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A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise
Arnulf Jentzen, Michael Roeckner
TL;DR
Simulating Milstein schemes for multidimensional SODEs is often difficult because of iterated stochastic integrals, and the paper considers the analogous challenge for nonlinear SPDEs with multiplicative trace-class noise. It derives and analyzes an infinite-dimensional Milstein-type method that is easy to simulate. For the stated problem, its complexity is about O(ε^-2), compared with about O(ε^-8/3) for implicit Euler.
Problem
Milstein schemes for multidimensional SODEs are often difficult to simulate because of iterated stochastic integrals, and the paper examines whether an infinite-dimensional analogue can be simulated efficiently for nonlinear SPDEs with multiplicative trace-class noise.
Source: §1 · pp. 2–3Method
The paper derives an infinite-dimensional Milstein-type scheme using an iterated-integral identity that reduces the correction to a simple product.
Source: §1 · pp. 3, 8Results
About O(ε^-2) operations and standard normal random variables achieve precision ε with the new method, compared with about O(ε^-8/3) for implicit Euler.
Source: §1 · p. 9Takeaways & Limitations
The method’s reduced complexity offers an instrument for simulating nonlinear SPDEs with multiplicative trace-class noise.
Source: §1 · p. 9The stated SPDE setting uses H = L2((0, 1), R), smooth f and b with globally bounded derivatives, and a smooth initial function vanishing at both endpoints.
Source: §1 · p. 3
Abstract
from arXiv · showhide
A new numerical method for stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense the infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise.
1 Introduction
The introduction frames strong approximation of nonlinear SPDEs with multiplicative trace-class noise as a computational-complexity problem. It develops an infinite-dimensional Milstein-type scheme whose simplified simulation reduces the stated complexity relative to the implicit Euler method.
1 Introduction
- The approximation problem seeks an H-valued, measurable numerical output meeting a prescribed precision with the least computational effort.Effort counts arithmetic and trigonometric operations, function evaluations, and standard normal random variables.Source: §1 · p. 4
- The implicit Euler reference combines spatial convergence of order 3/2− with temporal convergence of order 1/2, motivating N^3 time steps.The introduction describes this choice as asymptotically optimal for that scheme.Source: §1 · p. 6
- Taylor expansion of the diffusion coefficient in the mild equation, with recursive insertion of a lower-order approximation, provides a route to the higher-order scheme.The introduction notes that a direct Itô-formula strategy is unavailable in general because the SPDE need not have a strong solution to which the formula applies.Source: §1 · p. 7
- The iterated high-dimensional stochastic integral makes a direct infinite-dimensional Milstein analogue appear difficult to simulate.In multidimensional SODEs, Milstein’s middle term can contain 8,000 summands when d = m = 20.Source: §1 · pp. 3, 7
- An iterated-integral identity reduces the correction term to a simple product, so the resulting scheme uses finite-dimensional Wiener-process increments.The spatial approximation projects the state space to an N-dimensional subspace; the time discretization uses N^2 steps.Source: §1 · p. 8
- O(N^3 log(N)) operations and random variables compute the new scheme, compared with O(N^4 log(N)) for implicit Euler.The new scheme uses N^2 time steps rather than N^3 while retaining the same convergence rate as the Euler scheme.Source: §1 · pp. 6, 9
- For problem (9) of SPDE (7), the new algorithm (15) achieves precision ε > 0 using about O(ε^-2) computational operations and standard normal random variables, versus about O(ε^-8/3) for linear implicit Euler.Source: §1 · pp. 6, 9
- The simplification relies on the noise operator acting as a multiplication operator, which yields commutative noise and simplifies the scheme even in infinite dimensions.The introduction attributes the reduction to the structure of B(v), rather than to b being linear.Source: §1 · p. 9
2 Setting and assumptions
The analysis assumes an evolution equation on Hilbert spaces with trace-class Wiener forcing, regular drift and diffusion coefficients, and an integrable initial state. Under Assumptions 1–4, these conditions ensure a unique solution.
2 Setting and assumptions
- The framework uses separable Hilbert spaces, a trace-class covariance operator, and an orthonormal eigenbasis for the linear operator’s domain spaces.The spaces V_r are defined as domains of fractional powers of −A.Source: §2 · p. 10
- The drift F is twice continuously Fréchet differentiable, with bounded first and second derivatives on V_β.Source: §2 · p. 10
- The diffusion B has bounded first and second derivatives, and B′(v)B(v) is symmetric; simultaneous diagonalization of A and Q is not assumed.Source: §2 · pp. 10–11
- The initial value lies in V_γ with a finite fourth moment, and the stated assumptions yield a unique solution.Source: §2 · p. 11
3 Numerical scheme and main result
The scheme uses finite-dimensional state and noise projections and Wiener increments. Its convergence estimate separates state-space, noise-space, and temporal discretization errors.
3 Numerical scheme and main result
- The scheme projects the state onto a finite-dimensional Galerkin space and truncates the noise using finite subsets of its eigenmodes.Source: §3 · p. 12
- The update uses only increments of the truncated Wiener processes.Source: §3 · p. 12
- The root-mean-square error is bounded by a constant times separate state-space, noise-space, and temporal discretization terms.The state-space term decreases with the Galerkin dimension N, the noise-space term with the retained noise modes K, and the temporal term with the number of steps M.Source: §3 · p. 13
4 Proof of Theorem 1
The proof of Theorem 1 divides the error analysis into temporal and noise discretization estimates, Lipschitz estimates, and a proof of an iterated-integral identity. These estimates are combined to bound the difference between the exact and numerical solutions.
4 Proof of Theorem 1
- The proof combines the estimates from its six parts to bound the difference between the exact and numerical solutions.The proof applies Gronwall’s lemma in its final step.Source: §4 · pp. 16, 17, 20, 21
4.6 Lipschitz estimates
- The Lipschitz estimates use global Lipschitz continuity of the drift and diffusion coefficients.The proof applies this continuity to the coefficients F and B.Source: §4 · p. 17, §4.6 · pp. 36–37
4.2 Noise discretization error: Proof of (38)
- The cited passage states the result for every m ∈ {0, 1, . . . , M} and every M, K ∈ N, which shows (38).Source: §4.2 · p. 28
4.7 Proof of (30): Iterated integral identity
- The proof establishes iterated-integral identity (30), which is used to rewrite the numerical solution.The rewritten form supports the comparison with the exact solution.Source: §4 · p. 14, §4.7 · p. 41