Yogi A. Erlangga

A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

Compressive simultaneous full-waveform simulation

The fact that the computational complexity of wavefield simulation is proportional to the size of the discretized model and acquisition geometry and not to the complexity of the simulated wavefield is a major impediment within seismic imaging. By turning simulation into a compressive sensing problem, where simulated data are recovered from a relatively small number of independent simultaneous sources, we remove this impediment by showing that compressively sampling a simulation is equivalent to compressively sampling the sources, followed by solving a reduced system. As in compressive sensing, this reduces sampling rate and hence simulation costs. We demonstrate this principle for the time-harmonic Helmholtz solver. The solution is computed by inverting the reduced system, followed by recovering the full wavefield with a program that promotes sparsity. Depending on the wavefields sparsity, this approach can lead to significant cost reductions, particularly when combined with the implicit preconditioned Helmholtz solver, which is known to converge even for decreasing mesh sizes and increasing angular frequencies. These properties make our scheme a viable alternative to explicit time-domain finite differences.

A new iterative solver for the time-harmonic wave equation

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian

Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to provide an optimal value for the shift, and to explain the meshdependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.

Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems

We propose a multilevel projection-based method for acceleration of Krylov subspace methods. The projection is constructed in a similar way as in deflation but shifts small eigenvalues to the largest one instead of to zero. In contrast with deflation, however, the convergence rate of a Krylov method combined with this new projection method is insensitive to the inaccurate solve of the Galerkin matrix, which with some particular choice of deflation subspaces is closely related to the coarse-grid solve in multigrid or domain decomposition methods. Such an insensitivity allows the use of inner iterations to solve the Galerkin problem. An application of a Krylov subspace method to the associated Galerkin system with the projection preconditioner leads to a multilevel, nested Krylov iteration. In this multilevel projection Krylov subspace method, information about small eigenvalues to be projected is contained implicitly in the Galerkin system associated with the matrix of the linear system to be solved. These small eigenvalues, from a Krylov method point of view, are responsible for slow convergence. In terms of projection methods, this is conceptually similar to multigrid but different in the sense that in multigrid the projection is done by the smoother. Furthermore, with the only condition being that the deflation matrices are full rank, we have in principle more freedom in choosing the deflation subspace. Intergrid transfer operators used in multigrid are some of the possible candidates. We present numerical results from solving the Poisson equation and the convection-diffusion equation, both in two dimensions. The latter represents the case where the related matrix of coefficients is nonsymmetric. By using a simple piecewise constant interpolation as the basis for constructing the deflation subspace, we obtain the following results: (i)

Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices

h

Curvelet-based migration preconditioning and scaling

-independent convergence for the Poisson equation and (ii) almost independent of

Algebraic Multilevel Krylov Methods

h

Seismic waveform inversion with Gauss‐Newton‐Krylov method

and the Peclet number for the convection-diffusion equation.