Numerical solutions of the Helmholtz equation with Robbins boundary conditions are researched using the quadratic spline collocation method. By reordering the unknowns, we obtain a 2×2 block linear system where the two diagonal sub-matrices are block tridiagonal. For the obtained linear system, we introduce a two-step preconditioner using the block polynomial preconditioner. Theoretical analysi...

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a generalized Stokes interface problem. Applicati...

We present an efficient preconditioner for the orbital minimization method when the Hamiltonian is discretized using planewaves (i.e., pseudospectral method). This novel preconditioner is based on an approximate Fermi operator projection by pole expansion, combined with the sparsifying preconditioner to efficiently evaluate the pole expansion for a wide range of Hamiltonian operators. Numerical...

A preconditioned AOR iterative method is proposed with the preconditioner I + S∗ αβ. Some comparison theorems are given when the coefficient matrix of linear system A is an irreducible L−matrix. The convergence rate of AOR iterative method with the preconditioner I + S∗ αβ is faster than the convergence rate with the preconditioner I + Sα by Li et al. Numerical example verifies comparison theor...

A sparse mesh-neighbour based approximate inverse preconditioner is proposed for a type of dense matrices whose entries come from the evaluation of a slowly decaying free space Green’s function at randomly placed points in a unit cell. By approximating distant potential fields originating at closely spaced sources in a certain way, the preconditioner is given properties similar to, or better th...

This paper introduces the recursive sweeping preconditioner for the numerical solu4 tion of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner 5 with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping precondi6 tioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. 7 Compared ...

Purpose: Design of a preconditioner for fast and efficient parallel imaging and compressed sensing reconstructions. Theory: Parallel imaging and compressed sensing reconstructions become time consuming when the problem size or the number of coils is large, due to the large linear system of equations that has to be solved in l1 and l2-norm based reconstruction algorithms. Such linear systems can...

this paper deals with scrutinizing the convergence properties of iterative methods to solve linear system of equations. recently, several types of the preconditioners have been applied for ameliorating the rate of convergence of the accelerated overrelaxation (aor) method. in this paper, we study the applicability of a general class of the preconditioned iterative methods under certain conditio...

The study focuses primarily on Vandermonde-like matrix systems. The idea is to express Vandermonde and Vandermonde-like matrix systems as the problems related to Krylov Matrices. The connection provides a different angle to view the Vandermondelike systems. Krylov subspace methods are strongly related to polynomial spaces, hence a nice connection can be established using LU factorization as pro...

This paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices A. By resorting to rational functions of A, the algorithm decomposes the spectrum of A into two disjoint regions and approximates the restriction of A−1 on these regions separately. We show a systematic way to construct these rational functions so that they can be applied stably and inexp...