Operator Splitting Around Euler-Maruyama Scheme and High Order Discretization of Heat Kernels

2019 ◽  
Author(s):  
Yuga Iguchi ◽  
Toshihiro Yamada
Keyword(s):  
Operator Splitting ◽  
Heat Kernels ◽  
High Order

scholarly journals Operator splitting around Euler-Maruyama scheme and high order discretization of heat kernels

2020 ◽  
Author(s):  
Toshihiro Yamada ◽  
Yuga Iguchi
Keyword(s):  
Commutative Algebra ◽  
Computational Algorithm ◽  
Diffusion Processes ◽  
Operator Splitting ◽  
Heat Kernels ◽  
Heat Equations ◽  
Discretization Method ◽  
Splitting Scheme ◽  
Order Operator ◽  
Numerical Examples

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Some regularity estimates for convolution semigroups on a group of polynomial growth

2004 ◽  
Vol 77 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Nick Dungey
Keyword(s):  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Kernels ◽  
High Order ◽  
Convolution Semigroup ◽  
Geometric Condition ◽  
Convolution Semigroups ◽  
Regularity Estimates ◽  
Convolution Powers ◽  
Gaussian Estimates

AbstractWe study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

scholarly journals Optimized Operator-Splitting Methods in Numerical Integration of Maxwell's Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Z. X. Huang ◽  
X. L. Wu ◽  
W. E. I. Sha ◽  
B. Wu
Keyword(s):  
Numerical Integration ◽  
Time Domain ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Operator Splitting ◽  
Splitting Methods ◽  
High Order ◽  
Numerical Results ◽  
Operator Splitting Methods ◽  
The Time Domain

Optimized operator splitting methods for numerical integration of the time domain Maxwell's equations in computational electromagnetics (CEM) are proposed for the first time. The methods are based on splitting the time domain evolution operator of Maxwell's equations into suboperators, and corresponding time coefficients are obtained by reducing the norm of truncation terms to a minimum. The general high-order staggered finite difference is introduced for discretizing the three-dimensional curl operator in the spatial domain. The detail of the schemes and explicit iterated formulas are also included. Furthermore, new high-order Padé approximations are adopted to improve the efficiency of the proposed methods. Theoretical proof of the stability is also included. Numerical results are presented to demonstrate the effectiveness and efficiency of the schemes. It is found that the optimized schemes with coarse discretized grid and large Courant-Friedrichs-Lewy (CFL) number can obtain satisfactory numerical results, which in turn proves to be a promising method, with advantages of high accuracy, low computational resources and facility of large domain and long-time simulation. In addition, due to the generality, our optimized schemes can be extended to other science and engineering areas directly.

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