Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise

We consider a higher-order Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for stochastic partial differential equations is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered and the so called effective order of convergence is introduced which allows to directly compare various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of stochastic partial differential equations, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results

An Analysis of the Milstein Scheme for SPDEs without a Commutative Noise Condition

In order to approximate solutions of stochastic partial differential equations (SPDEs) that do not possess commutative noise, one has to simulate the involved iterated stochastic integrals. Recently, two approximation methods for iterated stochastic integrals in infinite dimensions were introduced in C. Leonhard and A. R\"o{\ss}ler: Iterated stochastic integrals in infinite dimensions: approximation and error estimates, Stoch. Partial Differ. Equ. Anal. Comput., 7(2): 209-239 (2019). As a result of this, it is now possible to apply the Milstein scheme by Jentzen and R\"ockner: A Milstein scheme for SPDEs, Found. Comput. Math., 15(2): 313-362 (2015) to equations that need not fulfill the commutativity condition. We prove that the order of convergence of the Milstein scheme can be maintained when combined with one of the two approximation methods for iterated stochastic integrals. However, we also have to consider the computational cost and the corresponding effective order of convergence for a meaningful comparison with other schemes. An analysis of the computational cost shows that, in dependence on the equation, a combination of the Milstein scheme with both of the two methods may be the preferred choice. Further, the Milstein scheme is compared to the exponential Euler scheme and we show for different SPDEs depending on the parameters describing, e.g., the regularity of the equation, which one of the schemes achieves the highest effective order of convergence

A Derivative-Free Milstein Type Approximation Method for SPDEs covering the Non-Commutative Noise case

Higher order schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations that need not to fulfill a commutativity condition for the noise term and which can flexibly be combined with some approximation method for the involved iterated integrals. Recently, the authors introduced two algorithms to simulate such iterated stochastic integrals; these clear the way for the implementation of the proposed higher order scheme. We prove the mean-square convergence of the introduced derivative-free Milstein type scheme which attains the same order as the original Milstein scheme. The original scheme, however, is definitely outperformed when the computational cost is taken into account additionally, that is, in terms of the effective order of convergence. We derive the effective order of convergence for the derivative-free Milstein type scheme analytically in the case that one of the recently proposed algorithms for the approximation of the iterated stochastic integrals is applied. Compared to the exponential Euler scheme and the original Milstein scheme, the proposed derivative-free Milstein type scheme possesses at least the same and in most cases even a higher effective order of convergence depending on the particular SPDE under consideration. These analytical results are illustrated and confirmed with numerical simulations