Convergence and complexity study of GMRES variants for solving multi-frequency elastic wave propagation problems

A Theoretical Investigation of Channel Wave Multipath Propagation in a Coal Seam

“…Multipath fading develops as a result of the multipath propagation of channel waves at the time of reception of the signal by the receiver. is can have a significant impact on the received signal's quality [19][20][21]. It is possible to deduce the path difference between the multipath signal and the direct signal, along with the time delay formula in the multipath signal, by considering the direct wave based on the path formula and multipath propagation model given above.…”

“…η β denotes the interface reflection coefficient. Equations (19), (21), and (22) can be solved simultaneously so that E is obtained, which is expressed as follows:…”

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

“…where A ∈ C n×n is a large, sparse and nonsingular matrix, σ i ∈ C are t shifts given at once, I is the n × n identity matrix, x (i) and b are solutions and right-hand side vectors of the t linear systems, respectively. Problem (1.1) arises in implicit numerical solutions of partial differential (PDEs) [1,2] and fractional differential equations (FDEs) [3,4], in control theory [5,6], large-scale eigenvalue computations [7], quantum chromodynamics (QCD) applications [8] and in other computational science problems [9][10][11]. Krylov subspace methods are an efficient alternative to sparse direct methods for solving a sequence of multi-shifted linear systems, owing to the shift-invariance property where we denote by K m (A, b) := span{b, Ab, .…”

“…By a suitable choice of the initial vectors x (i) 0 , for example take x (i) 0 = 0, the solution of systems (1.1) requires a single Krylov basis [12,13]. This approach has shown to effectively reduce storage and algorithmic costs in the analysis of realistic QCD, PageRank and multi-frequency elastic wave propagation problems [9][10][11].…”

An efficient two-level preconditioner for multi-frequency wave propagation problems

“…[12,19] for an analysis in the Helmholtz case. We do not consider this aspect in this paper but note that the MSSS preconditioner developed in [9] allows to apply the inverse of (2) fast, even for large frequencies [8]. We conclude with numerical examples obtained from a finite element discretization of the time-harmonic visco-elastic wave equation at multiple wave frequencies.…”