Applying splitting methods with complex coefficients to the numerical integration of unitary problems

关于Trotter-Suzuki分裂的均匀半经典可观测量误差界：一个简单的代数证明

Uniform semiclassical observable error bound of Trotter-Suzuki splitting: a simple algebraic proof

本文通过代数方法证明了在半经典极限下，Trotter化方案中某些可观测量的误差与时间步长无关，无需依赖 semiclassical 机器。

向前-向后分裂法的字段指南及其FASTA实现

A Field Guide to Forward-Backward Splitting with a FASTA Implementation

本文介绍了向前-向后分裂法，重点讨论了实际应用中的实现问题，包括步长选择、加速、停止条件和初始化，并通过数值实验比较不同方法的有效性。

Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present …

A randomized operator splitting scheme inspired by stochastic optimization methods

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an a…

Splitting numerical integration for matrix completion

Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical gradient descent within the framework of optimization on manifolds. In particular, we reformulate an unconstrained optimization problem on a low-rank manifold into a differential dynamic system. We develop a splitting numerical integration method by applying a splitting integration scheme to the dynamic system. We conduct the conve…

Convergence of a splitting method for a general interest rate model

We prove mean-square convergence of a novel numerical method, the tamed-splitting method, for a generalized Ait-Sahalia interest rate model. The method is based on a Lamperti transform, splitting and applying a tamed numerical method for the nonlinearity. The main difficulty in the analysis is caused by the non-globally Lipschitz drift coefficients of the model. We examine the existence, uniqueness of the solution and boundedness of moments for the transformed SDE.We then prove bounded moments and inverses moments for the numerical approximation. The tamed-splitting method is a hybrid method i…

动态低秩近似的投影分裂积分器

A projector-splitting integrator for dynamical low-rank approximation

提出一种基于低秩流形切空间正交投影分裂的显式低计算成本积分器，并证明其具有标准积分器不具备的鲁棒性，可用于自适应秩调整和低秩矩阵优化中的截断。

On the stability of robust dynamical low-rank approximations for hyperbolic problems

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a $L^2$ stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then a…

A pseudo-spectral splitting method for linear dispersive problems with transparent boundary conditions

The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the, necessarily finite, computational domain. To obtain an efficient numerical scheme we discretize space using a spectral method. This allows us to drastically reduce the number of grid points required for a given accuracy. Applying a fully implicit time integrator, however, would require us to invert full matrices. This is addressed by performing an operator spli…

Superiorization vs. Accelerated Convex Optimization: The Superiorized/Regularized Least-Squares Case

We conduct a study and comparison of superiorization and optimization approaches for the reconstruction problem of superiorized/regularized least-squares solutions of underdetermined linear equations with nonnegativity variable bounds. Regarding superiorization, the state of the art is examined for this problem class, and a novel approach is proposed that employs proximal mappings and is structurally similar to the established forward-backward optimization approach. Regarding convex optimization, accelerated forward-backward splitting with inexact proximal maps is worked out and applied to bot…