When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is highly indefinite and thus notoriously difficult to solve, in fact increasingly so at higher frequency. The exact controllability approach [1] instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period, the controllability method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. The original energy functional used to penalize the departure from periodicity is strictly convex only for sound-soft scattering problems. To extend the controllability approach to general boundaryvalue problems governed by the Helmholtz equation, new penalty functionals are proposed, which are numerically efficient. Numerical experiments for wave scattering from sound-soft and sound-hard obstacles, inclusions, but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods.Date: May 2, 2018.(CM) transforms the problem back to the time domain, where it seeks a periodic solution yp¨, tq with (known) period T " 2π{ω of the corresponding time-dependent wave equation. The unknown initial conditions, v 0 " yp¨, 0q and v 1 " y t p¨, 0q, that yield the desired periodic solution are then determined by minimizing a convex cost functional, J 1 pv 0 , v 1 q, which penalizes the departure from periodicity. Akin to a shooting method, the controllability approach iteratively solves the least-squares optimization problem with a standard conjugate gradient (CG) iteration [10]. Each CG iteration then requires the solution of a forward and a backward wave equation together with the solution of a symmetric and positive definite linear system independent of ω, both easily solved using standard numerical methods. Hence, the CM-CG approach solely relies on standard numerical algorithms, which are not only robust with respect to ω but also easy to parallelize. In [11], Bardos and Rauch proved the uniqueness of the minimizer for sound-soft exterior Helmholtz problems. They also proposed an alternative functional, J 8 pv 0 , v 1 q, which is unconditionally coercive even for trapping obstacles. Later Koyama proved convergence of the CM-CG method based on J 1 for sound-soft wave scattering from a disk [12].The CM-CG method in [1,8] relied on a piecewise linear finite element (FE) discretization in space and the second-order leapfrog scheme in time. Low-order FE discretizations, however, are notoriously prone to the pollution effect [13]. Moreover, local mesh refinement imposes a severe CFL stability constraint on explicit time integration, as the maximal time-step is dictated by the smallest element in the mesh. Recently, Heikkola et al. [14,15...
The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures.
Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.
We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.
Conventional methods to solve the time-harmonic elastic wave equations usually rely on either direct solvers or iterative solvers. The former are very efficient for treating multiple right hand side problems, as the matrix factorization needs to be done only once for all the right hand sides. However, it suffers from a significant shortcoming associated with high memory consumption and lack of scalability. The latter are matrix-free, and therefore much lighter in memory and scalable. However, dedicated preconditioners are required to converge these methods. The efficiency of existing preconditioners quickly deteriorates as the frequency increases. Another approach to compute time-harmonic solution to elastic wave equations is to consider timedomain solvers. Instead of computing the stationary solution, which convergence is shown to be dependent on the presence of trapped waves and complex wave phenomenon, we develop here a numerical strategy based on a controllability method. The method has been recently analyzed in the frame of acoustic propagation and we extend it here in the frame of linear elasticity. We rely on a spectral element space discretization and a fourth order Runge Kutta time integration. We present the basic properties and formulation of the method, before investigating its scalability and its memory requirement on canonical three-dimensional numerical experiments. The method is shown to be scalable for a problem involving approximately 250 millions degrees of freedom up to more than fifteen hundred computational units.
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