sparse grid satisfies ${\mathcal{H}}_{\Lambda_{n}}\subset\SS_{\theta}$. Under
the assumptions of Theorem 2, for any $\frac{2}{3}<\tau<1$,

$\displaystyle\left\lVert\bm{m}-{\mathcal{I}}_{\Lambda_{n}}\bm{m}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\leq
C_{\tau,p}\left(\\#{\mathcal{H}}_{\Lambda_{n}}\right)^{1-1/\tau}.$

where $C_{\tau,p}$ is a function of $\tau$, $p$ but is dimension-independent.
In particular,

$\displaystyle
C_{\tau,p}=C^{\frac{1}{\tau}}\exp\left(\frac{1}{\tau}C_{2}^{\tau}p^{1-\tau}\frac{2^{2(-(p+1)\tau+1)}}{1-2^{-(p+1)\tau+1}}\frac{1}{1-2^{1-\left(\frac{1}{2}-\delta\right)p\tau}}\right),$

where in turn $C$ is defined in Lemma 27 and $C_{2}$ is defined in Lemma 23.

######  Remark 30 (On optimality of the convergence rate $-\frac{1}{2}$).

The best convergence rate with respect to the number of collocation nodes
predicted by the theorem is $-\frac{1}{2}$ and is obtained for
$\tau=\frac{2}{3}$. This is the same as the convergence rate of the parametric
truncation with respect to the number of parameters: Denoting
$\bm{m}({\bm{y}})$ the parametric solution for
${\bm{y}}\in{\mathbb{R}}^{{\mathbb{N}}}$ and by
$\bm{m}_{N}({\bm{y}})\coloneqq\bm{m}((y_{1},\dots y_{N},0,0,\dots))$, for any
$N\in{\mathbb{N}}$, its $N$-dimensional truncation, one can show that

$\displaystyle\left\lVert\bm{m}-\bm{m}_{N}\right\rVert_{L^{2}_{\mu}({\mathcal{X}},L^{2}([0,T],H^{1}(D)))}\lesssim
N^{-1/2}.$

Since it is not possible to have less that 1 collocation node per dimension,
the sparse grid algorithm achieves the optimal approximation rate.

In particular, piecewise quadratic approximation ($p=3$) has optimal
convergence rate and using $p>3$ does not improve the convergence rate (but
may improve the constant $C_{\tau,p}$). For the same reason, sparse grid
interpolation based on other 1D interpolations schemes (e.g. global
polynomials) cannot give a better convergence rate (but may improve the
constant).

######  Remark 31.

Given an approximation
$\bm{m}_{\Lambda}({\bm{y}})={\mathcal{I}}_{\Lambda}[\bm{m}]({\bm{y}})$ of the
solution to the parametric LLG problem (15), it is easy to sample an
approximate random solution of the _random_ LLG problem (5) too. One has to
sample i.i.d. standard normal random variables
$\bm{Y}=(Y_{i})_{i=1}^{N_{\Lambda}}$ and evaluate
$\bm{m}_{\Lambda}(Y_{1},\dots,Y_{N_{\Lambda}})$. Here
$N_{\Lambda}\in{\mathbb{N}}$ is the support size of the multi-index set
$\Lambda$:
$N_{\Lambda}\coloneqq\min{\left\\{n\in{\mathbb{N}}:\forall\bm{\nu}\in\Lambda,\
\operatorname{\rm supp}(\bm{\nu})\leq n\right\\}}$. Equivalently,
$N_{\Lambda}$ is the number of active parameters in the sparse grid
interpolant ${\mathcal{I}}_{\Lambda}$. The root-mean-square error is naturally
the same as the one we estimated in the previous theorem

$\displaystyle\sqrt{{\mathbb{E}}_{\bm{Y}}\left\lVert\bm{m}(\bm{Y})-\bm{m}_{\Lambda}(\bm{Y})\right\rVert_{\mathbb{M}}}=\left\lVert\bm{m}-{\mathcal{I}}_{\Lambda}\bm{m}\right\rVert_{L^{2}_{\mu}(\SS_{\theta},\mathbb{M})}.$

We can also approximately sample from the random solution of the _stochastic_
PDE (2):

  1. (1)

Sample a Wiener process $W$

  2. (2)

Compute the first $N_{\Lambda}$ coordinates
$\bm{Y}=(Y_{1},\dots,Y_{N_{\Lambda}})$ of its Lèvy-Ciesielsky expansion
$W(t)=\sum_{i=1}^{\infty}Y_{i}\eta_{i}(t)$

  3. (3)

Compute $\bm{m}_{\Lambda}(\bm{Y})$, the approximate solution to the random LLG
problem (5)

  4. (4)

Finally compute the inverse Doss-Sussmann transform

$\displaystyle\bm{M}_{\Lambda}(W,t,x)\coloneqq
e^{W(t)G}\bm{m}_{\Lambda}(\bm{Y};t,x).$

Recall that a convenient expression for $e^{WG}$ is available in the third
line of (3). The approximation error is again comparable to the one found in
the previous theorem. Indeed, denoting $\left\lVert\cdot\right\rVert$ the
root-mean-square error, the Doss-Sussmann transform implies

$\displaystyle\sqrt{{\mathbb{E}}_{W}\left\lVert\bm{M}-\bm{M}_{\Lambda}\right\rVert_{\mathbb{M}}}$
$\displaystyle=\left\lVert
e^{WG}\left(\bm{m}-\bm{m}_{\Lambda}\right)\right\rVert.$

Recall then the third line of (3) to write

$\displaystyle\left\lVert
e^{WG}\left(\bm{m}-\bm{m}_{\Lambda}\right)\right\rVert=\left\lVert\bm{m}-\bm{m}_{\Lambda}+\sin(W)G\left(\bm{m}-\bm{m}_{\Lambda}\right)-\cos(W)G^{2}\left(\bm{m}-\bm{m}_{\Lambda}\right)\right\rVert.$

Finally, the triangle inequality and the fact that $\bm{g}\in L^{\infty}(D)$
yield

$\displaystyle\left\lVert\bm{M}-\bm{M}_{\Lambda}\right\rVert$
$\displaystyle\leq\left(1+\left\lVert\bm{g}\right\rVert_{L^{\infty}(D)}+\left\lVert\bm{g}\right\rVert_{L^{\infty}(D)}^{2}\right)\left\lVert\bm{m}-\bm{m}_{\Lambda}\right\rVert.$

###  7.4. Numerical tests

We test numerically the convergence of the sparse grid method defined above.
Since no exact sample path of the solution is available, we approximate them
with the space and time approximation from [1]. This method is based on the
_tangent plane scheme_ and has the advantage of solving one _linear_ elliptic
problem per time step with finite elements. The time-stepping is based on a
BDF formula. The method is high-order for both the finite elements and BDF
discretization.

We consider the problem on the 2D domain $D=[0,1]^{2}$ with $z=0$. The final
time is $T=1$. The noise coefficient is defined as

(56) $\displaystyle\bm{g}(\bm{x})=\left(-\frac{1}{2}\cos(\pi
x_{1}),-\frac{1}{2}\cos(\pi x_{2}),\sqrt{1-\left(\frac{1}{2}\cos(\pi
x_{1})\right)^{2}-\left(\frac{1}{2}\cos(\pi x_{2})\right)^{2}}\right).$

Observe that $\partial_{n}\bm{g}=0$ on $\partial D$ and that $|\bm{g}|=1$ on
$D$. The initial condition is $\bm{m}_{0}=(0,0,1)$.

The space discretization is order 1 on a structured triangular mesh with
$2048$ elements and mesh-size $h>0$. The time discretization is order 1 on
$256$ equispaced timesteps of size $\tau$. We use piecewise affine sparse
grid, corresponding to $p=2$. As for the multi-index selection, we compare two
strategies:

  * •

The basic profit from Section 7.2. We recall that

$\displaystyle\tilde{{\mathcal{P}}}_{\bm{\nu}}=\prod_{i:\nu_{i}=1}2^{-\frac{1}{2}\ell(i)}\prod_{i:\nu_{i}>1}\left(2^{\nu_{i}+\frac{1}{2}\ell(i)}\right)^{-p}\left(\prod_{i}p2^{\nu_{i}}\right)^{-1}\qquad\forall\bm{\nu}\in{\mathcal{F}},$

where $\ell(i)=\lceil\log_{2}(i)\rceil$. Compared to (51), here we have set
$C_{1}=C_{2}=\epsilon=1$ and $\alpha=0$ for simplicity.

  * •

A modified version of the improved profit from Section 7.3, namely

(57)
$\displaystyle{\mathcal{P}}_{\bm{\nu}}=\prod_{i:\nu_{i}=1}2^{-\frac{3}{2}\ell(i)}\prod_{i:\nu_{i}>1}\left(2^{\nu_{i}+\frac{1}{2}\ell(i)}\right)^{-p}\left(\prod_{i}p2^{\nu_{i}}\right)^{-1}\qquad\forall\bm{\nu}\in{\mathcal{F}},$

where again $\ell(i)=\lceil\log_{2}(i)\rceil$. Compared to Section 7.3, we
have set $C_{1}=C_{2}=\gamma=1$ and neglected the factor $r_{\ell}(\bm{\nu})$.

We estimate the approximation error of the sparse grid approximations with

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\left\lVert\bm{m}_{\tau
h}({\bm{y}}_{i})-{\mathcal{I}}_{\Lambda}[\bm{m}_{\tau
h}]({\bm{y}}_{i})\right\rVert_{L^{2}(0,T,H^{1}(D))},$

where $N=1024$, $\left({\bm{y}}_{i}\right)_{i=1}^{N}$ are i.i.d. standard
normal samples of dimension $2^{10}$ each. $\bm{m}_{hk}({\bm{y}}_{i})$ denotes
the corresponding space and time approximations of the sample paths.

Observe that if the time-step size is $\tau=2^{-n}$, then the parameter-to-
finite element solution map depends only on the first $n+1$ levels of the LC.
In our case, $n=8$, so the maximum relevant level is $L=9$, i.e. 512
dimensions. In the following numerical examples we always approximate fewer
dimensions, which means that the time-discretization error is negligible
compared to the parametric approximation error.

$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{-1}$$10^{0}$Number of collocation points
$Q$ErrorP1 basic profitP1 improved profitP2 improved
profit$Q^{-1/4}$$Q^{-1/2}$

$10^{1}$$10^{2}$$10^{3}$$10^{0}$$10^{1}$$10^{2}$Number of collocation points
$Q$Number of dimensionsP1 basic profitP1 improved profitP2 improved profit$Q$

$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{-1}$$10^{0}$Number of collocation points
$Q$Error$\tau=2^{-6}$$\tau=2^{-7}$$\tau=2^{-8}$$Q^{-1/4}$$Q^{-1/2}$

Figure 3. Approximation of ${\bm{y}}\mapsto\bm{m}({\bm{y}})$. Top: Error vs.
number of collocation nodes; Bottom left: Number of effective dimensions vs.
number of collocation nodes. Bottom right: Comparison of convergence of the
sparse grid approximation ($p=3$, i.e. piecewise quadratic) for different
space and time discretization parameters. In both case time-step and mesh size
are related by $h=8\tau$.

Results are displayed in Figure 3. In the left plot, we observe that using
basic profits leads to a sub-algebraic convergence rate which decreases as the
number of approximated dimensions increases. Conversely, improved profits
leads to a robust algebraic convergence of order about $\frac{1}{2}$.
Piecewise _quadratic_ interpolation is optimal as predicted in Section 7.3 and
it delivers the same convergence rate as piecewise linear interpolation.
Hence, the restriction in Theorem 29 is possibly an artifact of the proof. In
view of Remark 30, it seems unnecessary to test higher polynomial degrees. In
the right plot, we observe that the number of active dimensions (i.e., those
dimension which are seen by the sparse-grid algorithm) grows similarly for all
methods, with the basic profit having a slightly higher value.

Let us conclude the section with a study of the dependence on space and time
approximation parameters. We verify numerically that the approximation power
of the method does not degrade when space and time approximations are refined,
see the bottom right plot in Figure 3.

##  8\. Multi-level sparse grid collocation

In the present section, we show how the sparse grid scheme defined and studied
in this work can be combined with a method for space and time approximation to
define a fully discrete approximation scheme. Here we employ again the
linearly implicit BDF-finite element scheme from [1].

Given $\tau>0$, consider $N_{\tau}$ equispaced timesteps on $[0,T]$. Given
$h>0$, define a quasi-uniform triangulation ${\mathcal{T}}_{h}$ of the domain
$D$ with mesh-spacing $h$. Denote, for any ${\bm{y}}\in\SS_{\theta}$,
$\bm{m}_{\tau h}({\bm{y}})$ the corresponding space and time approximation of
$\bm{m}({\bm{y}})$. Assume that there exists a constant $C_{\rm FE}>0$
independent of $h$ or $\tau$ such that

$\displaystyle\left\lVert\bm{m}-\bm{m}_{\tau,h}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\leq
C_{\rm FE}(\tau+h).$

Moreover, we assume that the computational cost (number of floating-point
operations) of computing a single $\bm{m}_{\tau h}({\bm{y}})$ is proportional
to

$\displaystyle C_{\rm sample}(\tau,h)=\tau^{-1}h^{-d}.$

Indeed, the numerical scheme requires, at each timestep, solving a linear
system of size proportional to the number of elements of ${\mathcal{T}}_{h}$,
which in turn is proportional to $h^{-d}$. The latter operation can be
executed with empirical linear complexity using GMRES with multigrid
preconditioning. See also [38], for mathematically rigorous preconditioning
strategies for LLG.

Theorem 29 shows that there exists $C_{\rm SG}>0$ and $0<r<\frac{1}{2}$ such
that, denoting ${\mathcal{I}}_{\Lambda}$ the sparse grid interpolant and
${\mathcal{H}}_{\Lambda}$ the corresponding sparse grid,

$\displaystyle\left\lVert\bm{m}-{\mathcal{I}}_{\Lambda}[\bm{m}]\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\leq
C_{\rm SG}\left(\\#{\mathcal{H}}_{\Lambda}\right)^{-r}.$

A _Single Level_ approximation of $\bm{m}$ can be defined as

$\displaystyle\bm{m}_{\Lambda,\tau,h}^{\rm
SL}\coloneqq{\mathcal{I}}_{\Lambda}\left[\bm{m}_{h\tau}\right].$

The cost of computing the single level approximation is

$\displaystyle C^{\rm
SL}_{\Lambda,\tau,h}\coloneqq\\#{\mathcal{H}}_{\Lambda}C_{\rm sample}(\tau,h)$

The approximation accuracy can be estimated as

$\displaystyle\left\lVert\bm{m}-\bm{m}_{\Lambda,\tau,h}^{\rm
SL}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}$
$\displaystyle\leq\left\lVert\bm{m}-{\mathcal{I}}_{\Lambda}[\bm{m}]\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}+\left\lVert{\mathcal{I}}_{\Lambda}[\bm{m}-\bm{m}_{\tau,h}]\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}$
$\displaystyle\leq C_{\rm
SG}\left(\\#{\mathcal{H}}_{\Lambda}\right)^{-r}+C_{\rm stab}C_{\rm
FE}(h+\tau).$

where $C_{\rm stab}>0$ is the stability constant of the sparse grid
interpolation operator. A quasi-optimal single level approximation requires
balancing the three approximation parameters $\Lambda,\tau$ and $h$ so that
the summands in the previous estimate have similar values. This choice leads
to the following error estimate with respect to the cost $C^{\rm
SL}_{\Lambda,\tau,h}$

(58) $\displaystyle\left\lVert\bm{m}-\bm{m}_{\Lambda,\tau,h}^{\rm
SL}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\lesssim\left(C^{\rm
SL}_{\Lambda,\tau,h}\right)^{-\frac{1}{\frac{1}{r}+(d+1)}}.$

A _Multilevel_ approximation of $\bm{m}$ can be defined following [55]. Let
$K\geq 0$ and consider a sequence of approximation parameters
$\left(\Lambda_{k}\right)_{k=0}^{K}$, $\left(\tau_{k}\right)_{k=0}^{K}$ and
$\left(h_{k}\right)_{k=0}^{K}$. Denote $\bm{m}_{k}=\bm{m}_{\tau_{k},h_{k}}$
for $0\leq k\leq K$ and $\bm{m}_{-1}\equiv 0$. Define the multilevel
approximation as

$\displaystyle\bm{m}_{K}^{\rm
ML}\coloneqq\sum_{k=0}^{K}{\mathcal{I}}_{\Lambda_{k}}\left[\bm{m}_{K-k}-\bm{m}_{K-k-1}\right].$

The computational cost is proportional to

$\displaystyle C^{\rm
ML}_{K}=\sum_{k=0}^{K}\\#{\mathcal{H}}_{\Lambda_{k}}C_{\rm
sample}(\tau_{K-k},h_{K-k}).$

To guarantee approximation, we require the following assumption on the sparse
grid approximation of differences: For any $0\leq k\leq K$,

$\displaystyle\left\lVert\bm{m}_{k}-\bm{m}_{k-1}-{\mathcal{I}}_{\Lambda}[\bm{m}_{k}-\bm{m}_{k-1}]\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\leq
C_{\rm SG}\left(\\#{\mathcal{H}}_{\Lambda}\right)^{-r}(h_{k}+\tau_{k}).$

For the multilevel-approximation to be quasi-optimal, all terms in the
multilevel expansion shall have similar magnitude to the $K$-th (finest) time
and space approximation. To this end, we choose the multi-index sets
$\Lambda_{k}$ so that

(59) $\displaystyle\left(\\#{\mathcal{H}}_{\Lambda_{K-k}}\right)^{-r}\leq
C_{\rm FE}\left(C_{\rm
SG}(K+1)\right)^{-1}\frac{\tau_{K}+h_{K}}{\tau_{k}+h_{k}}.$

As a consequence, the multilevel error can be estimated as

$\displaystyle\left\lVert\bm{m}-\bm{m}_{K}^{\rm
ML}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\leq 2C_{\rm
FE}(\tau_{K}+h_{K}).$

The error can be related to the computational cost as done in [55]. We obtain
the improved error-to-cost relation

(60) $\displaystyle\left\lVert\bm{m}-\bm{m}_{K}^{\rm
ML}\right\rVert_{L^{2}_{\mu}(\SS_{\theta};\mathbb{M})}\lesssim\left(C^{\rm
ML}_{K}\right)^{-\frac{1}{d+1}}.$

We compare numerically single- and multilevel schemes on the following example
of relaxation dynamics with thermal noise. The domain is $D=[0,1]^{2}$ with
$z=0$. The final time is $T=1$. The noise coefficient $\bm{g}$ is set to one
fifth of the coefficient defined in (56). The initial condition $\bm{m}^{0}$
coincides with (56). The time and space approximations are both of order 1.
The sparse grid scheme is piecewise linear and the multi-index sets are built
using the improved profit (57) from the previous numerical experiments.
Observe that, in the following convergence tests refinement leads
automatically to an increase of the number of approximated parameters and a
reduction of the parametric _truncation_ error. We consider $K=5$ and define
$\tau_{k}=2^{-k-2}$, $h_{k}=2^{-k}$, and $\Lambda_{k}$ using the same profit-
maximization as in the previous section.

For the single level approximation, we choose $\Lambda_{k}$ minimal such that
$\\#{\mathcal{H}}_{\Lambda_{k}}>2^{2k}$. The last choice corresponds to
assuming that the sparse grid approximation converges with order
$r=\frac{1}{2}$ with respect to the number of collocation nodes. We compute a
sequence of single level approximations $\bm{m}^{\rm
SL}_{\Lambda_{k},\tau_{k},h_{k}}$ for $k=0,\dots K$ and report the results in
Figure 4.

For the multilevel approximation, we follow formula (59). The constants
$C_{\rm FE}\approx 0.7510$, $C_{\rm SG}\approx 0.1721$ and $r\approx 0.4703$
are determined with short sparse grid and finite element convergence tests. We
obtain

K | $\\#{\mathcal{H}}_{\Lambda_{0}}$ | $\\#{\mathcal{H}}_{\Lambda_{1}}$ | $\\#{\mathcal{H}}_{\Lambda_{2}}$ | $\\#{\mathcal{H}}_{\Lambda_{3}}$ | $\\#{\mathcal{H}}_{\Lambda_{4}}$ | $\\#{\mathcal{H}}_{\Lambda_{5}}$
---|---|---|---|---|---|---
0 | 1 |  |  |  |  |
1 | 1 | 3 |  |  |  |
2 | 1 | 3 | 10 |  |  |
3 | 1 | 4 | 18 | 82 |  |
4 | 2 | 7 | 27 | 131 | 602 |
5 | 2 | 10 | 42 | 193 | 887 | 1500

The last figure $1500$ is chosen smaller than the one required by the formula
(4082) to guarantee reasonable computational times. Again results are reported
in Figure 4.

Since the solution in closed form is not available, we approximate it with a
reference solution. We consider $128$ Monte Carlo samples of $W$ and
approximate the corresponding sample paths in space and time with timestep
$\tau_{\rm ref}=2^{-9}$ and mesh size $h_{\rm ref}=2^{-7}$. Computing the
error for the single- and multilevel approximation requires first sampling the
interpolants on the Monte Carlo sample parameters and then interpolating in
the reference space.

The convergence test confirms that the multilevel method is superior to the
single level method.

$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$10^{7}$$10^{8}$$10^{9}$$10^{10}$$10^{-1.5}$$10^{-1}$$10^{-0.5}$Compuations
cost CErrorSingle levelMultilevel$C^{-1/3}$$C^{-1/5}$ Figure 4. Comparison of
single- and multilevel approximations based on piecewise polynomial sparse
grids for the parametric approximation and the linearly implicit BDF-finite
elements methods from [1] for time and space approximation.

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