Accelerated finite elements schemes for parabolic stochastic partial differential equations

Abstract: For a class of finite elements approximations for linear stochastic parabolic PDEs it is proved that one can accelerate the rate of convergence by Richardson extrapolation. More precisely, by taking appropriate mixtures of finite elements approximations one can accelerate the convergence to any given speed provided the coefficients, the initial and free data are sufficiently smooth.

• The paper introduces Richardson extrapolation to systematically accelerate FEM approximations for parabolic SPDEs.
• It derives a power series expansion for FEM solutions, exploiting algebraic symmetry that cancels odd-order error terms.
• The approach achieves arbitrarily high strong convergence rates under high regularity, enhancing simulations in filtering and finance.

Accelerated Finite Element Schemes for Linear Parabolic Stochastic PDEs

Problem Formulation and Motivation

The paper "Accelerated finite elements schemes for parabolic stochastic partial differential equations" (1812.02225) addresses the spatial discretization and numerical approximation of linear stochastic parabolic PDEs using finite element methods (FEM). These SPDEs arise in contexts such as nonlinear filtering of partially observed diffusion processes, with strong relevance in applied probability, mathematical finance, and engineering. High-dimensionality and the need for high-accuracy simulations make accelerating convergence of numerical schemes a practical imperative. The core focus of the paper is to analyze Richardson extrapolation to systematically enhance the convergence rate of FEM approximations for such SPDEs under sufficient smoothness conditions on data and coefficients.

Methodological Framework

The studied SPDEs are of the form

$$du_t(x) = \left[\mathcal{L}_t u_t(x) + f_t(x)\right]dt + \left[\mathcal{M}_t^{\rho} u_t(x) + g_t^{\rho}(x)\right] dW_t^{\rho},$$

posed on $[0,T]\times\mathbb{R}^d$, with $\mathcal{L}_t$ a uniformly parabolic, second-order differential operator and $\mathcal{M}_t^{\rho}$ a first-order operator associated with the noise. The coefficients and the free terms are assumed sufficiently smooth (in an $H^m$ sense), and parabolicity conditions guarantee well-posedness in $L_2(\Omega\times \mathbb{R}^d)$.

For spatial discretization, the authors consider a standard family of FEMs based on a symmetric, compactly supported "mother" finite element $\psi$. The FEM space is constructed on a uniform lattice, and functions are represented as

$$u^h_t(x) = \sum_{y\in \mathbb{G}_h} U^h_t(y) \psi^h_y(x),$$

where $U^h_t(y)$ are adapted processes corresponding to grid points and $\psi^h_y$ denote the rescaled and shifted mother element. The FEM leads to an infinite-dimensional stochastic system for the coefficients, whose practical implementation is justified via domain localization and truncation.

Power Series Expansions and Richardson Extrapolation

A critical technical result is that, under sufficient regularity assumptions (coefficients, data, initial conditions), the FEM approximations admit an expansion at mesh size $h = 0$:

$$U^h_t(x) = v^{(0)}_t(x) + \sum_{j=1}^k v^{(j)}_t(x) \frac{h^j}{j!} + r^h_t(x),$$

where $v^{(0)}$ is the exact solution sampled on the grid, $v^{(j)}$ are computable correction terms depending only on the SPDE data, and the remainder $r^h_t$ satisfies strong error bounds. Moreover, all odd expansion terms vanish ($v^{(j)} = 0$ for odd $j$) due to symmetry, which is a marked structural property arising from the symmetric choice of basis and mesh.

Given such an expansion, Richardson extrapolation can be applied: appropriate linear combinations of FEM solutions at mesh sizes $h, h/2, \ldots, h/2^{\bar{J}}$ are formed using coefficients solving a Vandermonde system to annihilate lower-order bias terms and thus accelerate the convergence order to arbitrarily high $J+1$ as allowed by data regularity. The construction shows that only $\lfloor J/2 \rfloor + 1$ meshes are needed due to the vanishing of odd terms.

Main Results

The primary theorem (Theorem 2 in the paper) establishes that for coefficients and data in $H^m$ with $m > 2J + d/2 + 2$, the FEM approximation $u^h$ admits the above power series expansion up to order $J$, and the strong (i.e., pathwise) error between the extrapolated FEM solution $\bar{u}^h$ and the exact solution $u$ in $L_2(\Omega; L^2)$ satisfies

$$E\sup_{t\in[0,T]} \sum_{x\in\mathbb{G}_h} |u_t(x) - \bar{u}^h_t(x)|^2 |h|^{d} \leq N |h|^{2J+2} E \mathfrak{K}_m^2,$$

where $N$ is independent of $h$, and $\mathfrak{K}_m$ involves Sobolev norms of the initial data and free terms.

Key claims and features:

• The rate of convergence obtained via Richardson extrapolation can be made arbitrarily high, contingent on the regularity of data/coefficient functions.
• Only $\lfloor J/2 \rfloor + 1$ FEM solutions at dyadically refined meshes are required for acceleration up to order $J+1$.
• The result generalizes finite-difference Richardson acceleration for SPDEs ([GK, 2010]) to FEMs, with explicit algebraic conditions (on $\psi$ and the mesh) ensuring the vanishing of lower-order terms.
• The expansion and acceleration are proven for strong convergence (in contrast to weak convergence acceleration results that are more common in SDE/SPDE contexts).
• The theoretical results extend to localized (bounded) domains with suitable modifications.

Analytical Techniques

The proof involves:

• Rigorous derivation of the power series expansion for the FEM solution operators, with explicit computation of expansion coefficients and identification of the structural vanishing of odd terms.
• Inductive construction of higher-order correction terms via differentiation with respect to mesh parameter $h$ and careful treatment of stochastic integrals under the Itô calculus.
• A hierarchy of Sobolev embedding and error inequalities for controlling discretization, expansion, and truncation errors.
• Delicate algebraic compatibility and invertibility conditions on the mother element $\psi$ and mesh set $\Lambda$ ensure necessary properties for both expansion and stability. Three explicit examples are constructed (1D linear, tensor-product, and triangular FEMs in 2D), and the verification of these conditions is supplied in full rigor.

Implications and Future Directions

The principal practical implication is the possibility to construct high-order, accelerated FEM solvers for SPDEs, with controlled computational complexity (via reduced stencil mixing and only moderate mesh refinement), provided that the SPDE data are sufficiently smooth. This advances numerical SPDE simulation capabilities, offering a clear methodology for order-raising strategies akin to well-established deterministic PDE paradigms.

The results have particular significance for high-dimensional filtering and estimation problems, where spatial resolution is computationally expensive and variance reduction is crucial. The approach can be combined with localization, truncation, and fully discretized (in space-time) schemes.

Theoretical implications include:

• Validation of FEM power series expansions for stochastic parabolic equations, paralleling classical deterministic results but in a substantially more delicate probabilistic setting.
• Explicit demonstration of bias cancellation via extrapolation, revealing the algebraic underpinnings of odd-order correction vanishing in symmetric schemes.

Limitations and extensions:

• The acceleration holds for spatial (mesh) discretization. Accelerating time-discretization bias (e.g., for Euler–Maruyama schemes) via Richardson extrapolation is shown to be infeasible for strong errors (supporting previous findings).
• The approach relies critically on high regularity in input data; for lower-regularity or discontinuous coefficients, higher-order convergence cannot be guaranteed.
• While error localization (for practical implementation in bounded domains) is addressed, further work is required for impurity-dominated and non-smooth coefficient SPDEs.

Potential future research includes extending these results to nonlinear stochastic PDEs, adaptive mesh refinement, higher-order time integration, and analysis in Banach-valued or rough path frameworks.

Conclusion

This work establishes a rigorous framework for accelerating finite element approximations of linear parabolic SPDEs via Richardson extrapolation, yielding arbitrarily high strong convergence rates under smoothness conditions. The interplay of algebraic symmetry, analytic structure, and stochastic calculus enables the precise quantification of discretization error and exposes deep analogies and distinctions relative to the deterministic PDE context. The methodology and results present a foundational advance for both theoretical analysis and high-accuracy numerical simulation of SPDEs (1812.02225).