相关论文: Parareal algorithm via Chebyshev-Gauss spectral co…
An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the…
Distributed algorithms to solve linear equations in multi-agent networks have attracted great research attention and many iteration-based distributed algorithms have been developed. The convergence speed is a key factor to be considered for…
We introduce a scalable approach to Gaussian process inference that combines spatio-temporal filtering with natural gradient variational inference, resulting in a non-conjugate GP method for multivariate data that scales linearly with…
A weighted version of the parareal method for parallel-in-time computation of time dependent problems is presented. Linear stability analysis for a scalar weighing strategy shows that the new scheme may enjoy favorable stability properties…
In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and…
We show that a pseudospectral representation of the wavefunction using multiple spatial domains of variable size yields a highly accurate, yet efficient method to solve the time-dependent Schr\"odinger equation. The overall spatial domain…
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional…
Consider the classical problem of solving a general linear system of equations $Ax=b$. It is well known that the (successively over relaxed) Gauss-Seidel scheme and many of its variants may not converge when $A$ is neither diagonally…
Although convergence of the Parareal and multigrid-reduction-in-time (MGRIT) parallel-in-time algorithms is well studied, results on their optimality is limited. Appealling to recently derived tight bounds of two-level Parareal and MGRIT…
We present a full implementation of the parareal algorithm---an integration technique to solve differential equations in parallel---in the Julia programming language for a fully general, first-order, initial-value problem. We provide a…
Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes…
This paper studies distributed continuous-time optimization for time-varying quadratic cost functions with uncertain parameters. We first propose a centralized adaptive optimization algorithm using partial information of the cost function.…
We develop a distributed Block Chebyshev-Davidson algorithm to solve large-scale leading eigenvalue problems for spectral analysis in spectral clustering. First, the efficiency of the Chebyshev-Davidson algorithm relies on the prior…
We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm, recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently solving subdiffusion equations with heterogeneous coefficients in long time. This…
In the context of distributed deep learning, the issue of stale weights or gradients could result in poor algorithmic performance. This issue is usually tackled by delay tolerant algorithms with some mild assumptions on the objective…
A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the…
This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a…