Related papers: Lanczos with compression for symmetric matrix Lyap…
Because of the vast volume of data being produced by today's scientific simulations, lossy compression allowing user-controlled information loss can significantly reduce the data size and the I/O burden. However, for large-scale cosmology…
The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are…
This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is…
We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming…
Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for…
In this paper, we consider finding a low-rank approximation to the solution of a large-scale generalized Lyapunov matrix equation in the form of $A X M + M X A = C$, where $A$ and $M$ are symmetric positive definite matrices. An algorithm…
We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a…
Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a…
Continuing the previous initiatives arXiv: 2207.05347 and arXiv: 2212.06180, we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm…
Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate…
In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the…
We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements, {\em without restricting to variational ansatzes}. The lattice of size $N$ is partitioned into two subclusters. At…
We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation $\underline{A}\underline{X}+\underline{X}\underline{A}^{T}=\underline{P}$, where the coefficient matrices $\underline{A}$ and…
While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum…
In this paper, we consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and…
We develop a novel convergence analysis of the classical deterministic block Krylov methods for the approximation of $h$-dimensional dominant subspaces and low-rank approximations of matrices $ A\in\mathbb K^{m\times n}$ (where $\mathbb…
In this paper, several Kaczmarz-type numerical methods for solving the matrix equation $AX=B$ and $XA=C$ are proposed, where the coefficient matrix $A$ may be full rank or rank deficient. These methods are iterative methods without matrix…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors…
In the first part of the paper we consider accelerated first order optimization method for convex functions with $L$-Lipschitz-continuous gradient, that is able to automatically adapts to problems which satisfies Polyak-{\L}ojasiewicz…