Related papers: Low-rank generalized alternating direction implici…

In this paper, a Gauss-Seidel method with oblique direction (GSO) is proposed for finding the least-squares solution to a system of linear equations, where the coefficient matrix may be full rank or rank deficient and the system is…

We develop computational methods for approximating the solution of a linear multi-term matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and…

This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a…

In this paper, we consider a class of nonconvex-linear minimax problems on Riemannian manifolds, which find wide applications in machine learning and signal processing. For solving this class of problems, we develop a flexible Riemannian…

An efficient method, preconditioned conjugate gradient method with a filtering function (PCG-F), is proposed for solving iteratively the Dirac equation in 3D lattice space for nuclear systems. The filtering function is adopted to avoid the…

Analytic interpolation problems with rationality and derivative constraints occur in many applications in systems and control. In this paper we present a new method for the multivariable case, which generalizes our previous results on the…

A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytic general solution is presented,…

We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned…

A q-Gauss-Newton algorithm is an iterative procedure that solves nonlinear unconstrained optimization problems based on minimization of the sum squared errors of the objective function residuals. Main advantage of the algorithm is that it…

The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an…

In this work we generalize the Jacobi-Davidson method to the case when eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits advantages of the original Jacobi-Davidson method, has lower complexity…

We propose a Riemannian version of Nesterov's Accelerated Gradient algorithm (RAGD), and show that for geodesically smooth and strongly convex problems, within a neighborhood of the minimizer whose radius depends on the condition number as…

In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. In [1], a low rank numerical method is proposed to correct the modeling…

Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the…

Linear discriminant analysis (LDA) is a fundamental classification and dimension reduction method that achieves Bayes optimality under Gaussian mixture, but often struggles in high-dimensional settings where the covariance matrix cannot be…

Learning of low-rank matrices is fundamental to many machine learning applications. A state-of-the-art algorithm is the rank-one matrix pursuit (R1MP). However, it can only be used in matrix completion problems with the square loss. In this…

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration…

In this paper, we address the problem of solving infinite-dimensional harmonic algebraic Lyapunov and Riccati equations up to an arbitrary small error. This question is of major practical importance for analysis and stabilization of…

Fully implicit Runge-Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but high-order IRK methods are not commonly used in practice with numerical PDEs due to the difficulty…

A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are…