Related papers: Iterative solutions to the steady state density ma…
We discuss the numerical solution methods available when solving for the steady-state density matrix of a time-independent open quantum optical system, where the system operators are expressed in a suitable basis representation as sparse…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…
Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix $N$ becomes very large. There remains…
The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising…
In the case of quantum systems interacting with multiple environments, the time-evolution of the reduced density matrix is described by the Liouvillian. For a variety of physical observables, the long-time limit or steady state solution is…
Solving large, sparse linear systems is a fundamental workload in scientific computing and engineering simulations, often dominating runtime and energy consumption in high-performance computing (HPC) applications. In this work, we explore…
We propose a two-level nested preconditioned iterative scheme for solving sparse linear systems of equations in which the coefficient matrix is symmetric and indefinite with relatively small number of negative eigenvalues. The proposed…
This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures,…
We propose a two-level iterative scheme for solving general sparse linear systems. The proposed scheme consists of a sparse preconditioner that increases the skew-symmetric part and makes the main diagonal of the coefficient matrix as close…
We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix $A$ and a possibly dense, rank deficient matrix of the form $\gamma UU^T$, where $\gamma > 0$…
In this paper we consider linear systems with dense-matrices which arise from numerical solution of boundary integral equations. Such matrices can be well-approximated with $\mathcal{H}^2$-matrices. We propose several new preconditioners…
Large-scale dynamic inverse problems are often ill-posed due to model complexity and the high dimensionality of the unknown parameters. Regularization is commonly employed to mitigate ill-posedness by incorporating prior information and…
This paper introduces the sparsifying preconditioner for the pseudospectral approximation of highly indefinite systems on periodic structures, which include the frequency-domain response problems of the Helmholtz equation and the…
We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a…
We present a general variational approach to determine the steady state of open quantum lattice systems via a neural network approach. The steady-state density matrix of the lattice system is constructed via a purified neural network ansatz…
The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems by a bounded scatterer. This paper presents the directional preconditioner for the iterative solution of linear systems of the…
We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution…
In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular,…
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space…