Generalized Optimal AMG Convergence Theory Ahsan Ali

# Generalized Optimal AMG Convergence Theory for Nonsymmetric and Indefinite
Problems ††thanks: This work was funded by NSF grant DMS-2110917.

Ahsan Ali Department of Mathematics and Statistics, University of New Mexico
<EMAIL_ADDRESS>
###### Abstract

Algebraic multigrid (AMG) is known to be an effective solver for many sparse
symmetric positive definite (SPD) linear systems. For SPD systems, the
convergence theory of AMG is well-understood in terms of the $A$-norm but in a
nonsymmetric setting such an energy norm is non-existent. For this reason,
convergence of AMG for nonsymmetric systems of equations remains an open area
of research. Existing nonsymmetric AMG algorithms in this setting mostly rely
on heuristics motivated by SPD convergence theory. In the SPD setting, the
classical form of optimal AMG interpolation provides a useful insight in
determining the two grid convergence rate of the method. In this work, we
discuss a generalization of the optimal AMG convergence theory targeting
nonsymmetric problems by constructing a $2\times 2$ block symmetric indefinite
system so that the Petrov–Galerkin AMG process for the nonsymmetric matrix $A$
can be recast as a Galerkin AMG process for a symmetric indefinite system. We
show that using this generalization of the optimal interpolation theory, one
can obtain the same identity for the two-grid convergence rate as that derived
in the SPD setting for optimal interpolation. We also provide supporting
numerical results for the convergence result and nonsymmetric advection-
diffusion problems.

###### keywords:

algebraic multigrid, optimal interpolation, two-grid, nonsymmetric,
indefinite.

In collaboration with: James Brannick, Karsten Kahl, Oliver A. Krzysik, Jacob
B. Schroder, and Ben S. Southworth

##  1 Introduction

Algebraic Multigrid (AMG) [3, 4, 20] emerged as a powerful and efficient
iterative solver for solving large-scale linear systems arising from various
applications, including computational fluid dynamics, structural mechanics,
and other scientific simulations. When typical discretization approaches such
as finite difference, finite volume, or finite element are applied in the
numerical approximation to partial differential equations, the resulting
approximation can swiftly grow to several million, billions, or even more,
numbers of unknowns. AMG methods are effective to solve such large sparse
linear system of equations

(1) $A\bf{x}=\bf{b},$

where $A\in\mathbb{R}^{n\times n}$ is a sparse symmetric or nonsymmetric
matrix with $\bf{x},\bf{b}$ $\in\mathbb{R}^{n}$. AMG is one of the most widely
used linear solvers due to its potential for $\mathcal{O}(n)$ computational
cost when solving a linear system with $n$ number of degrees of freedoms.

Convergence of AMG is well-motivated when $A$ is symmetric positive definite
(SPD), which frequently arises from the discretization of elliptic partial
differential equations (PDEs) [11, 12, 14, 23, 17, 22, 6]. When $A$ is
nonsymmetric, the AMG framework has been extended to nonsymmetric matrices
using a number of different AMG techniques [21, 9, 7, 13, 15, 16, 1]. While
substantial progress has been made in the development of nonsymmetric AMG
solvers, there is still need for a reliable theoretical motivation to
investigate the robustness of existing, as well as new methods. Overall, this
relative lack of AMG convergence theory in the nonsymmetric environment
contributes to lack of robustness for nonsymmetric solvers, when compared to
the SPD case, and thus motivates this work.

The efficacy of AMG as a solver is dependent upon the complementary
relationship between relaxation and coarse-grid correction. Relaxation (also
known as smoothing) involves the application of a relaxation method (e.g.
Gauss-Seidel) which damps some error (typically low-energy in the SPD case),
whereas the coarse-grid correction refers to a complementary process that
damps the error not reduced by relaxation. To construct the coarse grid and
the related transfer operators, we first split the $n$ unknowns of $A$ into a
classical AMG C-F partition of $n_{f}$ fine nodes and $n_{c}$ coarse nodes.
This splitting is disjoint, and if we reorder $A$ by F-points followed by
C-points, we arrive at

(2) $A=\left[\begin{array}[]{cc}A_{ff}&A_{fc}\\\ A_{cf}&A_{cc}\\\
\end{array}\right],$

where for example, $A_{ff}$ corresponds to entries in $A$ between two
F-points. Let $P:\mathbb{R}^{n_{c}}\mapsto\mathbb{R}^{n}$ be the interpolation
operator that maps corrections from the coarse-level and
$R:\mathbb{R}^{n}\mapsto\mathbb{R}^{n_{c}}$ be the restriction operator that
maps residuals to the coarse-level where $n_{c}$ again denotes the dimension
of coarse space. In classical AMG, interpolation takes the form

(3) $P=\left[\begin{array}[]{c}W\\\ I\end{array}\right],$

where $W$ must be sparse (for efficiency) and represents interpolation from
the coarse grid to fine grid $F$-points. In the design of many classical AMG
techniques, the ideal interpolation operator [11] is defined as the unique
operator

(4) $P_{ideal}=\begin{bmatrix}-A_{ff}^{-1}A_{fc}\\\ I\end{bmatrix}.$

The error propagation matrix of the two-level method with both pre-smoothing
and post-smoothing steps is given by

(5) $E_{TG}=(I-(M^{*})^{-1}A)(I-P(RAP)^{-1}RA)(I-M^{-1}A),$

where $A_{c}=RAP$ is the coarse grid operator and $M$ denotes the smoother.
The operator $E_{TG}$ describes how error changes from iteration $k$ to
iteration $k+1$ and generally we want its norm to be less than 1, indicating a
contraction. In pursuit of this goal, we want to generating a stable
interpolation matrix $P$ with both $P$ and $A_{c}$ being sparse matrices such
that an approximation property holds. If the process is done recursively,
i.e., AMG is applied to $RAP$, the a multilevel hierarchy is achieved.

Different AMG approaches offer different guidelines to construct appropriate
interpolation operator $P$. Among the most notable and widely accepted
guidelines are the so-called ideal [8, 24] and optimal [23, 6] forms of
interpolation. In this paper, we focus on the idea of optimal interpolation
primarily motivated from [6]. The aim of optimal interpolation is to capture
the modes that relaxation leaves behind (i.e., the algebraically smoothest
modes) in $\operatorname{span}(P)$. The derivation of optimal interpolation
matrix $P_{\sharp}$ and it’s corresponding classical AMG form allows us to
gain a deeper understanding of AMG convergence from a theoretical perspective.
In this work, we discuss a generalization of the optimal interpolation based
AMG framework by taking a nonsymmetric discretization matrix and the
constructing a $2\times 2$ block symmetric indefinite system. The goal is to
demonstrate that, as in the SPD setting, the identity for determining the
convergence rate of the resulting two-grid approach is the same for the block
$2\times 2$ case.

The arrangement of this paper is as follows. First we present a derivation of
Kaczmarz relaxation [18, 3] in order to define the corresponding form of the
symmetrized smoother of $A$. After that, we review the two-level AMG theory
based on optimal interpolation and the convergence identity for optimal
interpolation in the SPD case. The important contributions of our work then
follow and are the construction of the $2\times 2$ block symmetric indefinite
system and the generalization of the optimal interpolation AMG theory to this
case, which are addressed in the main contribution section. Finally, we
present the supporting numerical outcomes by examining a nonsymmetric
advection-diffusion operator, along with concluding remarks and
recommendations for further research.

##  2 Kaczmarz Relaxation and Symmetrized Smoother

###  2.1 Kaczmarz in Matrix Form

Kaczmarz relaxation has been observed to be historically used as a smoother in
two-level classical AMG convergence theory [3]. Convergence of the symmetrized
smoother (which is always the case with Kaczmarz or Block-Kaczmarz) is another
essential requirement of theory. These facts motivate our choice of Kaczmarz,
despite Kaczmarz’s lack of popularity as a smoother for AMG solvers in
practice. Given an equation

(6) $A\mathbf{x}=\mathbf{b},$

where $A$ is an $n\times n$ nonsingular matrix, define a new unknown vector
$\mathbf{y}$ such that

(7) $A^{*}\mathbf{y}=\mathbf{x},$

where $A^{*}$ is the transpose of $A$. The equation obtained for $\mathbf{y}$
is

(8) $AA^{*}\mathbf{y}=\mathbf{b}.$

The matrix $AA^{*}$ is symmetric positive definite for any $A$ which is
nonsingular. Hence Gauss-Seidel relaxation for equation (8) will converge.
Therefore Kaczmarz is basically Gauss-Seidel iterations on the linear system
$AA^{*}\mathbf{y}=\mathbf{b}$, where $\mathbf{x}=A^{*}\mathbf{y}$. Note that,
we emphasize using $AA^{*}$ instead of $A^{*}A$ based on [5]. Let us consider
$\widetilde{A}=AA^{*}$. Then the system becomes,
$\widetilde{A}\mathbf{y}=\mathbf{b}$. We now decompose $\widetilde{A}$ as
$\widetilde{A}=\widetilde{D}+\widetilde{L}+\widetilde{U}$. Then using Gauss-
Seidel, the solution to $\widetilde{A}y=b$ can be obtained iteratively via

$\mathbf{y}^{(k+1)}=(\widetilde{D}+\widetilde{L})^{-1}(\mathbf{b}-\widetilde{U}\mathbf{y}^{(k)})$

This equation can be further modified as:

$\mathbf{y}^{(k+1)}=[I-(\widetilde{D}+\widetilde{L})^{-1}\widetilde{A}]\mathbf{y}^{(k)}+(\widetilde{D}+\widetilde{L})^{-1}\mathbf{b}$

Then using $\widetilde{A}=AA^{*}$ and (7) we obtain,

$\displaystyle\mathbf{y}^{(k+1)}$
$\displaystyle=\mathbf{y}^{(k)}-(\widetilde{D}+\widetilde{L})^{-1}AA^{*}\mathbf{y}^{(k)}+(\widetilde{D}+\widetilde{L})^{-1}\mathbf{b}$
$\displaystyle A^{*}\mathbf{y}^{(k+1)}$
$\displaystyle=A^{*}\mathbf{y}^{(k)}-A^{*}(\widetilde{D}+\widetilde{L})^{-1}AA^{*}\mathbf{y}^{(k)}+A^{*}(\widetilde{D}+\widetilde{L})^{-1}\mathbf{b}$
$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=\mathbf{x}^{(k)}-A^{*}(\widetilde{D}+\widetilde{L})^{-1}A\mathbf{x}^{(k)}+A^{*}(\widetilde{D}+\widetilde{L})^{-1}\mathbf{b}$
$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=[I-A^{*}(\widetilde{D}+\widetilde{L})^{-1}A]\mathbf{x}^{(k)}+A^{*}(\widetilde{D}+\widetilde{L})^{-1}\mathbf{b}$
$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=[I-A^{*}N^{-1}A]\mathbf{x}^{(k)}+A^{*}N^{-1}\mathbf{b}$
$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=[I-(N(A^{*})^{-1})^{-1}A]\mathbf{x}^{(k)}+(N(A^{*})^{-1})^{-1}\mathbf{b}$
(9) $\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=[I-M^{-1}A]\mathbf{x}^{(k)}+M^{-1}\mathbf{b}$

where $N=\widetilde{D}+\widetilde{L}$ and $M=N(A^{*})^{-1}$. The section on
numerical results makes use of this particular version of the smoother $M$.

###  2.2 Symmetrized Smoother

We define the smoother error propagator by $I-M^{-1}A$ and assume that the
smoother is convergent (in the SPD case, in energy norm $||\cdot||_{A}$) so
that $M+M^{*}-A$ is SPD. The following two operators defined in [12] are
related to the smoother $M$:

(10) $\widetilde{M}=M^{*}(M^{*}+M-A)^{-1}M,$ (11)
$\overline{M}=M(M^{*}+M-A)^{-1}M^{*}$

Note that $I-\widetilde{M}^{-1}A=(I-M^{-1}A)(I-M^{*-1}A)$, therefore
$\widetilde{M}$ is the symmetrized version of $M$. Similar properties holds
for $\overline{M}$ . If $M$ is symmetric then $\widetilde{M}=\overline{M}$. In
general, $\widetilde{M}$ is used for error analysis and $\overline{M}$ is used
in the definition of preconditioners. In our context, we use $\widetilde{M}$
as the symmetrized smoother following [8].

##  3 Two-level Theory Review

In SPD theory, the following identity for the convergence rate is often used
[6]:

$||E_{TG}(P)||_{A}^{2}=1-\displaystyle\frac{1}{\sup_{\mathbf{v}}\kappa(P,\mathbf{v})},\quad\kappa(P,\mathbf{v})=\displaystyle\frac{||(I-\Pi_{\widetilde{M}}(P))\mathbf{v}||_{\widetilde{M}}^{2}}{||\mathbf{v}||_{A}^{2}}.$

Some key assumptions in this theory:

  * •

$A\in\mathbb{R}^{n\times n}$ is a symmetric and positive definite (SPD) matrix

  * •

Convergence is considered in the well defined energy-norm or $A$-norm defined
as $||\mathbf{x}||_{A}^{2}=\langle A\mathbf{x},\mathbf{x}\rangle$

  * •

$\Pi_{\widetilde{M}}(P)=P(P^{T}\widetilde{M}P)^{-1}P^{T}\widetilde{M}$ is the
$\langle\cdot,\cdot\rangle_{\widetilde{M}}$ orthogonal projection on
$\operatorname{range}(P)$

Now the two grid convergence theory based on optimal interpolation is given by
the following lemma [6]:

######  Lemma 3.1.

Let $P:\mathbb{R}^{n_{c}}\to\mathbb{R}^{n}$ be full rank and let
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$ and
$\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{n}$ denote the eigenvalues and
orthonormal (with respect to $\langle\cdot,\cdot\rangle_{\widetilde{M}}$)
eigenvectors of the generalized eigenvalue problem

$A\mathbf{x}=\lambda\widetilde{M}\mathbf{x}.$

Then the minimal convergence rate of the two-grid method is given by

$||E_{TG}(P_{\sharp})||_{A}^{2}=1-\displaystyle\frac{1}{\kappa_{\sharp}},\quad\kappa_{\sharp}:=\displaystyle{\inf_{\operatorname{dim}(\operatorname{range}(P))\,=\,n_{c}}\sup_{\mathbf{v}}\kappa(P,\mathbf{v})}=\displaystyle\frac{1}{\lambda_{n_{c}+1}},$

where the optimal interpolation operator $P_{\sharp}$ satisfies

(12)
$\operatorname{range}(P_{\sharp})=\operatorname{range}\left(\begin{pmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n_{c}}\end{pmatrix}\right).$

The detailed proof of this lemma can be found in [6] where it uses the the
definition of $A$-norm and Courant-Fischer Min-max representation.

##  4 Main Contribution

Our ultimate objective is to use the same concept of optimal interpolation to
develop two grid convergence theory and to extend the aforesaid SPD theory to
the study of nonsymmetric problems. We pay particular attention to examining
this classical AMG version of optimal interpolation and contrasting it with
the so-called ideal interpolation in a real-world context. We therefore employ
a finite-dimensional complex space to introduce all the operators, since we
are generalizing the preceding theoretical observation for a broader class of
discretization operators. We start by letting the singular value decomposition
of a non-hermitian matrix be $A=U\Sigma V^{*}$, where
$\Sigma=\text{diag}\,(\sigma_{1},\ldots,\sigma_{n})$,
$U=[u_{1}|\ldots|u_{n}]\in\mathbb{C}^{n\times n}$ and
$V=[v_{1}|\ldots|v_{n}]\in\mathbb{C}^{n\times n}$ are unitary matrices. Then
the left and right singular vectors satisfy the equations:

(13) $\displaystyle Av_{i}$ $\displaystyle=\sigma_{i}u_{i},$ (14)
$\displaystyle A^{*}u_{i}$ $\displaystyle=\sigma_{i}v_{i}.$

These equations are equivalent to the hermitian and indefinite eigenvalue
problem:

(15) $\begin{bmatrix}0&A\\\ A^{*}&0\end{bmatrix}\begin{bmatrix}U&U\\\
V&-V\end{bmatrix}=\begin{bmatrix}U&U\\\
V&-V\end{bmatrix}\begin{bmatrix}\Sigma&0\\\ 0&-\Sigma\end{bmatrix}.$

The inspiration for the aforementioned construction of a hermitian indefinite
matrix comes from [7]. We take into consideration the unknown vector
$\begin{bmatrix}\mathbf{z}\\\ \mathbf{x}\end{bmatrix}$ and the right hand side
vector $\begin{bmatrix}\mathbf{\hat{b}}\\\ \mathbf{b}\end{bmatrix}$ in order
to confirm that the $2\times 2$ block matrix in (15) can be used to represent
the original matrix equation (1). The matching block system is then
$\begin{bmatrix}0&A\\\ A^{*}&0\end{bmatrix}\begin{bmatrix}\mathbf{z}\\\
\mathbf{x}\end{bmatrix}=\begin{bmatrix}\mathbf{\hat{b}}\\\
\mathbf{b}\end{bmatrix}$ suggests that $A\bf{x}=\bf{b}$ and
$A^{*}\bf{z}=\bf\hat{b}$.

Based on the $2\times 2$ block hermitian matrix in (15) we propose the
following generalized optimal interpolation theory:

######  Lemma 4.1.

Let $A\in\mathbb{C}^{k\times k}$ be nonsingular hermitian or non-hermitian
matrix. In $n=2k$-dimensional space, consider a block matrix of the form
$\cal{A}=\begin{bmatrix}0&A\\\ A^{*}&0\end{bmatrix}$ which consists of $k$
positive and $k$ negative eigenvalues. Let
$P:\mathbb{C}^{n_{c}}\to\mathbb{C}^{n}$ be full rank and let
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$ and
$\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{n}$ denote the eigenvalues and
orthonormal (with respect to the symmetrized smoother $\widetilde{M}$)
eigenvectors respectively of the generalized eigenvalue problem

$\mathcal{A}\,\mathbf{x}=\lambda\widetilde{M}\mathbf{x},$

and $\widetilde{M}$ is defined as above for our symmetrized smoother. Then the
spectral radius of the two-grid error transfer operator is given by

(16) $\rho\left(E_{TG}(P_{\sharp})\right)=1-\lambda_{n_{c}+1}$

where the optimal interpolation operator $P_{\sharp}$ satisfies

$\operatorname{range}(P_{\sharp})=\operatorname{range}\left(\begin{pmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n_{c}}\end{pmatrix}\right).$

######  Proof 4.2.

Consider the generalized eigenvalue problem for the $2\times 2$ Hermitian
indefinite matrix (15) as

(17) $\mathcal{A}\,V=\widetilde{M}V\Lambda,$

where the columns of $V=[\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{n}]$
are eigenvectors and the diagonal elements of
$\Lambda:=\operatorname{diag}([\lambda_{1},\lambda_{2},...,\lambda_{n}])$ are
the eigenvalues such that
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$. The symmetrized
smoother $\widetilde{M}$ is defined as
$\widetilde{M}=M^{*}(M^{*}+M-\mathcal{A})^{-1}M$ and the orthonormality of the
eigenvector matrix is ensured through the $\widetilde{M}$ inner product so
that $V^{*}\widetilde{M}V=I$. The error propagation matrix of the two-level
method reads as following:

$E_{TG}(P)=(I-P(P^{*}\mathcal{A}P)^{-1}P^{*}\mathcal{A})(I-\widetilde{M}^{-1}\mathcal{A})=(I-P\mathcal{A}_{c}^{-1}P^{*}\mathcal{A})(I-\widetilde{M}^{-1}\mathcal{A})$

where $P\in\mathbb{C}^{n\times n_{c}}$ is the interpolation matrix, and
$\mathcal{A}_{c}=P^{*}\mathcal{A}P$ is the Galerkin variational coarse-level
matrix. Define the optimal interpolation operator $P_{\sharp}$ which satisfies
$\operatorname{range}(P_{\sharp})=\operatorname{range}\left(\begin{pmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n_{c}}\end{pmatrix}\right)$
and its $\widetilde{M}$ orthogonal complement as $\mathcal{S}_{\sharp}$ so
that
$\operatorname{range}(\mathcal{S}_{\sharp})=\operatorname{range}\left(\begin{pmatrix}\mathbf{v}_{n_{c}+1}&\mathbf{v}_{n_{c}+2}&\cdots&\mathbf{v}_{n}\end{pmatrix}\right).$
Therefore, from (17), we have

$\displaystyle\mathcal{A}\,V$ $\displaystyle=\widetilde{M}V\Lambda$
$\displaystyle\mathcal{A}\,\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n}\end{bmatrix}$
$\displaystyle=\widetilde{M}\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n}\end{bmatrix}\Lambda$
$\displaystyle\mathcal{A}\,\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n_{c}}|\mathbf{v}_{n_{c}+1}&\cdots&\mathbf{v}_{n}\end{bmatrix}$
$\displaystyle=\widetilde{M}\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n_{c}}|\mathbf{v}_{n_{c}+1}&\cdots&\mathbf{v}_{n}\end{bmatrix}\Lambda$
$\displaystyle\mathcal{A}\,\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}$
$\displaystyle=\widetilde{M}\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}\Lambda.$

Also note that

(18) $\mathcal{A}P_{\sharp}=\widetilde{M}P_{\sharp}\Lambda_{n_{c}}.$

We start by taking the smoother as follows:

(19)
$(I-\widetilde{M}^{-1}\mathcal{A})V=V-\widetilde{M}^{-1}\mathcal{A}V=V-V\Lambda=V(I-\Lambda)$

assuming that the symmetrized smoother $\widetilde{M}$ is non-singular. Now,

$\displaystyle E_{TG}(P_{\sharp})V$
$\displaystyle=\left(I-P_{\sharp}(P_{\sharp}^{*}\mathcal{A}P_{\sharp})^{-1}P_{\sharp}^{*}\mathcal{A}\right)(I-\widetilde{M}^{-1}\mathcal{A})V$
$\displaystyle=(I-P_{\sharp}(P_{\sharp}^{*}\mathcal{A}P_{\sharp})^{-1}P_{\sharp}^{*}\mathcal{A})V(I-\Lambda)\quad[\text{using}\,\,\eqref{need2}]$
$\displaystyle=(I-P_{\sharp}(P_{\sharp}^{*}\widetilde{M}P_{\sharp}\Lambda_{n_{c}})^{-1}P_{\sharp}^{*}\mathcal{A})V(I-\Lambda)\quad[\text{using}\,\,\eqref{need}]$
$\displaystyle=(I-P_{\sharp}\Lambda_{n_{c}}^{-1}P_{\sharp}^{*}\mathcal{A})V(I-\Lambda)\quad[\text{since}\,\,P_{\sharp}^{*}\widetilde{M}P_{\sharp}=I_{n_{c}}]$
$\displaystyle=(V-P_{\sharp}\Lambda_{n_{c}}^{-1}P_{\sharp}^{*}\mathcal{A}V)(I-\Lambda)$
$\displaystyle=(V-P_{\sharp}\Lambda_{n_{c}}^{-1}P_{\sharp}^{*}\widetilde{M}V\Lambda)(I-\Lambda)\quad[\text{using}\,\,\eqref{main_th}]$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-P_{\sharp}\Lambda_{n_{c}}^{-1}P_{\sharp}^{*}\widetilde{M}\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}\Lambda\right)(I-\Lambda)$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-P_{\sharp}\Lambda_{n_{c}}^{-1}\begin{bmatrix}P_{\sharp}^{*}\widetilde{M}P_{\sharp}|P_{\sharp}^{*}\widetilde{M}\mathcal{S}_{\sharp}\end{bmatrix}\Lambda\right)(I-\Lambda)$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-P_{\sharp}\Lambda_{n_{c}}^{-1}\begin{bmatrix}I_{n_{c}}|\,0\end{bmatrix}\Lambda\right)(I-\Lambda)$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-P_{\sharp}\Lambda_{n_{c}}^{-1}\begin{bmatrix}\Lambda_{n_{c}}|\,0\end{bmatrix}\right)(I-\Lambda)$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-P_{\sharp}\begin{bmatrix}I_{n_{c}}|\,0\end{bmatrix}\right)(I-\Lambda)$
$\displaystyle=\left(\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}-\begin{bmatrix}P_{\sharp}|\,0\end{bmatrix}\right)(I-\Lambda)$
$\displaystyle=\begin{bmatrix}0\,|\mathcal{S}_{\sharp}\end{bmatrix}(I-\Lambda).$

Therefore,
$E_{TG}(P_{\sharp})\begin{bmatrix}P_{\sharp}|\mathcal{S}_{\sharp}\end{bmatrix}=\begin{bmatrix}0\,|\mathcal{S}_{\sharp}\end{bmatrix}(I-\Lambda)$,
which clearly shows that the action of $E_{TG}(P_{\sharp})$ over the matrix of
eigenvectors $V$ reduces the first $n_{c}$ columns of $V$ to zero vectors. For
this reason when we consider the eigenvalue matrix $1-\Lambda$, then the
eigenvalues corresponding to first $n_{c}$ eigenvectors
$\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{n_{c}}$ can be discarded.
Moreover since
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n_{c}}\leq\lambda_{n_{c}+1}\leq\cdots\leq\lambda_{n}$
therefore $1-\lambda_{1}\geq 1-\lambda_{2}\geq\cdots\geq 1-\lambda_{n_{c}}\geq
1-\lambda_{n_{c}+1}\geq\cdots\geq 1-\lambda_{n}$. Hence, the eigenvalues of
$E_{TG}(P_{\sharp})$ are given by $1-\lambda_{n_{c}+1}\geq\cdots\geq
1-\lambda_{n}$. Therefore the spectral radius of two grid error propagator
matrix is given by $\rho\left(E_{TG}(P_{\sharp})\right)=1-\lambda_{n_{c}+1}.$

One of the most important aspects of the foregoing proof is that, without even
introducing an $A$-norm, we can immediately demonstrate the preservation of
the required identity (16) using a direct proof approach. Using optimal
interpolation, we were able to demonstrate the crucial element of two grid
convergence theory, even though an energy norm does not exist for our
nonsymmetric $A$.

##  5 Numerical Results

To verify the proposed theory, we present numerical evidence by considering a
constant advection with diffusion problem, which generates nonsymmetric test
matrics:

(20a) $\displaystyle-\alpha\nabla\cdot\nabla u+\mathbf{b}(x,y)\cdot\nabla u$
$\displaystyle=f\quad\mbox{for }\Omega=[-1,1]^{2},$ (20b) $\displaystyle u$
$\displaystyle=0\quad\mbox{on }\partial\Omega,$

where $\alpha$ is the diffusion constant and $\mathbf{b}(x,y)$ is the
advection. For this problem we consider
$\mathbf{b}(x,y)=[\sqrt{2/3},\,\sqrt{1/3}]$. The discretization is first-order
upwind discontinuous Galerkin (DG) for advection and the interior-penalty
method for the DG discretization of diffusion. Using PyMFEM [2, 19], examples
1 (diffusion) and 9 (advection) were used to generate the discretization. We
arbitrarily choose the diffusion constant $\alpha=0.1$ so that we have an
advection-diffusion mixed problem.

For this example, our finding is that the discretization matrix here is
nonsymmetric but it consists of positive eigenvalues (so positive definite).
The corresponding $2\times 2$ block matrix is symmetric but not positive
definite. The sparsity pattern of the $2\times 2$ block matrix is shown in
figure 1. Note that the symmetrized block smoother $\widetilde{M}$ has the
same properties i.e. symmetric but not positive definite.

In order to carry out the numerical experiment, we employ the power iteration
method [10], which accepts $E_{TG}$ and $\mathcal{A}$ as input parameters. We
then compute the error reduction rate
$\lambda_{k}=\displaystyle\frac{||e_{k+1}||}{||e_{k}||}$, along with the
residual reduction rate
$\lambda_{\mathcal{A}_{k}}=\displaystyle\frac{||\mathcal{A}e_{k+1}||}{||\mathcal{A}e_{k}||}=\displaystyle\frac{||r_{k+1}||}{||r_{k}||}$
so that for various choices of coarse points $n_{c}$, we can compare those
rates with the theoretical result $1-\lambda_{n_{c}+1}$. The agreement between
theory and the reduction rates is shown in 2, where it is evident that both
reduction rates coincide with the theoretical result $1-\lambda_{n_{c}+1}$,
and that the convergent rates decrease with an increase in the number of
coarse points $n_{c}$. Given that the nonsymmtric operator $A$ in this example
has a dimension of $256\times 256$, the associated $2\times 2$ symmetric
indefinite block matrix $\mathcal{A}$ is twice as large, or $512\times 512$.
The same numerical experiment was performed with a symmetric Poisson operator
$A$ and with a random sparse nonsymmetric operator $A$ with varying degrees of
freedom, and the identical $2\times 2$ block matrix $\mathcal{A}$ was created.
We find that the hypothesis and the reduction rates agree similarly
independent of the problem size.

Figure 1: Sparsity pattern of the $2\times 2$ block matrix $\mathcal{A}$ where
the nonsymmetric operator $A$ used in $\mathcal{A}_{0,1}$ and
$\mathcal{A}_{1,0}$ block is constructed through the finite element
discretization of steady-state nonsymmetric advection diffusion problem.
Figure 2: Comparison between theoretical prediction and computed reduction
rates using Kaczmarz relaxation method for $2\times 2$ block symmetric
indefinite system constructed based on a nonsymmetric operator. We use 500
iterations of the power method to compute the reduction rates.

##  6 Conclusions

By building a $2\times 2$ block symmetric indefinite system, we present in
this paper a generalization of the optimal AMG convergence theory that can be
applied to nonsymmetric situations. In order to support the derived identity
in the resultant two-grid AMG convergence theory employing optimal
interpolation, we offer a straightforward proof for the proposed theory as
well as numerical evidence. Our next effort will be focused on applying this
expansion of optimal interpolation theory to symmetric indefinite problems
originating from saddle point systems. In the context of studying Lattice
Quantum Chromodynamics(QCD) [7], our upcoming work will also include the
analysis of the same optimal interpolation theory for the non-hermitian Wilson
discretization of the 2D-Dirac equation.

## References

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# What can we learn from TMD measurements?111 Authored by Jefferson Science
Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S.
Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce this manuscript for U.S. Government purposes.

Alessandro Bacchetta

###### Abstract

Transverse-momentum-dependent parton distribution and fragmentation functions
describe the partonic structure of the nucleon in a three-dimensional momentum
space. They are subjects of flourishing theoretical and experimental activity.
They provide novel and intriguing information on hadronic structure, including
evidence of the presence of partonic orbital angular momentum.

###### Keywords:

parton distribution functions, semi-inclusive DIS, transverse momentum

###### :

12.38.-t, 13.60.-r, 13.88.+e

TMDs is an acronym for Transverse Momentum Distributions or Transverse
Momentum Dependent parton distribution functions, also called unintegrated
parton distribution functions. Most of our knowledge of the inner structure of
nucleons is encoded in parton distribution functions (PDFs). We introduce them
to describe hard scattering processes involving nucleons. The presence of a
hard probe in these processes — e.g., in DIS the photon with virtuality
$Q^{2}$ — identifies a longitudinal direction, and a plane perpendicular to
that, the transverse plane. Intuitively, standard collinear PDFs describe the
probability to find in a fast-moving nucleon a parton with a specific fraction
of the nucleon’s longitudinal momentum. TMDs describe also the probability
that the parton has a specific transverse momentum. They are therefore a
natural extension of standard PDFs from one to three dimensions in momentum
space.

Although useful from the intuition point of view, the probabilistic
interpretation of PDFs and TMDs has some technical problems and is not
strictly needed Collins (2003). What is essential is that PDFs and TMDs can be
defined in a formally clear way, through the application of factorization
theorems. They reveal crucial aspects of the dynamics of confined partons,
they can be extracted from experimental data, and allow us to make prediction
for hard-scattering experiments involving nucleons. In this sense, the
information contained in TMDs is as important as that contained in standard
PDFs.

The main difference between collinear PDFs and TMDs is that the latter do not
appear in totally inclusive processes. For instance, they do not appear in
totally inclusive DIS, but they are needed when semi-inclusive DIS is studied
and the transverse momentum of one outgoing hadron, $P_{h\perp}$, is measured.
They are necessary to describe Drell–Yan processes when the transverse
momentum of the virtual photon, $q_{T}$, is measured.

Factorization for processes involving TMDs has been worked out explicitly at
leading twist (twist 2) and one-loop order and argued to hold at all orders
Collins and Soper (1981); Ji et al. (2005). For instance, in unpolarized semi-
inclusive DIS we can measure the structure function $F_{UU,T}$, which in the
region $P_{h\perp}^{2}\ll Q^{2}$ can be expressed as Bacchetta et al. (2008a)

$\displaystyle F_{UU,T}$
$\displaystyle=\bigl{|}H\bigl{(}x\zeta^{1/2},z^{-1}\zeta_{\smash{h}}^{1/2},\mu_{F}\bigr{)}\bigr{|}^{2}\,\sum_{a}x\,e_{a}^{2}\int
d^{2}\boldmath{p}_{T}\,d^{2}\boldmath{k}_{T}\,d^{2}\boldmath{l}_{T}\,$
$\displaystyle\times\delta^{(2)}\bigl{(}\boldmath{p}_{T}-\boldmath{k}_{T}+\boldmath{l}_{T}-\boldmath{P}_{h\perp}/z\bigr{)}\,f_{1}^{a}(x,p_{T}^{2};\zeta,\mu_{F})\,D_{1}^{a}(z,k_{T}^{2};\zeta_{h},\mu_{F})\,U(l_{T}^{2};\mu_{F})\,.$
(1)

Apart from the transverse-momentum-dependent PDFs and fragmentation functions,
the formula contains the soft factor $U$, a nonperturbative and process-
independent object.

For the specific case of unpolarized observables integrated over the azimuthal
angle of the measured transverse momentum, the analysis is usually performed
in $b$-space in the Collins–Soper–Sterman framework Collins et al. (1985). The
region of $P_{h\perp}^{2}\gg M^{2}$, or $b^{2}\ll 1/M^{2}$, can be calculated
perturbatively, but when $P_{h\perp}^{2}\approx M^{2}$ a nonperturbative
component has to be introduced and its parameters must be fitted to
experimental data. This component is usually assumed to be a flavor-
independent Gaussian Landry et al. (2003).

At present, especially for azimuthally-dependent structure functions,
phenomenological analyses are often carried out using the tree-level
approximated expression

$\displaystyle F_{UU,T}$ $\displaystyle=\sum_{a}x\,e_{a}^{2}\int
d^{2}\boldmath{p}_{T}\,d^{2}\boldmath{k}_{T}\,\delta^{(2)}\bigl{(}\boldmath{p}_{T}-\boldmath{k}_{T}-\boldmath{P}_{h\perp}/z\bigr{)}\,f_{1}^{a}(x,p_{T}^{2})\,D_{1}^{a}(z,k_{T}^{2})\,.$
(2)

Also in this case, the transverse-momentum dependence of the partonic
functions is assumed to be a flavor-independent Gaussian D’Alesio and Murgia
(2004). The tree-level approximation and the Gaussian assumption are known to
be inadequate at $P_{h\perp}^{2}\gg M^{2}$, but they could still effectively
describe the physics at $P_{h\perp}^{2}\approx M^{2}$. Especially for low-
energy experiments, this is where the bulk of the data is.

The definition of quark TMDs is Collins (2003); Ji et al. (2005) (taking the
example of the fully unpolarized distribution of a quark with flavor $a$)

$f_{1}^{a}(x,p_{T}^{2};\zeta,\mu_{F})=\int\frac{d\xi^{-}d^{2}\boldmath{\xi}_{T}}{(2\pi)^{3}}\;e^{ip\cdot\xi}\,\langle
P|\bar{\psi}^{a}(0)\,{\cal L}^{v\dagger}_{(\pm\infty,0)}\,\gamma^{+}{\cal
L}^{v}_{(\pm\infty,\xi)}\,\psi^{a}(\xi)|P\rangle\bigg{|}_{\xi^{+}=0}.$ (3)

The Wilson lines, ${\cal L}$, guarantee the gauge invariance of the TMDs. They
depend on the gauge vector $v$ and contain also components at infinity running
in the transverse direction. A remarkable property of TMDs is that the
detailed shape of the Wilson line is process-dependent. This immediately leads
to the conclusion that TMDs are not universal. However, the situation is not
hopeless and the predictive power of TMD factorization is not completely
destroyed, for the following reasons

  * •

For transverse-momentum-dependent fragmentation functions, the shape of the
Wilson line appears to have no influence on physical observables Metz (2002);
*Collins:2004nx; *Yuan:2008yv; *Gamberg:2008yt; *Meissner:2008yf.

  * •

In SIDIS and Drell–Yan, the difference between the Wilson line consists in a
simple direction reversal and leads to calculable effects, namely a simple
sign reversal of all T-odd TMDs Collins (2002).

  * •

In hadron-hadron collisions to hadrons, standard universality cannot be
applied. It is however conceivable that only a manageable number of TMDs with
distinct Wilson lines are needed, preserving part of the predictive power of
the formalism Collins and Qiu (2007); *Vogelsang:2007jk.

  * •

If we consider specific transverse-momentum-weighted observables instead of
unintegrated observables, it should be possible to obtain factorized
expressions in terms of transverse moments of TMDs multiplied by calculable,
process-dependent factors Bacchetta et al. (2005); *Bomhof:2006ra;
*Bacchetta:2007sz.

Our understanding of TMDs and their extraction from data has made giant steps
in the last years, thanks to new theoretical ideas and experimental
measurements. In the near future, more experimental data are expected from
HERMES, COMPASS, BELLE and JLab.

When the spin of the nucleon and that of the quark are taken into account,
eight twist-2 functions can be introduced. They are listed in Tab. What can we
learn from TMD measurements?111 Authored by Jefferson Science Associates, LLC
under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains a
non-exclusive, paid-up, irrevocable, world-wide license to publish or
reproduce this manuscript for U.S. Government purposes.. As with collinear
PDFs, extracting TMDs calls for global fits to semi-inclusive DIS, Drell–Yan,
and $e^{+}e^{-}$-annihilation data. Care has to be taken when considering the
peculiar universality properties of TMDs. At the moment, we have some
information only about the two functions in the first column of the table:
$f_{1}$ (unpolarized function) and $f_{1T}^{\perp}$ (Sivers function).

Twist-2 transverse-momentum-dependent distribution functions. The U,L,T
correspond to unpolarized, longitudinally polarized and transversely polarized
nucleons (rows) and quarks (columns). Functions in boldface survive transverse
momentum integration. Functions in gray cells are T-odd.

Ultimately, the knowledge of TMDs should allow us to build tomographic images
of the inner structure of the nucleon in momentum space, complementary to the
impact-parameter space tomography that can be achieved by studying generalized
parton distribution functions (GPDs). An example of tomographical images of
the nucleon based on a model calculation of TMDs Bacchetta et al. (2008b) is
presented in Fig. 1.

|
---|---
|
Figure 1: Momentum-space tomographic “images” of the up quarks in a nucleon
obtained from a model calculation of TMDs Bacchetta et al. (2008b). The circle
with the arrow indicates the nucleon and its spin orientation. The distortion
in the lower panels is due to the Sivers function. In the future, it should be
possible to reconstruct these images from experimental data.

TMDs measurements should allow us to address some intriguing questions, e.g.,

  * •

Are there differences between the TMDs of different quark flavors (and of
gluons)? We know that collinear PDFs are different, not only in normalization,
but also in shape. We can expect that also the transverse momentum
distribution is different. See Ref. Mkrtchyan et al. (2008) for an example of
an experimental analysis of this issue.

  * •

How does the transverse momentum dependence change with $x$? Such a dependence
has already been introduced to describe data at low $x$ Landry et al. (2003).

  * •

Does the transverse momentum dependence of fragmentation functions change for
different quark flavors and different produced hadrons?

  * •

Are there reasons to abandon a Gaussian ansatz? We know that this assumption
fails at high transverse momentum, but there are no compelling reasons to take
a Gaussian shape even for the low-transverse-momentum, nonperturbative region.

The last item of the list is connected also to another fundamental issue that
makes TMDs interesting, i.e., the observation of partonic orbital angular
momentum. In nonrelativistic quantum mechanics, it is well known that
wavefunctions with orbital angular momentum vanish at zero momentum. This is a
general statement independent of the specific potential in which the
wavefunction is computed. This feature is reflected also in TMDs:
contributions from partons with nonzero angular momentum have to vanish at
zero transverse momentum (and therefore cannot be described by a simple
Gaussian). In general, a downturn of a TMD going to zero transverse momentum
can signal the presence of nonzero orbital angular momentum. While this
effects could barely be visible in unpolarized TMDs, certain combinations of
polarized TMDs could filter out more clearly the configurations with nonzero
orbital angular momentum. Fig. (2) shows an example of this phenomenon, using
a model calculation for illustration purposes.

Figure 2: An illustration of how the presence of orbital angular momentum can
influence the shape of TMDs. The model calculation shows different
combinations of the $f_{1}$ and $g_{1}$ TMDs for $u$ and $d$ quark at
$x=0.02$. The downturns for $p_{T}^{2}\to 0$ are due to the presence of
orbital angular momentum.

Apart from the details of their shape, all the TMDs that are not boldface in
Tab. What can we learn from TMD measurements?111 Authored by Jefferson Science
Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S.
Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce this manuscript for U.S. Government purposes. vanish
in the absence of orbital angular momentum due to angular momentum
conservation. Measuring any one of them to be nonzero is already an
unmistakable indication of the presence of partonic orbital angular momentum.
We know already from other sources (in particular the measurement of nucleons’
anomalous magnetic moments) that partonic orbital angular momentum is not
zero, however TMDs have the advantage that they can be flavor-separated and
that they are $x$ dependent. Thus, they allow us to say if orbital angular
momentum is present for each quark flavor and for gluons, and at each value of
$x$.

If stating that a fraction of partons have nonzero orbital angular momentum is
relatively simple, it is not easy to make a quantitative estimate of the net
partonic orbital angular momentum using TMDs. Any statement in this direction
is bound to be model-dependent. Generally speaking, TMDs have to be computed
in a model and the parameters of the model have to be fixed to reproduce the
TMDs extracted from data. Then, the total orbital angular momentum can be
computed in the model. Unfortunately, it is possible that two models describe
the data equally well, but give two different values for the total orbital
angular momentum.

As an example of a procedure of this kind, let us take the measurement of the
Sivers function. We know that the proper way to measure the quark total
angular momentum is by measuring the combination Ji (1997)

$J^{a}=\int_{0}^{1}dx\,x\,\Bigl{(}H^{a}(x,0,0)+E^{a}(x,0,0)\Bigr{)},$ (4)

where the definition of the generalized parton distribution $E$ in terms of
light-cone wavefunctions is

$\begin{split}E(x,0,0)=\lim_{q_{T}\to
0}\biggl{(}-\frac{1}{q_{x}-iq_{y}}\int\frac{d^{2}p_{T}}{16\pi^{2}}\,\Bigl{[}&\psi^{+\ast}_{+}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{+}\bigl{(}x,p_{T}+(1-x)q_{T}\bigr{)}\\\
&+\psi^{+\ast}_{-}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{-}\bigl{(}x,p_{T}+(1-x)q_{T}\bigr{)}\Bigr{]}\biggr{)}.\end{split}$
(5)

On the other hand, the definition of the Sivers function in terms of light-
cone wavefunctions can be written as

$f_{1T}^{\perp}(x,p_{T})=\frac{1}{16\pi^{3}}{\rm
Im}\Bigl{[}\psi^{+\ast}_{+}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{+}\bigl{(}x,p_{T}\bigr{)}+\psi^{+\ast}_{-}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{-}\bigl{(}x,p_{T}\bigr{)}\Bigr{]}.$
(6)

In spite of the similarities between the two expressions and the fact that the
same light-cone wavefunctions are involved, in general there is no
straightforward connection between the Sivers function and the GPD $E$
Meissner et al. (2007). Nevertheless, in a certain class of spectator models
it turns out that Burkardt and Hwang (2004); *Lu:2006kt

$f_{1T}^{\perp a}(x)=-L(x)\,E^{a}(x,0,0).$ (7)

Exploiting this very simple relation and using for illustration purposes the
results of the Sivers function fit from Ref. Arnold et al. (2008) we obtain

$\frac{E^{a}(x,0,0)}{E^{u}(x,0,0)}=\frac{f_{1T}^{\perp a}(x)}{f_{1T}^{\perp
u}(x)}=\frac{A_{a}}{A_{u}}\,\frac{f_{1}^{a}(x)}{f_{1}^{u}(x)},$ (8)

where (error estimates do not take into account parameter correlations)

$\displaystyle\frac{A_{d}}{A_{u}}$ $\displaystyle=-1.8\pm 0.2,$
$\displaystyle\frac{A_{\bar{u}}}{A_{u}}$ $\displaystyle=-1.1\pm 0.1,$
$\displaystyle\frac{A_{\bar{d}}}{A_{u}}$ $\displaystyle=1.3\pm 0.2,$
$\displaystyle\frac{A_{s}}{A_{u}}=-\frac{A_{\bar{s}}}{A_{u}}$
$\displaystyle=-4.8.$ (9)

Although assumption-based, the above analysis shows that the measurement of
the Sivers function can be used to give interesting constraints on the GPD $E$
and ultimately on the amount of total orbital angular momentum for each
flavor.

In summary, TMDs open new dimensions in the exploration of the partonic
structure of the nucleon. They require challenging extensions of the standard
formalism used for collinear parton distribution functions, leading us to a
deeper understanding of QCD. Among other things, they give evidence of the
presence of partonic orbital angular momentum and, with model assumptions,
they can help constraining its size.

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