Related papers: Exact splitting methods for semigroups generated b…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also…
We propose stochastic splitting algorithms for solving large-scale composite inclusion problems involving monotone and linear operators. They activate at each iteration blocks of randomly selected resolvents of monotone operators and,…
In this paper, a class of finite difference numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The…
An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length…
We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$ and an error tolerance…
In this paper, we introduce the new construction of fractional derivatives and integrals with respect to a function, based on a matrix approach. We believe that this is a powerful tool in both analytical and numerical calculations. We begin…
We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of…
Numerical solving the Schr\"odinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose…
This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes,…
While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we…
We propose a methodology for studying the performance of common splitting methods through semidefinite programming. We prove tightness of the methodology and demonstrate its value by presenting two applications of it. First, we use the…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time…
In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is…
A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in…