相关论文: Matrix- and tensor-oriented numerical schemes for …
In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this…
The coupled nonlinear space fractional Ginzburg-Landau (CNLSFGL) equations with the fractional Laplacian have been widely used to model the dynamical processes in a fractal media with fractional dispersion. Due to the existence of…
Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau…
In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and…
We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and…
In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In…
In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix…
A new integration scheme, combining the stability and the precision of usual pseudo-spectral codes with the locality of finite differences methods, is introduced. It turns out to be particularly suitable for the study of front and…
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial…
We show that the discrete operator stemming from the time and space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines…
In this paper, using a pseudospectral approach, we develop operational matrices based on the shifted Chebyshev polynomials to approximate numerically Caputo fractional derivatives and Riemann-Liouville fractional integrals. In order to make…
Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce…
This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical result on…
In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of…
In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of…
In this paper we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
We present a framework for accelerating a spectrum of machine learning algorithms that require computation of bilinear inverse forms $u^\top A^{-1}u$, where $A$ is a positive definite matrix and $u$ a given vector. Our framework is built on…
In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula…
In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep…