}\left(-t\right)dt=$ (5.9)
$\displaystyle=\frac{1}{\lambda^{3/2}}\cdot\frac{d^{\left(1-3\alpha\right)/2}}{\sqrt{2}\left(1+\alpha\right)^{3/2}}\int\limits_{-d}^{0}e^{-2\lambda
t^{1+\alpha}}f^{2}\left(t\right)dt.$

Thus, (5.9) implies

$\int\limits_{-d}^{0}e^{-2\lambda
t^{1+\alpha}}\left(\int\limits_{t}^{0}f\left(\tau\right)d\tau\right)^{2}dt\leq\frac{1}{\lambda^{3/2}}\cdot\frac{d^{\left(1-3\alpha\right)/2}}{\sqrt{2}\left(1+\alpha\right)^{3/2}}\int\limits_{-d}^{0}e^{-2\lambda
t^{1+\alpha}}f^{2}\left(t\right)dt.$ (5.10)

Summing up (5.8) and (5.10), we obtain the target estimate (4.1). $\square$

##  6 Proof of Theorem 4.3

To simplify the presentation, we prove this theorem only for the case (2.3).
The case (2.2) is simpler than (2.3), see (2.5). It follows from (2.3), (2.5)
and [28, Lemma 3.2] that

$\int\limits_{Q_{T}}\left(\int\limits_{\Omega}K_{2}\left(x,y\right)z\left(y,t\right)dy\right)^{2}\varphi_{\lambda}\left(x,t\right)dxdt\leq
C_{1}\int\limits_{Q_{T}}z^{2}\varphi_{\lambda}dxdt,\mbox{ }\forall z\in
L_{2}\left(Q_{T}\right).$ (6.1)

Let
$\left(u_{1},m_{1}\right),\left(u_{2},m_{2}\right)\in\overline{B\left(R\right)}$
be two arbitrary pairs of functions. Denote
$\left(w,q\right)=\left(u_{2},m_{2}\right)-\left(u_{1},m_{1}\right).$ Then
triangle inequality, (3.13), (3.14) and (4.15) imply

$\left(w,q\right)\in\overline{B_{0}\left(2R\right)}.$ (6.2)

Also,

$J_{\lambda,\beta}\left(u_{2},m_{2}\right)-J_{\lambda,\beta}\left(u_{1},m_{1}\right)=J_{\lambda,\beta}\left(u_{1}+w,m_{1}+q\right)-J_{\lambda,\beta}\left(u_{1},m_{1}\right).$
(6.3)

Using (6.3), we analyze now functionals $J_{1,\lambda}$ and $J_{2,\lambda}$ in
(3.18).

###  6.1 Analysis of $J_{1,\lambda}$

First, we consider the operator $L_{1}\left(u_{1}+w,m_{1}+q\right)$ in (3.8)
and separate its linear $L_{1,\mbox{ lin}}$ and nonlinear
$L_{1,\mbox{nonlin}}$ parts with respect to $\left(w,q\right).$ We drop here
dependence of $L_{1}$ on $v_{0},p_{0}$ for brevity. We have:

$\left.\begin{array}[]{c}L_{1}\left(u_{1}+w,m_{1}+q\right)=L_{1}\left(u_{1},m_{1}\right)+\\\
+L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)+L_{1,\mbox{nonlin}}\left(u,m,w,q\right).\end{array}\right.$
(6.4)

Hence,

$\left.\begin{array}[]{c}\left[L_{1}\left(u_{1}+w,m_{1}+q\right)\right]^{2}-\left[L_{1}\left(u_{1},m_{1}\right)\right]^{2}=\\\
=2L_{1}\left(u_{1},m_{1}\right)L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)+\\\
+\left[L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)\right]^{2}+\\\
+2L_{1}\left(u_{1},m_{1}\right)L_{1,\mbox{nonlin}}\left(u,m,w,q\right)+\\\
+2L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)L_{1,\mbox{nonlin}}\left(u,m,w,q\right)+\\\
+\left[L_{1,\mbox{nonlin}}\left(u,m,w,q\right)\right]^{2}.\end{array}\right.$
(6.5)

We have:

$\left.\begin{array}[]{c}L_{1,\mbox{lin}}\left(u,m,w,q\right)=w_{t}+\Delta
w{-r\nabla w}\left(\int\limits_{T/2}^{t}\nabla u_{1}(x,\tau)d\tau+\nabla
v_{0}\left(x\right)\right)-\\\ -{r\nabla u}_{1}\int\limits_{T/2}^{t}\nabla
w(x,\tau)d\tau-\\\
-f\left(x\right)\left(w\left(x,t\right)-\int\limits_{T/2}^{t}w_{t}\left(x,\tau\right)d\tau\right)\int\limits_{\Omega}K\left(x,y\right)m\left(y,t\right)dy-\\\
-f\left(x\right)\left[\left(u\left(x,t\right)-\int\limits_{T/2}^{t}u_{t}\left(x,\tau\right)d\tau\right)+F\left(x\right)\right]\int\limits_{\Omega}K\left(x,y\right)q\left(y,t\right)dy-\\\
-sq-s_{t}\int\limits_{T/2}^{t}q(x,\tau)d\tau,\mbox{ }\left(x,t\right)\in
Q_{T}.\end{array}\right.$ (6.6)

It is space consuming to write the explicit form of the term $L_{1,\mbox{
nonlin}}\left(u,m,w,q\right).$ Nevertheless, using (2.7), (3.8),
(3.11)-(3.13), (4.15), (6.2), (6.4)-(6.6) and Cauchy-Schwarz inequality we
obtain after routine manipulations:

$\left.\begin{array}[]{c}\left[L_{1}\left(u_{1}+w,m_{1}+q\right)\right]^{2}-\left[L_{1}\left(u_{1},m_{1}\right)\right]^{2}-\\\
-2L_{1}\left(u_{1},m_{1}\right)L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)\geq\\\
\geq\left(1/2\right)\left(w_{t}+\Delta w\right)^{2}-C_{1}\left(\nabla
w\right)^{2}-\\\ -C_{1}\left[\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}+\left(\int\limits_{T/2}^{t}w_{t}d\tau\right)^{2}\right]-C_{1}\left(\int\limits_{\Omega}K_{2}\left(x,y\right)q\left(y,t\right)dy\right)^{2}.\end{array}\right.$
(6.7)

Consider the functional $J_{1,\lambda,\mbox{lin}}\left(u_{1},m_{1},w,q\right)$
acting on $\left(w,q\right)\in H_{0}.$ This is a linear and bounded functional
defined as:

$\left.\begin{array}[]{c}J_{1,\lambda,\mbox{lin}}\left(u_{1},m_{1},w,q\right)=2\int\limits_{Q_{T}}L_{1}\left(u_{1},m_{1}\right)L_{1,\mbox{lin}}\left(u_{1},m_{1},w,q\right)\varphi_{\lambda}\left(x,t\right)dxdt,\\\
J_{1,\lambda,\mbox{lin}}\left(u_{1},m_{1},w,q\right):H_{0}\rightarrow\mathbb{R}.\end{array}.\right.$
(6.8)

Therefore, by the Riesz theorem, there exists unique point
$J_{1,\lambda,\mbox{lin}}^{\prime}\left(u_{1},m_{1}\right)\in H_{0}$ such that

$J_{1,\lambda,\mbox{lin}}\left(u_{1},m_{1},w,q\right)=\left[J_{1,\lambda}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right],\mbox{
}\forall\left(w,q\right)\in H_{0},$ (6.9)

see Remark 3.1 for $\left[,\right]$). Using (6.4)-(6.6) and considerations,
which are similar to any of our above cited works on the convexification [4,
21, 22, 23, 29], we can prove that
$J_{1,\lambda,\mbox{lin}}^{\prime}\left(u_{1},m_{1}\right)$ is actually the
Fréchet derivative of the functional
$J_{1,\lambda,\mbox{lin}}\left(u_{1},m_{1}\right):\overline{B\left(R\right)}\rightarrow\mathbb{R}$
at the point $\left(u_{1},m_{1}\right).$ The Lipschitz continuity property
(4.4) of $J_{1,\lambda}^{\prime}\left(u,m\right)$ is rather easy to prove,
similarly with the proof of either Theorem 3.1 of [4] or Theorem 5.3.1 of
[23]. Hence, we omit the proof of (4.4).

Hence, by (3.18) and (6.7)-(6.9)

$e^{-2\lambda
b^{2}}\left\\{J_{1,\lambda}\left(u_{1}+w,m_{1}+q\right)-J_{1,\lambda}\left(u_{1},m_{1}\right)-\left[J_{1,\lambda}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\right\\}\geq$
$\geq\left(1/2\right)e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left(w_{t}+\Delta
w\right)^{2}\varphi_{\lambda}dxdt-C_{1}e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(\nabla w\right)^{2}\varphi_{\lambda}dxdt-$
$-C_{1}e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left[\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}+\left(\int\limits_{T/2}^{t}w_{t}d\tau\right)^{2}\right]\varphi_{\lambda}dxdt-$
(6.10) $-C_{1}e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(\int\limits_{\Omega}K_{2}\left(x,y\right)q\left(y,t\right)dy\right)^{2}\varphi_{\lambda}dxdt.$

Let $\lambda_{0}\geq 1$ be the number of Theorem 4.2. Apply now Carleman
estimate (4.2) to the first term in the second line of (6.10). Also, use
(6.1). We obtain that there exists a sufficiently large number
$\lambda_{1}=\lambda_{1}\left(\Omega,\alpha,\mu,T,M,c,R\right)\geq\lambda_{0}$
such that

$e^{-2\lambda
b^{2}}\left\\{J_{1,\lambda}\left(u_{1}+w,m_{1}+q\right)-J_{1,\lambda}\left(u_{1},m_{1}\right)-\left[J_{1,\lambda}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\right\\}\geq$
$\geq\frac{C_{1}}{\lambda}e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(w_{t}^{2}+\sum\limits_{i,j=1}^{n}w_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+$
$+C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left(\lambda\left(\nabla
w\right)^{2}+\lambda^{3}w^{2}\right)\varphi_{\lambda}dxdt-$ (6.11)
$-C_{1}e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left[\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}+\left(\int\limits_{T/2}^{t}w_{t}d\tau\right)^{2}\right]\varphi_{\lambda}dxdt-$
$-C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}q^{2}\varphi_{\lambda}dxdt-$
$-C_{1}\left(\left\|w\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|w\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right],$
$\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1}.$

We now need to estimate from the above the term with the Volterra integrals in
(6.11):

$\int\limits_{Q_{T}}\left[\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}+\left(\int\limits_{T/2}^{t}w_{t}d\tau\right)^{2}\right]\varphi_{\lambda}dxdt.$
(6.12)

If we would have only the term with $\nabla w,$ then the standard Carleman
Weight Function with $1+\alpha=2$ in (3.16) would be sufficient Indeed, in
this case we would have [19, Lemma 3.1.1], [23, Lemma 3.1.1]

$\left.\begin{array}[]{c}\int\limits_{Q_{T}}\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}\exp\left[2\lambda\left(x_{1}^{2}-\left(t-T/2\right)^{2}\right)\right]dxdt\leq\\\
\leq C_{1}\left(1/\lambda\right)\int\limits_{Q_{T}}\left(\nabla
w\right)^{2}\exp\left[2\lambda\left(x_{1}^{2}-\left(t-T/2\right)^{2}\right)\right]dxdt,\end{array}\right.$
(6.13)

also see Remark 4.1. Next, since we have $+\lambda\left(\nabla w\right)^{2}$
in (6.11) and since $\lambda>>1/\lambda$ for sufficiently large $\lambda,$
then the term in the second line of (6.13) would be dominated in (6.11) by the
term with $+\lambda\left(\nabla w\right)^{2}.$ Thus, we are especially
concerned with the second term in (6.12) since the term with $w_{t}^{2}$ is
multiplied by the multiplier $1/\lambda$ in (6.11), which tends to zero as
$\lambda\rightarrow\infty.$ In fact, we need Theorem 4.1 exactly due to our
concern with $w_{t}$.

Using (3.16) and Theorem 4.1, we obtain

$\int\limits_{Q_{T}}\left[\left(\int\limits_{T/2}^{t}\left(\nabla
w\right)d\tau\right)^{2}+\left(\int\limits_{T/2}^{t}w_{t}d\tau\right)^{2}\right]\varphi_{\lambda}dxdt\leq\frac{C_{1}}{\lambda^{3/2}}\int\limits_{Q_{T}}\left[\left(\nabla
w\right)^{2}+w_{t}^{2}\right]\varphi_{\lambda}dxdt.$

Hence, since

$\frac{1}{\lambda^{3/2}}<<\frac{1}{\lambda}\mbox{ and
}\frac{1}{\lambda^{3/2}}<<\lambda,\mbox{ }\forall\lambda\geq\lambda_{1},$

then (6.11) becomes:

$\left.\begin{array}[]{c}e^{-2\lambda b^{2}}S\left(J_{1,\lambda}\right)=\\\
=e^{-2\lambda
b^{2}}\left\\{J_{1,\lambda}\left(u_{1}+w,m_{1}+q\right)-J_{1,\lambda}\left(u_{1},m_{1}\right)-\left[J_{1,\lambda}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\right\\}\geq\\\
\geq\left(C_{1}/\lambda\right)e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(w_{t}^{2}+\sum\limits_{i,j=1}^{n}w_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+\\\
+C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left(\lambda\left(\nabla
w\right)^{2}+\lambda^{3}w^{2}\right)\varphi_{\lambda}dxdt-\\\
-C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}q^{2}\varphi_{\lambda}dxdt-\\\
-C_{1}\left(\left\|w\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|w\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right],\\\
\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1}.\end{array}\right.$ (6.14)

###  6.2 Analysis of $\lambda^{3/2}J_{1,\lambda}+J_{2,\lambda}$ and of
$J_{\lambda,\beta}$

Using formula (3.9) and considerations, which are completely similar with ones
of the previous subsection, we obtain the following analog of (6.14):

$\left.\begin{array}[]{c}e^{-2\lambda b^{2}}S\left(J_{2,\lambda}\right)=\\\
=e^{-2\lambda
b^{2}}\left\\{J_{2,\lambda}\left(u_{1}+w,m_{1}+q\right)-J_{2,\lambda}\left(u_{1},m_{1}\right)-\left[J_{2,\lambda}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\right\\}\geq\\\
\geq\left(C_{1}/\lambda\right)e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(q_{t}^{2}+\sum\limits_{i,j=1}^{n}q_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+\\\
+C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left(\lambda\left(\nabla
q\right)^{2}+\lambda^{3}q^{2}\right)\varphi_{\lambda}dxdt-\\\
-C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left(\Delta
w\right)^{2}\varphi_{\lambda}dxdt-e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(\nabla w\right)^{2}\varphi_{\lambda}dxdt-\\\
-C_{1}\left(\left\|q\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|q\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\exp\left[-2\lambda\mu\left(T/2\right)^{1+\alpha}\right],\\\
\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1}.\end{array}\right.$ (6.15)

Multiply both sides of (6.14) by $\lambda^{3/2}$ and sum up with (6.15). We
obtain

$\left.\begin{array}[]{c}e^{-2\lambda
b^{2}}\left[\lambda^{3/2}S\left(J_{1,\lambda}\right)+S\left(J_{2,\lambda}\right)\right]\geq\\\
\geq C_{1}e^{-2\lambda
b^{2}}\sqrt{\lambda}\int\limits_{Q_{T}}\left(w_{t}^{2}+\sum\limits_{i,j=1}^{n}w_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+\\\
+e^{-2\lambda
b^{2}}\left(C_{1}/\lambda\right)\int\limits_{Q_{T}}\left(q_{t}^{2}+\sum\limits_{i,j=1}^{n}q_{x_{i}x_{j}}^{2}\right)\varphi_{\lambda}dxdt+\\\
+C_{1}e^{-2\lambda b^{2}}\int\limits_{Q_{T}}\left[\lambda^{5/2}\left(\nabla
w\right)^{2}+\lambda^{9/2}w^{2}+\lambda\left(\nabla
q\right)^{2}+\lambda^{3}q^{2}\right]\varphi_{\lambda}dxdt-\\\
-C_{1}\left(\left\|w\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|w\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\lambda^{3/2}\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right]-\\\
-C_{1}\left(\left\|q\left(x,T\right)\right\|_{H^{1}\left(\Omega\right)}^{2}+\left\|q\left(x,0\right)\right\|_{H^{1}\left(\Omega\right)}^{2}\right)\lambda^{3/2}\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right],\\\
\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1}.\end{array}\right.$ (6.16)

Consider now an arbitrary number $\gamma\in\left(0,1\right).$ Since by the
last line of (2.1) $Q_{\gamma T}\subset Q_{T},$ then, making inequality (6.16)
stronger and using (3.11) and (3.12), we obtain

$\left.\begin{array}[]{c}\lambda^{3/2}\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right]\left\|\left(w,q\right)\right\|_{H}^{2}+\\\
+\left(\lambda^{3/2}S\left(J_{1,\lambda}\right)+S\left(J_{2,\lambda}\right)\right)\geq\\\
\geq C_{1}\left[\exp\left(-2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right)/\lambda\right]\left\|\left(w,q\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2},\\\
\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (6.17)

Consider now the functional $J_{\lambda,\beta}\left(u,m\right)$ in (3.18). By
(6.17)

$\left.\begin{array}[]{c}\lambda^{3/2}\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right]\left\|\left(w,q\right)\right\|_{H}^{2}+\\\
+J_{\lambda,\beta}\left(u_{1}+w,m_{1}+q\right)-J_{\lambda,\beta}\left(u_{1},m_{1}\right)-\left[J_{\lambda,\beta}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\geq\\\
\geq C_{1}\left\\{\exp\left[-2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]/\lambda\right\\}\left\|\left(w,q\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2}+\\\
+\beta\left\|\left(w,q\right)\right\|_{H}^{2},\\\
\forall\left(u_{1},m_{1}\right),\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (6.18)

Next, since
$\beta\in\left[2\exp\left[-\lambda\left(T/2\right)^{1+\alpha}\right],1\right),$
then
$\beta/2>\lambda^{3/2}\exp\left[-2\lambda\left(T/2\right)^{1+\alpha}\right]$
for $\lambda\geq\lambda_{1}.$ Hence, (6.18) implies

$\left.\begin{array}[]{c}J_{\lambda,\beta}\left(u_{1}+w,m_{1}+q\right)-J_{\lambda,\beta}\left(u_{1},m_{1}\right)-\left[J_{\lambda,\beta}^{\prime}\left(u_{1},m_{1}\right),\left(w,q\right)\right]\geq\\\
\begin{array}[]{c}\geq C_{1}\left\\{\exp\left[-2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]/\lambda\right\\}\left\|\left(w,q\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2}+\\\
+\left(\beta/2\right)\left\|\left(w,q\right)\right\|_{H}^{2},\\\
\forall\left(u_{1},m_{1}\right),\forall\left(u_{1}+w,m_{1}+q\right)\in\overline{B\left(R\right)},\mbox{
}\forall\lambda\geq\lambda_{1},\end{array}\end{array}\right.$

which proves the strong convexity property (4.5).

As soon as (4.5) is established, the existence and uniqueness of the minimizer
$\left(u_{\min,\lambda,\beta},m_{\min,\lambda,\beta}\right)$ of the functional
$J_{\lambda,\beta}\left(u,m\right)$ on the set $\overline{B\left(R\right)}$ as
well as inequality (4.6) follow immediately from a combination of Lemma 2.1
with Theorem 2.1 of [4] as well as from a combination of Lemma 5.2.1 with
Theorem 5.2.1 of [23]. $\square$

##  7 Proof of Theorem 4.4

It follows from (4.17) and (4.18) that Theorem 4.3 remains valid for the
functional $I_{\lambda,\beta}$ for all $\lambda\geq\lambda_{2}.$ For these
values of $\lambda,$ let
$\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)\in\overline{B_{0}\left(2R\right)}$
be the unique minimizer of the functional $I_{\lambda,\beta}$, which was found
in Theorem 4.3. Hence, using (4.5), we obtain

$\left.\begin{array}[]{c}I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)-I_{\lambda,\beta}\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)-\\\
-\left[I_{\lambda,\beta}^{\prime}\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right),\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)\right]\geq\\\
\geq C_{1}\exp\left[-2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]\times\\\
\times\left\|\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2},\\\
\forall\lambda\geq\lambda_{2},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (7.1)

By (4.6)

$\left.\begin{array}[]{c}-I_{\lambda,\beta}\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)-\\\
-\left[I_{\lambda,\beta}^{\prime}\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right),\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)\right]\leq
0.\end{array}\right.$

Hence, (7.1) implies

$\left.\begin{array}[]{c}C_{1}\exp\left[2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)\geq\\\
\geq\left\|\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2},\\\
\forall\lambda\geq\lambda_{2},\mbox{ }\forall\lambda\geq\lambda_{2},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (7.2)

Now, by (4.13)

$\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)=\left(u^{\ast},m^{\ast}\right)-G^{\ast}-\left[\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)+G-G\right].$
(7.3)

Recall that by (4.18)

$\left(\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}_{\min,\lambda,\beta}\right)+G=\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)\in\overline{B\left(3R\right)}.$

Hence, (7.3) implies

$\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)=\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)+\left(G-G^{\ast}\right).$
(7.4)

On the other hand, by (4.12)

$\left\|G-G^{\ast}\right\|_{H}\leq\delta.$ (7.5)

Hence, (7.4), (7.5) and Cauchy-Schwarz inequality imply

$\left.\begin{array}[]{c}\left\|\left(\widetilde{u}^{\ast}-\widetilde{u}_{\min,\lambda,\beta},\widetilde{m}^{\ast}-\widetilde{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2}\geq\\\
\geq\left(1/2\right)\left\|\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma
T}\right)}^{2}-\delta^{2}.\end{array}\right.$

Hence, using (7.2), we obtain

$\left.\begin{array}[]{c}C_{1}\exp\left[2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)+C_{1}\delta^{2}\geq\\\
\geq\left\|\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2},\\\
\forall\lambda\geq\lambda_{2},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (7.6)

Consider now
$I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right).$ By
(4.13) and (4.17)

$\left.\begin{array}[]{c}I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)=J_{\lambda,\beta}\left(\widetilde{u}^{\ast}+G_{1},\widetilde{m}^{\ast}+G_{2}\right)=\\\
=J_{\lambda,\beta}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right).\end{array}\right.$
(7.7)

Next, using (3.18), we obtain

$\left.\begin{array}[]{c}J_{\lambda,\beta}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)=\\\
=e^{-2\lambda
b^{2}}\lambda^{3/2}\int\limits_{Q_{T}}\left(L_{1}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right)^{2}\varphi_{\lambda}dxdt+\\\
+e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(L_{2}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right)^{2}\varphi_{\lambda}dxdt+\\\
+\beta\left\|\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right\|_{H}^{2}.\end{array}\right.$
(7.8)

It follows from (3.17), (4.10), (4.12) and (4.16) that

$\left.\begin{array}[]{c}e^{-2\lambda
b^{2}}\lambda^{3/2}\int\limits_{Q_{T}}\left(L_{1}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right)^{2}\varphi_{\lambda}dxdt+\\\
+e^{-2\lambda
b^{2}}\int\limits_{Q_{T}}\left(L_{2}\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right)^{2}\varphi_{\lambda}dxdt+\\\
+\beta\left\|\left(u^{\ast}+\left(G_{1}-G_{1}^{\ast}\right),m^{\ast}+\left(G_{2}-G_{2}^{\ast}\right)\right)\right\|_{H}^{2}\leq\\\
\leq C_{1}\lambda^{3/2}\delta^{2}+C_{1}\beta.\end{array}\right.$ (7.9)

Hence, by (7.7)-(7.9)

$I_{\lambda,\beta}\left(\widetilde{u}^{\ast},\widetilde{m}^{\ast}\right)\leq
C_{1}\lambda^{3/2}\delta^{2}+C_{1}\beta.$

Hence, (7.6) implies

$\left.\begin{array}[]{c}C_{1}\exp\left[3\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]\delta^{2}+C_{1}\beta\exp\left[2\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]\geq\\\
\geq\left\|\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2},\\\
\forall\lambda\geq\lambda_{2},\mbox{
}\forall\gamma\in\left(0,1\right).\end{array}\right.$ (7.10)

Recall that by (4.23) $\beta=2e^{-\lambda\left(T/2\right)^{1+\alpha}}.$ Hence,
using (7.10), we obtain

$\left.\begin{array}[]{c}\left\|\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}^{2}\leq\\\ \leq
C_{1}\exp\left[3\lambda\left(\gamma
T/2\right)^{1+\alpha}\right]\delta^{2}+C_{1}\exp\left[-\lambda\left(T/2\right)^{1+\alpha}\left(1-2\gamma^{1+\alpha}\right)\right].\end{array}\right.$
(7.11)

It follows from (4.20) that

$2\gamma^{1+\alpha}<1.$ (7.12)

Find $\lambda=\lambda\left(\delta,\rho,\alpha\right)$ such that

$\exp\left[3\lambda\gamma^{1+\alpha}\left(T/2\right)^{1+\alpha}\right]\delta^{2}=\exp\left[-\lambda\left(T/2\right)^{1+\alpha}\left(1-2\gamma^{1+\alpha}\right)\right].$
(7.13)

Hence,

$3\lambda\gamma^{1+\alpha}\left(T/2\right)^{1+\alpha}+\lambda\left(T/2\right)^{1+\alpha}\left(1-2\gamma^{1+\alpha}\right)=\ln\delta^{-2}.$
(7.14)

Hence, $\lambda=\lambda\left(\delta,\rho,\alpha\right)$ is as in (4.22). In
particular, in order to ensure that
$\lambda=\lambda\left(\delta,\rho,\alpha\right)$ $\geq\lambda_{2},$ we should
take the number
$\delta_{0}=\delta_{0}\left(\Omega,\alpha,\mu,T,M,c,R,\gamma\right)\in\left(0,1\right)$
so small that (4.21) holds, and we should also take
$\delta\in\left(0,\delta_{0}\right).$

Using (7.11)-(7.14) and (4.22), we obtain

$\left\|\left(u^{\ast}-\overline{u}_{\min,\lambda,\beta},m^{\ast}-\overline{m}_{\min,\lambda,\beta}\right)\right\|_{H^{2,1}\left(Q_{\gamma
T}\right)\times H^{2,1}\left(Q_{\gamma T}\right)}\leq
C_{1}\delta^{\left(1-2\gamma^{1+\alpha}\right)/\left(1+\gamma^{1+\alpha}\right)}.$
(7.15)

Note that by (7.12)

$\frac{1-2\gamma^{1+\alpha}}{1+\gamma^{1+\alpha}}\in\left(0,1\right).$

Hence, for any $\rho\in\left(0,1\right)$ we can choose
$\gamma=\gamma\left(\rho\right)$ as in (4.20). Substituting this
$\gamma\left(\rho\right)$ in (7.15), we obtain the first target estimate
(4.25). Next, using (3.6), (3.7), (4.11) and (4.25), we obtain the second
target estimate (4.26).

Assume now that (4.27) holds and prove (4.28). Below in this section 7
$\lambda=\lambda\left(\delta,\rho,\alpha\right)$ and as in (4.22) and $\beta$
is as in (4.23).

By (4.17), (4.19) and (4.24)

$\left.\begin{array}[]{c}J_{\lambda,\beta}\left(\widetilde{u}_{\min,\lambda,\beta}+G_{1},\widetilde{m}_{\min,\lambda,\beta}+G_{2}\right)=J_{\lambda,\beta}\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)\leq\\\
\leq
J_{\lambda,\beta}\left(\widetilde{u}+G_{1},\widetilde{m}+G_{2}\right),\mbox{
}\forall\left(\widetilde{u},\widetilde{m}\right)\in\overline{B_{0}\left(2R\right)}.\end{array}\right.$
(7.16)

Consider now an arbitrary point
$\left(u_{1},m_{1}\right)\in\overline{B\left(R\right)}.$ Then by (4.14) and
triangle inequality

$\left(\widetilde{u}_{1},\widetilde{m}_{1}\right)=\left(u_{1}-G_{1},m_{1}-G_{2}\right)\in\overline{B_{0}\left(2R\right)}.$
(7.17)

Hence, using (7.16) and (7.17), we obtain

$J_{\lambda,\beta}\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)\leq
J_{\lambda,\beta}\left(\widetilde{u}_{1}+G_{1},\widetilde{m}_{1}+G_{2}\right)=J_{\lambda,\beta}\left(u_{1},m_{1}\right).$

Hence, (4.27) and Theorem 4.3 imply that
$\left(\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)$
is the unique minimizer of the functional $J_{\lambda,\beta}\left(u,m\right)$
on the set $\overline{B\left(R\right)},$ which proves (4.28).

We now prove uniqueness of our CIP. Set $\delta=0.$ Also, for an arbitrary
number $\rho\in\left(0,1\right),$ set $\gamma=\gamma\left(\rho\right)$ as in
(4.20). Then (4.25) and (4.26) imply
$\left(k^{\ast},u^{\ast},m^{\ast}\right)=\left(k_{\min,\lambda,\beta},\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)$
in the set $\Omega\times Q_{\gamma T}\times Q_{\gamma T}.$ Since the triple
$\left(k_{\min,\lambda,\beta},\overline{u}_{\min,\lambda,\beta},\overline{m}_{\min,\lambda,\beta}\right)$
is unique in this set, then the triple
$\left(k^{\ast},u^{\ast},m^{\ast}\right)$ is also unique in this set. In
particular, since the function $k^{\ast}=k^{\ast}\left(x\right),x\in\Omega,$
then this function is found uniquely, independently on $\gamma.$ Given the
latter, uniqueness of the vector function $\left(u^{\ast},m^{\ast}\right)$ in
the entire domain $Q_{T}\times Q_{T}$ follows immediately from Theorems 5 and
6 of [26]. $\square$

##  8 Numerical Studies

In this section we describe our numerical studies of the Minimization Problem
formulated in section 3. As it is always done in numerical studies of Ill-
Posed and Inverse Problems, we need to figure out how to numerically generate
the data for our problem. More precisely, we need to numerically generate the
observation data (2.8) of our CIP. On the first step, we take with the
coefficient $k(x)$ of our choice and numerically generate the pair of
functions $(v(x,t),p(x,t))$, which solves MFGS (2.6). This vector function
$(v(x,t),p(x,t)),$ generates the observation data (2.8). On the second step,
we “pretend” that we do not know the coefficient $k(x)$ and solve the
Minimization Problem with the observation data (2.8). Next, we compare the
resulting computed solution $k_{\mbox{comp}}(x)$ with the $k(x)$ given in the
first step.

Our new procedure for data generation for MFGS (2.6) is described in
subsection 8.1, and the numerical tests of computing the function
$k_{\mbox{comp}}(x)$ are described in subsection 8.2.

###  8.1 Numerical data generation

Choose a sufficiently smooth function $v^{\ast\ast}(x,t)$. Then we solve the
following modification of the second equation in (2.6):

$p_{t}^{\ast\ast}(x,t)-\Delta
p^{\ast\ast}(x,t){-\mbox{div}(r(x)p^{\ast\ast}(x,t)\nabla
v^{\ast\ast}(x,t))}=0,\quad\left(x,t\right)\in Q_{T},$ (8.1)

with the initial condition

$p^{\ast\ast}(x,0)=p_{0}^{\ast\ast}(x),\quad x\in\Omega,$ (8.2)

and the Dirichlet boundary condition

$p^{\ast\ast}(x,t)\mid_{S_{T}}=g_{0,2}^{\ast\ast}(x,t),\quad\left(x,t\right)\in
S_{T},$ (8.3)

where $p_{0}^{\ast\ast}(x)$ and $g_{0,2}^{\ast\ast}(x,t)$ are functions of our
choice. The initial boundary value problem (8.1)-(8.3) is a conventional one
and we solve it by the Finite Difference Method. Next, we choose the
coefficient $k\left(x\right),$ which we want to reconstruct.

Given the pair $\left(p^{\ast\ast}(x,t),k\left(x\right)\right),$ we now need
to ensure that the function ${v^{\ast\ast}(x,t)}$ satisfies the first equation
(2.6). To do this, we define the function $s(x,t).$ Assume that
$p^{\ast\ast}(x,t)\neq 0$ in $\overline{Q}_{T}.$ This can always be achieved
by a proper choice of functions $p_{0}^{\ast\ast}(x)$ and
$g_{0,1}^{\ast\ast}(x,t)$ in (8.2), (8.3) and the use of the maximum principle
for parabolic equations. Then we define the function $s(x,t)$ as:

$\displaystyle\left.s(x,t)=\left[p^{\ast\ast}(x,t)\right]^{-1}\left[v_{t}^{\ast\ast}+\Delta
v^{\ast\ast}{-r(\nabla
v^{\ast\ast})^{2}/2-}k\left(x\right)\int\limits_{\Omega}K\left(x,y\right)p^{\ast\ast}\left(y,t\right)dy\right],\right.$

see (2.5) for the integral operator. Thus, this is our numerical method for
the generation of the solution
$\left(v^{\ast\ast}(x,t),p^{\ast\ast}(x,t)\right)$ of MFGS (2.6). Next, we use
the values of these two functions and their $\partial_{x_{1}}-$derivatives to
generate the target data (2.8) for our CIP.

###  8.2 Numerical arrangements

We have conducted numerical studies in the 2D case. In our numerical testing,
we took in (2.1), (2.6), (3.15), (3.16) and (3.18):

$a=1,\quad b=2,\quad A_{2}=0.5,\quad T=1,\quad r\left(x\right)\equiv
1,\quad\alpha=1/5,\quad\beta=0.001.$

Also, we have taken in (3.18)

$\beta\left\|\left(u,m\right)\right\|_{\widetilde{H}}^{2}=\beta\left(\left\|u\right\|_{H^{2}\left(Q_{T}\right)}^{2}+\left\|m\right\|_{H^{2}\left(Q_{T}\right)}^{2}\right).$
(8.4)

As to the kernel $K\left(x,y\right)$ in the global interaction term, we have
taken it as in (2.2),

$K_{1}\left(x,y\right)=k\left(x\right)\delta\left(x_{1}-y_{1}\right)\overline{K}_{1}\left(x,y_{2}\right),$
(8.5)

where $\overline{K}_{1}\left(x,y_{2}\right)$ is the Gaussian [37, subsection
4.2],

$\overline{K}_{1}\left(x,y_{2}\right)=\exp\left(-\frac{\left(x_{2}-y_{2}\right)^{2}}{\sigma^{2}}\right),\quad\sigma=0.2.$

We took the target coefficient $k(x)$ in (8.5), to be reconstructed, as:

$k(x)=\left\\{\begin{array}[]{cc}c_{a}=const.>0,&\mbox{inside the tested
inclusion,}\\\ 1,&\mbox{outside the tested inclusion.}\end{array}\right.$
(8.6)

Then we set:

$\mbox{correct inclusion/background contrast}=\frac{c_{a}}{1}.$ (8.7)

In the numerical tests below, we take $c_{a}=2,4,8$, and the inclusions with
the shapes of the letters ‘$A$’, ‘$\Omega$’ and ‘$SZ$’. To generate the
observation data in (2.8), we set in (8.1)-(8.3):

$v^{\ast\ast}(x,t)=0.1\cos(\pi x_{1})\sin(\pi x_{2})(t^{2}+1),\quad
p_{0}^{\ast\ast}(x)=x_{1}x_{2},\quad
g_{0,2}^{\ast\ast}\left(x,t\right)=(t+1)x_{1}x_{2}.$

Then we proceeded with data generation as in subsection 8.1.

Remark 8.1. _We choose letter-like shapes of inclusions because these are non-
convex shapes with voids. Shapes with such properties are traditionally hard
to image when solving inverse problems. Therefore, our results point towards
the robustness of our numerical technique. Furthermore, we image both: shapes
of inclusions and values of the unknown coefficient_ $k\left(x\right)$_,_
$x\in\Omega$_, see ( 8.6), (8.7)._

To solve the forward problem (8.1)-(8.3) for data generation, we have used the
spatial mesh sizes $1/80\times 1/80$ and the temporal mesh step size $1/320$.
In the computations of the Minimization Problem, the spatial mesh sizes were
$1/20\times 1/20$ and the temporal mesh step size was $1/10$. We have solved
problem (8.1)-(8.3) by the classic implicit scheme. To solve the Minimization
Problem, we have written operators $L_{1},L_{2}$ and the norm
$\left\|\cdot\right\|_{\widetilde{H}}$ in (8.4) in discrete forms of finite
differences and then minimized the functional $J_{\lambda,\beta}$ with respect
to the values of functions $u$ and $m$ at those grid points. As soon as its
minimizer $\left(u_{\min},m_{\min}\right)$ is found, the computed target
coefficient $k_{\mbox{comp}}\left(x\right)$ is found via an obvious analog of
(3.6), (3.7).

To guarantee that the solution of the problem of the minimization of the
functional $J_{\lambda,\beta}\left(u,m\right)$ in (3.18) satisfies the
boundary conditions (3.10), we adopt the Matlab’s built-in optimization
toolbox fmincon to minimize the discretized form of the functional
$J_{\lambda,\beta}\left(u,m\right)$. The iterations of fmincon stop when the
condition

$|\nabla J_{\lambda,\beta}\left(u,m\right)|<10^{-2}$

is met.

To exhibit the process for dealing with the Neumann boundary conditions in the
second line of (3.10) in the iterations of fmincon, we denote the discrete
points along $x_{1}$-direction as

$x_{1,i}=a+ih_{x_{1}},\quad i=0,1,\cdots,N_{x_{1}},,\quad
N_{x_{1}}=(b-a)/h_{x_{1}},\quad h_{x_{1}}=1/20.$ (8.8)

Then keepng in mind those Neumann boundary conditions, the discrete functions
$u,m$ in the iterations of fmincon should satisfy

$\left.\begin{array}[]{c}-4u(x_{1,N_{x_{1}-1}},x_{2},t)+u(x_{1,N_{x_{1}-2}},x_{2},t)=2h_{x_{1}}\partial_{t}g_{0,1}(b,x_{2},t)-3\partial_{t}g_{1,1}(b,x_{2},t),\\\
-4m(x_{1,N_{x_{1}-1}},x_{2},t)+m(x_{1,N_{x_{1}-2}},x_{2},t)=2h_{x_{1}}\partial_{t}g_{0,2}(b,x_{2},t)-3\partial_{t}g_{1,2}(b,x_{2},t).\end{array}\right.$
(8.9)

We note that formula (8.9) also contains the Dirichlet boundary conditions in
the first line of (3.10) as

$\left.\begin{array}[]{c}u(x_{1},x_{2},t)=\partial_{t}g_{0,1}(x_{1},x_{2},t),\mbox{
}m(x_{1},x_{2},t)=\partial_{t}g_{0,2}(x_{1},x_{2},t),\mbox{ }x\in
S_{T}.\end{array}\right.$ (8.10)

The starting point
$\left(u^{\left(0\right)},m^{\left(0\right)}\right)\left(x_{1},x_{2},t\right)$
of iterations of fmincon was chosen as:

$\eqalign{u^{(0)}(x_{1},x_{2},t)=&\frac{1}{2}\left(\frac{b-x_{1}}{b-a}\partial_{t}g_{0,1}\left(x_{1},x_{2},t\right)+\frac{x_{1}-a}{b-a}\partial_{t}g_{0,1}\left(x_{1},x_{2},t\right)\right)\\\
&+\frac{1}{2}\left(\frac{A_{2}-x_{2}}{2A_{2}}\partial_{t}g_{0,1}\left(x_{1},x_{2},t\right)+\frac{x_{2}+A_{2}}{2A_{2}}\partial_{t}g_{0,1}\left(x_{1},x_{2},t\right)\right),\\\
m^{(0)}(x_{1},x_{2},t)=&\frac{1}{2}\left(\frac{b-x_{2}}{b-a}\partial_{t}g_{0,2}\left(x_{1},x_{2},t\right)+\frac{x_{2}-a}{b-a}\partial_{t}g_{0,2}\left(x_{1},x_{2},t\right)\right)\\\
&+\frac{1}{2}\left(\frac{A_{2}-x_{2}}{2A_{2}}\partial_{t}g_{0,2}\left(x_{1},x_{2},t\right)+\frac{x_{2}+A_{2}}{2A_{2}}\partial_{t}g_{0,2}\left(x_{1},x_{2},t\right)\right).}$
(8.11)

In other words, we use linear interpolation inside the domain $Q_{T}$ of the
Dirichlet boundary conditions (3.10). Although it follows from (8.11) that the
starting point
$\left(u^{\left(0\right)},m^{\left(0\right)}\right)\left(x_{1},x_{2},t\right)$
satisfies only Dirichlet boundary conditions in the first line of (3.10) and
does not satisfy Neumann boundary conditions in the second line of (3.10),
still (8.9) and (8.10) imply that boundary conditions in both lines of (3.10)
are satisfied on all other iterations of fmincon.

We introduce the random noise in the observation data in (2.8) as follows:

$\eqalign{&\hskip
45.52458ptv_{0,\zeta}(x)=v_{0}(x)\left(1+\delta\zeta_{v,x}\right),\quad
p_{0,\zeta}(x)=p_{0}(x)\left(1+\delta\zeta_{p,x}\right),\\\
&g_{0,1,\zeta}(x,t)=g_{0,1}(x,t)\left(1+\delta\zeta_{0,1,x,t}\right),\quad
g_{0,2,\zeta}(x,t)=g_{0,2}(x,t)\left(1+\delta\zeta_{0,2,x,t}\right),\\\
&g_{1,1,\zeta}(x,t)=g_{1,1}(x,t)\left(1+\delta\zeta_{1,1,x,t}\right),\quad
g_{1,2,\zeta}(x,t)=g_{1,2}(x,t)\left(1+\delta\zeta_{1,2,x,t}\right),}$ (8.12)

where $\zeta_{v,x},\zeta_{p,x}$ are the uniformly distributed random variables
in the interval $[0,1]$ depending on the point $x\in\Omega$,
$\zeta_{0,1,x,t},\zeta_{0,2,x,t}$ are the uniformly distributed random
variables in the interval $[0,1]$ depending on the point $(x,t)\in S_{T}$, and
$\zeta_{1,1,x,t},\zeta_{1,2,x,t}$ are the uniformly distributed random
variables in the interval $[0,1]$ depending on the point $(x,t)\in\Gamma_{T}$.
In (8.12) $\delta=0.03,0.05$, which correspond to the $3\%$ and $5\%$ noise
levels respectively. The reconstruction from the noisy data is denoted as
$k_{\zeta}(x)$ given by the following analog of (3.6), (3.7):

$\left.\begin{array}[]{c}k_{\zeta}(x)=\left(u\left(x,t\right)-\int\limits_{T/2}^{t}u_{t}\left(x,\tau\right)d\tau\right)f_{\zeta}(x)+F_{\zeta}(x),\\\
f_{\zeta}(x)=\left[\int\limits_{\Omega_{1}}\overline{K}_{1}\left(x,\overline{y}\right)p_{0,\zeta}\left(x_{1},\overline{y}\right)d\overline{y}\right]^{-1},\mbox{
}\\\ F_{\zeta}\left(x\right)=\left(\Delta v_{0,\zeta}-r\left(\nabla
v_{0,\zeta}\right)^{2}/2-s\left(x,T/2\right)p_{0,\zeta}\right)\left(x\right)f_{\zeta}\left(x\right),\end{array}\right.$
(8.13)

where the subscript $\zeta$ means that these functions correspond to the noisy
data. Since we deal with first and second derivatives of noisy functions
$v_{0,\zeta}(x)$, $p_{0,\zeta}(x),g_{0,1,\zeta}(x,t)$, $g_{0,2,\zeta}(x,t)$,
$g_{1,1,\zeta}(x,t)$, $g_{1,2,\zeta}(x,t)$, we have to design a numerical
method to differentiate the noisy data. We use the natural cubic splines to
approximate the noisy observation data. Then we can use the derivatives of
those splines to approximate the derivatives of corresponding noisy
observation data. We generate the cubic splines $s_{v},s_{p}$ of functions
$v_{0,\zeta}(x),p_{0,\zeta}(x)$ with the spatial mesh grid size of $1/20\times
1/20$, and then calculate the first and second derivatives of $s_{v}$ to
approximate the first and second derivatives respect to $x,y$ of
$v_{0,\zeta}(x),p_{0,\zeta}(x)$. For the boundary data
$g_{i,j,\zeta}(x,t),i=0,1,j=1,2$, we generate the corresponding cubic splines
$S_{i,j},i=0,1,j=1,2$ in the temporal space with the temporal mesh grid size
of $1/10$, and then calculate the derivatives of $S_{i,j}$ to approximate the
first derivatives with respect to $t$ of functions $g_{i,j,\zeta}(x,t)$.

###  8.3 Numerical tests

Test 1. We test the case when the inclusion in (8.6) has the shape of the
letter ‘$A$’ with $c_{a}=2$. We use this test as a reference case to figure
out an optimal value of the parameter $\lambda$. The result is displayed in
Figure 1. We observe that the images have a low quality for $\lambda=0,1,2$.
Then the quality is improved with $\lambda=3,4$, and the reconstruction
quality deteriorates at $\lambda=10$. Hence, we choose $\lambda=3$ as the
optimal value.

Remark 8.2._The optimal value_ $\lambda=3,$_once chosen, is used in other
tests 2-4._

Figure 1: Test 1. The reconstructed coefficient $k(x)$, where the function
$k(x)$ is given in (8.6) with $c_{a}=2$ inside of the letter ‘A’. We test
different values of $\lambda$. The reconstructions are unsatisfactory when
$\lambda$ is too small, $\lambda=0,1,2$. Next, the reconstruction quality is
good when $\lambda=3,4$. On the other hand, the reconstruction quality
deteriorates at $\lambda=10$. Thus, we choose $\lambda=3$ as the optimal
value.

Test 2. We test the case when the inclusion in (8.6) has the shape of the
letter ‘$A$’ for different values of the parameter $c_{a}=4,8$ inside of the
letter ‘$A$’. Hence, by (8.7) the inclusion/background contrasts now are
respectively $4:1$ and $8:1$. Computational results are displayed on Figure 2.
One can observe that these images are accurate ones. In particular, the
computed inclusion/background contrasts are accurate.

Figure 2: Test 2. Exact (left) and reconstructed (right) coefficient $k(x)$
with $c_{a}=4$ (first row) and $c_{a}=8$ (second row) inside of the letter ‘A’
as in (8.6). The inclusion/background contrasts in (8.7) are respectively
$4:1$ and $8:1$. The reconstructions are accurate

Test 3. We test the case when the coefficient $k(x)$ in (8.6) has the shape of
the letter ‘$\Omega$’ with $c_{a}=2$ inside of it. Results are presented on
Figure 3. We again observe an accurate reconstruction.

Figure 3: Test 3. Exact (left) and reconstructed (right) coefficient $k(x)$,
where the function $k(x)$ is given in (8.6) with $c_{a}=2$ inside of the
letter ‘$\Omega$’. The reconstruction is accurate.

Test 4. We now test different values of the parameter $c_{a}=4,8$ inside of
the letter ‘$\Omega$’. The computational results are depicted on Figure 4. The
quality of these images and computed inclusion/background contrasts in (8.7)
are as good as the results of the test with the letter ‘$A$’ in Test 2.

Figure 4: Test 4. Exact (left) and reconstructed (right) coefficient $k(x)$
with $c_{a}=4$ (first row) and $c_{a}=8$ (second row) inside of the letter
‘$\Omega$’ as in (8.6). The inclusion/background contrasts in (8.7) are
respectively $4:1$ and $8:1$. The reconstructions are accurate.

Test 5. We test the reconstruction for the case when the inclusion in (8.6)
has the shape of two letters ‘SZ’ with $c_{a}=2$ in each of them. SZ are two
letters in the name of the city (Shenzhen) were the second and the third
authors reside. The results are exhibited on Figure 5.

Figure 5: Test 5. Exact (left) and reconstructed (right) coefficient $k(x)$,
where the function $k(x)$ is given in (8.6) with $c_{a}=2$ inside of two
letters ‘SZ’. The reconstruction is worse than the one for the case of the
single letter ‘$\Omega$’ in Figure 3. Nevertheless, the reconstruction are
still accurate in both letters. The computed inclusion/background contrasts in
(8.7) are accurately reconstructed in both letters.

Test 6. We consider the case of the random noisy data in (8.12) with
$\delta=0.03$ and $\delta=0.05$, i.e. with 3% and 5% noise level. We test the
reconstruction for the cases when the inclusion in (8.6) has the shape of
either the letter ‘A’ or the letter ‘$\Omega$’ with $c_{a}=2$. The results are
displayed on Figure 6. One can observe accurate reconstructions in all four
cases. In particular, the inclusion/background contrasts in (8.7) are
reconstructed accurately.

Figure 6: Test 6. Reconstructed coefficient $k(x)$ with the shape of letters
‘A’ (left) and ‘$\Omega$’ (right) with $c_{a}=2$ from noisy data (8.12) with
$\delta=0.03$ (top) and $\delta=0.05$ (bottom), i.e. with 3% and 5% noise
level. In all these four cases, both reconstructions and inclusion/background
contrasts in (8.7) are accurate.

##  9 Summary

For the first time, we have developed here a numerical method for a CIP for
MFGS (2.6) with the rigorously guaranteed global convergence property, see
Definition 1.2 for the term “global convergence”. In other words, its
convergence to the true solution is guaranteed if starting at an arbitrary
point of a bounded set, whose diameter is not required to be small. We have
proven the global convergence of our technique and explicitly provided its
convergence rate.

To computationally demonstrate robustness of our technique, we took quite
complicated shapes of tested inclusions. Our numerical results for both
noiseless and noisy data demonstrate that we can accurately reconstruct both
shapes of inclusions and values of the unknown coefficient both inside and
outside of them.

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