Operator Splitting Methods: A Probability-Oriented Introduction

Core Concepts

Operator splitting methods provide a framework for decomposing and approximating complex dynamical systems by representing them as a sum of simpler components. This survey introduces the basic concepts and results of operator splitting methods, with a focus on their applications in probability theory.

Abstract

The content provides a pedagogical introduction to operator splitting methods, covering the following key points: Motivation and Examples: The idea of operator splitting is illustrated through examples from various fields, such as advection-diffusion-reaction equations, stiff ODEs, and Schrödinger equations. Operator splitting methods aim to decompose a complex dynamical system into simpler components, allowing the use of specialized numerical methods for each component. Matrices and the Lie Product Formula: The Lie product formula for matrices is presented, which forms the basis for the discrete-time operator splitting schemes. The Strang splitting scheme, which achieves second-order accuracy, is introduced as a simple application of the Lie product formula. Semigroups and the Chernoff Product Formula: The theory of strongly continuous semigroups (C0-semigroups) and their generators is reviewed, providing the abstract framework for operator splitting methods. The Chernoff product formula is discussed, which allows the approximation of C0-semigroups using iterative compositions of simpler semigroups. The Trotter-Kato Formula: The Trotter-Kato formula is presented as a core result in the theory of operator splitting methods, connecting the convergence of semigroups, their resolvents, and their generators. Various examples illustrate the applications of the Trotter-Kato formula, including the derivation of the Feynman-Kac formula and the Girsanov theorem. The survey aims to provide a comprehensive introduction to the field of operator splitting methods, with a focus on the probabilistic perspective and the underlying mathematical foundations. It serves as a starting point for further exploration of the topic and its applications in probability theory and beyond.

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A quick probability-oriented introduction to operator splitting methods

by M.B. Vovchan... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2401.00123.pdf

Deeper Inquiries

How can operator splitting methods be extended to handle time-dependent boundary conditions and non-autonomous abstract Cauchy problems

To extend operator splitting methods to handle time-dependent boundary conditions and non-autonomous abstract Cauchy problems, we need to consider the evolution of the system over time with varying boundary conditions or time-dependent coefficients. For time-dependent boundary conditions, we can incorporate these changes into the splitting scheme by updating the boundary conditions at each time step. This involves adjusting the decomposition of the system into simpler components to account for the changing boundary conditions. In the case of non-autonomous abstract Cauchy problems, where the coefficients or operators vary with time, we can apply the Trotter-Kato formula to approximate the solution by splitting the problem into simpler components corresponding to different time intervals. By iteratively applying the splitting scheme with updated coefficients at each time step, we can approximate the solution to the non-autonomous problem.

What are the limitations of the standard Trotter-Kato formula, and how can the rate of convergence be improved under additional assumptions

The standard Trotter-Kato formula has limitations in terms of the rate of convergence, especially when dealing with unbounded operators or non-compact sets. To improve the rate of convergence under additional assumptions, one approach is to consider stronger topologies or norms that allow for a more refined analysis of the convergence behavior. By imposing additional conditions on the operators or the spaces in which they act, we can potentially enhance the convergence rate. Another way to improve the rate of convergence is to explore the use of different approximation schemes or numerical techniques in conjunction with the Trotter-Kato formula. For example, incorporating Yosida approximations or variational methods can provide alternative ways to approximate the solution and potentially achieve faster convergence rates under certain conditions.

What are the connections between operator splitting methods and other numerical techniques, such as Yosida approximations and variational methods, and how can these connections be leveraged to develop new approximation schemes

Operator splitting methods have connections to various numerical techniques, including Yosida approximations and variational methods, which can be leveraged to develop new approximation schemes. By integrating these techniques with operator splitting methods, we can enhance the accuracy and efficiency of the numerical solutions obtained. Yosida approximations, which involve approximating unbounded operators by bounded ones, can be used in conjunction with operator splitting to handle more complex systems with unbounded operators. This approach allows for a more tractable decomposition of the system into simpler components while maintaining accuracy in the approximation. Variational methods, such as minimizing functionals or optimizing certain criteria, can also be integrated with operator splitting to improve the overall numerical solution. By formulating the problem in a variational framework and applying operator splitting techniques, we can explore different optimization strategies and potentially achieve better convergence rates and accuracy in the solutions obtained.