相关论文: Subspace method for multiparameter-eigenvalue prob…

Emerging tensor network techniques for solutions of Partial Differential Equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultrafast numerical solutions of high-dimensional…

Given a large real symmetric, positive semidefinite m-by-m matrix, the goal of this paper is to show how a numerical approximation of the entropy, given by the sum of the entropies of the individual eigenvalues, can be computed in an…

Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…

The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the…

When applying eigenvalue decomposition on the quadratic term matrix in a type of linear equally constrained quadratic programming (EQP), there exists a linear mapping to project optimal solutions between the new EQP formulation where $Q$ is…

This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We…

The tensor-train (TT) format is a data-sparse tensor representation commonly used in high dimensional data approximations. In order to represent data with interpretability in data science, researchers develop data-centric skeletonized low…

In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the…

A numerical algorithm is proposed to deal with parametric eigenvalue problems involving non-Hermitian matrices and is exploited to find location of defective eigenvalues in the parameter space of non-Hermitian parametric eigenvalue…

A new algorithm for eigenvalue problems for the fractional Jacobi type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some…

Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or…

Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose…

The widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches…

Real-world physical systems, like composite materials and porous media, exhibit complex heterogeneities and multiscale nature, posing significant computational challenges. Computational homogenization is useful for predicting macroscopic…

In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of…

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition…

A key bottleneck in quantum machine learning is the computational cost of repeated quantum circuit evaluations during the inference phase. To address this, we present a framework for constructing fast, cheap, provably accurate classical…

In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS)…

We consider nonlinear eigenvalue problems to compute all eigenvalues in a bounded region on the complex plane. Based on domain decomposition and contour integrals, two robust and scalable parallel multi-step methods are proposed. The first…

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…