Showing papers in "SIAM Journal on Matrix Analysis and Applications in 2020"
TL;DR: This paper explores the well-known identification of the manifold of rank $p$ positive-semidefinite matrices of size $n$ with the quotient of the set of full-rank matrices by the orthogo...
Abstract: This paper explores the well-known identification of the manifold of rank $p$ positive-semidefinite matrices of size $n$ with the quotient of the set of full-rank $n$-by-$p$ matrices by the orthogo...
71 citations
TL;DR: This work develops a family of reformulations of an arbitrary consistent linear system into a stochastic problem governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices.
Abstract: We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix ...
67 citations
TL;DR: Lower bounds for the smallest singular value of a partial Fourier matrix with arbitrary "off-the-grid" nodes were shown in this paper, where the authors proved sharp lower bounds for a rectangular Vandermonde matrix with the nodes on the uni...
Abstract: We prove sharp lower bounds for the smallest singular value of a partial Fourier matrix with arbitrary "off the grid" nodes (equivalently, a rectangular Vandermonde matrix with the nodes on the uni...
35 citations
TL;DR: This paper proposes a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave eigenvalue of LaSalle's inequality.
Abstract: In this paper, we propose a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave e...
32 citations
TL;DR: This work proposes a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND -- sparsified Nested Dissection, based on nested dissection, sparsification and low-rank compression and demonstrates that a version using orthogonal factorization and block-diagonal scaling takes less CG iterations to converge than previous similar algorithms on various kinds of problems.
Abstract: We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification...
24 citations
TL;DR: Combining the stretegy of power scheme, random projection, and singular value decomposition, the computation of low multilinear rank approximations of tensors is derived.
Abstract: This paper is devoted to the computation of low multilinear rank approximations of tensors. Combining the stretegy of power scheme, random projection, and singular value decomposition, we derive a ...
22 citations
TL;DR: Probabilistic perturbation bounds as well as probabilistic roundoff error bounds for the sequential accumulation of the inner product are derived, giving a quantitative confirmation of Wilkinson's intuition.
Abstract: Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between two real $n$-vectors. We derive probabilistic pertur...
22 citations
TL;DR: It is shown that the realization with a maximal passivity radius is a normalized port-Hamiltonian one and its computation is linked to a particular solution of a linear matrix inequality that defines passivity of the transfer function.
Abstract: We construct optimally robust port-Hamiltonian realizations of a given rational transfer function that represents a passive system. We show that the realization with a maximal passivity radius is a...
21 citations
TL;DR: For real Hermitian tensors, this work gives a full characterization for them to have hermitian decompositions over the real field and also studies other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.
Abstract: Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is n...
21 citations
TL;DR: An expansion of the generalized block Krylov subspace framework of [Electron. Anal., 47 (2017), pp. 100--126] allows the use of low-rank modifications of the Krylovsubspace framework.
Abstract: We analyze an expansion of the generalized block Krylov subspace framework of [Electron. Trans. Numer. Anal., 47 (2017), pp. 100--126]. This expansion allows the use of low-rank modifications of th...
20 citations
TL;DR: A new Riemannian modified Polak--Ribiere--Polyak conjugate gradient algorithm is presented to construct the reduced systems of quadratic-bilinear systems.
Abstract: This paper presents a new Riemannian modified Polak--Ribiere--Polyak conjugate gradient algorithm to construct the reduced systems of quadratic-bilinear systems. We eliminate the orthogonality and ...
TL;DR: The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space and is studied for decompositions.
Abstract: The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions asso...
TL;DR: The epsilon alternating least squares ($\epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal.
Abstract: The epsilon alternating least squares ($\epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matric...
TL;DR: An iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors is proposed.
Abstract: We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solut...
TL;DR: In the construction of rank-structured matrix representations of dense kernel matrices, a heuristic compression method, called the proxy point method, has been used in practice to efficiently compress matrix representations.
Abstract: In the construction of rank-structured matrix representations of dense kernel matrices, a heuristic compression method, called the proxy point method, has been used in practice to efficiently compu...
TL;DR: This work considers the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors, and computing the low-rank tensor approximation of the moment tensor implicitly using O(pnr) operations per iteration and no extra memory.
Abstract: We consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian m...
TL;DR: In this article, it was shown that for some kernel matrices corresponding to well-separated point sets, fast analytical low-rank approximation can be achieved via the use of proxy points.
Abstract: It has been known in potential theory that, for some kernel matrices corresponding to well-separated point sets, fast analytical low-rank approximation can be achieved via the use of proxy points. ...
TL;DR: This work deals with the minimization of the minimizations of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values.
Abstract: We deal with the minimization of the ${\mathcal H}_\infty$-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks ...
TL;DR: This paper proposes a novel eigenvalue-based approach to solving the unbalanced orthogonal Procrustes problem by making effective use of the necessary condition for the global minimizer and using it to solve the problem.
Abstract: In this paper, we propose a novel eigenvalue-based approach to solving the unbalanced orthogonal Procrustes problem. By making effective use of the necessary condition for the global minimizer and ...
TL;DR: This work analyzes some fixed point iterations, including Newton's iteration, for the computation of the solution of Quasi--Birth-Death stochastic processes and provides a structured perturbation analysis for the solution.
Abstract: Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative so...
TL;DR: This work considers the classical problem of approximate joint diagonalization of matrices and proposes a versatile Riemannian optimization problem, which can be cast as an optimization problem on the general linear group.
Abstract: We consider the classical problem of approximate joint diagonalization of matrices, which can be cast as an optimization problem on the general linear group. We propose a versatile Riemannian optim...
TL;DR: Canonical Polyadic Decomposition (CPD) as discussed by the authors represents a third-order tensor as the minimal sum of rank-1 terms, and it has found many concrete applications in telec...
Abstract: Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the minimal sum of rank-1 terms. Because of its uniqueness properties the CPD has found many concrete applications in telec...
TL;DR: The problem of finding the closest stable matrix for a dynamical system has many applications and is studied for both continuous and discrete-time systems and the corresponding optimization problem.
Abstract: The problem of finding the closest stable matrix for a dynamical system has many applications. It is studied for both continuous and discrete-time systems and the corresponding optimization problem...
TL;DR: The solution of matrices with a $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained opt-based methods, and many more.
Abstract: The solution of matrices with a $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained opt...
TL;DR: In this paper, an algorithm for the least square solution of a rectangular linear system with arbitrary ill-conditioned matrix was proposed. But this algorithm assumes that a complementary matrix is known.
Abstract: We introduce an algorithm for the least squares solution of a rectangular linear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We assume that a complementary matrix $Z$ is known s...
TL;DR: For a fixed symmetric matrix and symmetric perturbation, the authors developed purely deterministic bounds on how invariant subspaces of symmetric matrices can differ when measured by a suitable "rowwise" m...
Abstract: For a fixed symmetric matrix $A$ and symmetric perturbation $E$ we develop purely deterministic bounds on how invariant subspaces of $A$ and $A+E$ can differ when measured by a suitable “rowwise” m...
TL;DR: It is shown what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed and the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the converge of block GM RES.
Abstract: It is well-established that any nonincreasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with $A$ ha...
TL;DR: In this article, the first globally convergent algorithms for computing the Kreiss constant of a matrix to arbitrary accuracy were established, and three different iterations for continuous-time Kreiss constants were proposed.
Abstract: We establish the first globally convergent algorithms for computing the Kreiss constant of a matrix to arbitrary accuracy. We propose three different iterations for continuous-time Kreiss constants...
TL;DR: In this article, the authors proposed a directional compression technique for the Helmholtz equation that can handle large dense matrices that can only be handled if efficient compression techniques are used.
Abstract: Boundary element methods for the Helmholtz equation lead to large dense matrices that can only be handled if efficient compression techniques are used. Directional compression techniques can reach ...
TL;DR: Using spectral indications from the analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are neither Toeplitz nor weakly diagonally dominant corresponding to the fractional Laplacian kernel and the constant kernel, respectively.
Abstract: Nonlocal problems have been used to model very different applied scientific phenomena which involve the fractional Laplacian when one looks at the Levy processes and stochastic interfaces. This pap...