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Stability-Corrected Wave Functions and Structure-Preserving Rational Krylov Methods for Large-Scale Wavefield Simulations on Open Domains
| Content Provider | Semantic Scholar |
|---|---|
| Author | Zimmerling, Jörn T. |
| Copyright Year | 2016 |
| Abstract | In a first-order formulation, simulating wavefield propagation on unbounded domains amounts to solving the large order dynamical system [A(s) + sI]u(s) = b(s) for all s ∈ Ω, (1) where Ω is the frequency interval of interest. In the above equation, A(s) is the spatially discretized first-order hyperbolic wave operator and u(s) and b(s) are the unknown field and known source vectors, respectively. We note that A(s) is frequency dependent in general, due to application of the coordinate stretching or Perfectly Matched Layer (PML) technique. This technique is included to simulate outward wave propagation towards infinity. Taking equation (1) as a starting point, we discuss two Krylov-based solution methods that solve wavefield problems on open domains. Some physical properties of the approximate solutions are discussed as well. The first method linearizes the discretized wave operator with respect to frequency by setting up a frequency independent PML that constructs a set of complex PML spatial step sizes for a given frequency interval Ω [1]. The resulting linearized wave operator A no longer explicitly depends on frequency, but has complex entries and is unstable as well. Fortunately, this matrix can still be used to compute stable time-domain or conjugate-symmetric frequency-domain wave field approximations. Frequency-domain approximations, for example, can be obtained by evaluating the stability-corrected wave function [2] u(s) = [r(A, s) + r(A∗, s)] b(s), (2) where the asterisk denotes complex conjugation and r(z, s) = η(z) z + s (3) is the filtered resolvent with η(z) the complex Heaviside function defined as η(z) = 1 for Re(z) > 0 and η(z) = 0 for Re(z) < 0. Direct evaluation is not feasible, however, since the order n of matrix A is simply too large. The field vector u(s) is therefore approximated by a polynomial Krylov reduced-order model um(s) of order m n. Such a model can be computed very efficiently via a three-term Lanczos-type recursion, since there exists a diagonal weighting matrix W such that ATW = WA. Details about the construction of the algorithm, the physical significance of the weighting matrix W , and some of the convergence properties of the above reduction scheme will be discussed. In the second Krylov reduction method, we do not linearize the wave operator with respect to frequency and we consider equation (1) directly. Specifically, we focus on rational Krylov subspace field approximations to the field vector u(s) satisfying equation (1). Such an approach may be particularly beneficial in case the wavefield response on Ω and at a particular receiver location is dominated by a few modes of the wavefield operator as is the case in many applications in optics, for example [3]. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://ens.ewi.tudelft.nl/pubs/remis17householder2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
