An Efficient Algorithm For Stochastic Differential Equations Based On Switching Idea

Statistical mechanics,quantum chemistry,machine learning,and other fields employ largescale stochastic differential equations.Most stochastic differential equations,on the other hand,are often unable to find an explicit expression for the solution.As a result,numerical computation has become a popular tool for studying stochastic differential equations.Not only is the convergence of numerical solutions a key research topic for large-scale stochastic differential equations,but so is the computational efficiency of numerical solutions.The convergence of two new efficient techniques for three types of large-scale stochastic differential equations in situations with substantial application backgrounds,such as the interacting particle system and the modified equations corresponding to the stochastic gradient descent method,is explored in this thesis.The three types of stochastic differential equations discussed in this thesis are the following.Firstly,to approach the interacting particle system piecewise,we first build a stochastic differential equation system with Markov switching as a perturbation equation.When simulating an interacting particle system with additive noise,the calculation of the interaction sum in the equation drift term takes up the majority of the calculation time.The random batch method can be thought of as a piecewise approximation of the original equation using a stochastic differential equation coupling discrete-time and finite-state Markov chains.The stochastic batch approach is enhanced in this study.The perturbation equation of piecewise approximation to the original equation is stochastic differential equations with continuous-time finite-state Markov chain.The advantage of this new approach is that it requires the same amount of computation as the stochastic batch method.At the same time,the state switching time does not need to be associated to the fixed time node.It increases the algorithm's versatility while also increasing its efficiency.In this paper,the algorithm is referred to as the random switching batch method.The approximation problem of stochastic differential equations can be easily extended using this approach.Second,a family of stochastic differential equations with diffusion term containing largescale summation term is discussed in this thesis,which originates from the stochastic gradient descent method,and diffusion terms explicitly contain a learning rate of the stochastic gradient descent method.We provides the convergence theory of random switching batch technique for solving the above equations based on the fundamental lemma defining the relationship between local and global error.The key to ensuring convergence is to match the learning rate and computation step size.Finally,a partial particle sampling approach is presented in this research.To reduce the computation workload,only a randomly selected portion of the components of the solution must be updated at each step of the simulation.This methodology,in contrast to the random switching batch technique,can use more general stochastic differential equations and can also be used in conjunction with the stochastic switched batch method.The stochastic C-stability and B-consistency theorems are used to verify the algorithm's convergence.