No abstract

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.

We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need to fulfill a commutativity condition for the noise term. The newly developed derivative-free Milstein type scheme differs significantly from schemes that are appropriate for the case of commutative noise. As a key result, the new derivative-free Milstein type scheme needs only two stages that are specifically tailored based on a technique that, compared to the original Milstein scheme, allows for a reduction of the computational complexity by one order of magnitude. Moreover, the proposed derivative-free Milstein scheme can flexibly be combined with some approximation method for the involved iterated stochastic integrals. As the main result, we prove the strong $$L^2$$ L 2 -convergence of the introduced derivative-free Milstein type scheme, especially if it is combined with any suitable approximation algorithm for the necessary iterated stochastic integrals. We carry out a rigorous analysis of the error versus computational cost and derive the effective order of convergence for the derivative-free Milstein type scheme in the case that the truncated Fourier series algorithm for the approximation of the iterated stochastic integrals is applied. As a further novelty, we show that the use of approximations of iterated stochastic integrals based on truncated Fourier series together with the proposed derivative-free Milstein type scheme improves the effective order of convergence compared to that of the Euler scheme and the original Milstein scheme. This result is contrary to well known results in the finite dimensional SDE case where the use of merely truncated Fourier series does not improve the effective order of convergence in the $$L^2$$ L 2 -sense compared to that of the Euler scheme.

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 [8]. As a result of this, it is now possible to apply the Milstein scheme by Jentzen and Röckner [2] 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.