Numerical Solution of Stochastic Partial Differential Equations with Correlated Noise

In this paper we investigate the numerical solution of stochastic partial differential equations (SPDEs) for a wider class of stochastic equations. We focus on non-diagonal colored noise instead of the usual space-time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen $\&$ Kloeden, we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate

A Milstein scheme for SPDEs

This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite dimensional commutativity condition. In particular, a suitable infinite dimensional analog of Milstein's algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation and stochastic reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some numerical simulations are remove

Solving Partial Differential Equations with Monte Carlo / Random Walk on an Analog-Digital Hybrid Computer

Current digital computers are about to hit basic physical boundaries with respect to integration density, clock frequencies, and particularly energy consumption. This requires the application of new computing paradigms, such as quantum and analog computing in the near future. Although neither quantum nor analog computer are general purpose computers they will play an important role as co-processors to offload certain classes of compute intensive tasks from classic digital computers, thereby not only reducing run time but also and foremost power consumption. In this work, we describe a random walk approach to the solution of certain types of partial differential equations which is well suited for combinations of digital and analog computers (hybrid computers). The experiments were performed on an Analog Paradigm Model-1 analog computer attached to a digital computer by means of a hybrid interface. At the end we give some estimates of speedups and power consumption obtainable by using future analog computers on chip.Comment: 9 pages, 7 figures. Proceeding for the MikroSystemTechnik Kongress 2023 (VDE Verlag MST Kongress 2023

Geophysical Fluid Dynamics

The workshop “Geophysical Fluid Dynamics” addressed recent advances in analytical, stochastic, modeling and computational studies of geophysical fluid models. Of central interest were the reduced geophysical models, that are derived by means of asymptotic and scaling techniques, and their investigations by methods from the above disciplines. In particular, contributions concerning the viscous and inviscid geostrophic models, the primitive equations of oceanic and atmospheric dynamics, tropical atmospheric models and their coupling to nonlinear dynamics of phase changes moisture, thermodynamical effects, stratifying effects, as well as boundary layers were presented and discussed