Solving forward and inverse Helmholtz equations via controllability methods

Waves are useful for probing an unknown medium by illuminating it with a source. To infer the characteristics of the medium from (boundary) measurements, for instance, one typically formulates inverse scattering problems in frequency domain as a PDE-constrained optimization problem. Finding the medium, where the simulated wave field matches the measured (real) wave field, the inverse problem requires the repeated solutions of forward (Helmholtz) problems. Typically, standard numerical methods, e.g. direct solvers or iterative methods, are used to solve the forward problem. However, large-scaled (or high-frequent) scattering problems are known being competitive in computation and storage for standard methods. Moreover, since the optimization problem is severely ill-posed and has a large number of local minima, the inverse problem requires additional regularization akin to minimizing the total variation. Finding a suitable regularization for the inverse problem is critical to tackle the ill-posedness and to reduce the computational cost and storage requirement. In my thesis, we first apply standard methods to forward problems. Then, we consider the controllability method (CM) for solving the forward problem: it 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 with the conjugate gradient (CG) method, the CMCG 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. Based on the original CM, introduced in 1994 by Bristeau et al., for sound-soft scattering problems, we extend the CMCG method to general boundary-value problems governed by the Helmholtz equation. Numerical results not only show the usefulness, robustness, and efficiency of the CMCG method for solving the forward problem, but also demonstrate remarkably accurate solutions. Second, we formulate the PDE-constrained optimization problem governed by the inverse scattering problem to reconstruct the unknown medium. Instead of a grid-based discrete representation combined with standard Tikhonov-type regularization, the unknown medium is projected to a small finite-dimensional subspace, which is iteratively adapted using dynamic thresholding. The adaptive (spectral) space is governed by solving several Poisson-type eigenvalue problems. To tackle the ill-posedness that the Newton-type optimization method converges to a false local minimum, we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy. Numerical examples illustrate the usefulness of the ASI approach, which not only efficiently and remarkably reduces the dimension of the solution space, but also yields an accurate and robust method

Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays

Discrete nonlinear Schrödinger equations have been used for many years to model the propagation of light in optical architectures whose refractive index profile is modulated periodically in the transverse direction. Typically, one considers a modal decomposition of the electric field where the complex amplitudes satisfy a coupled system that accommodates nearest neighbour linear interactions and a local intensity dependent term whose origin lies in the χ (3) contribution to the medium's dielectric response. In this presentation, two classic continuum configurations are discretized in ways that have received little attention in the literature: the ring cavity and counterpropagating waves. Both of these systems are defined by distinct types of boundary condition. Moreover, they are susceptible to spatial instabilities that are ultimately responsible for generating spontaneous patterns from arbitrarily small background disturbances. Good agreement between analytical predictions and simulations will be demonstrated

Adaptive spectral decompositions for inverse medium problems

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 (AS) 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

Parallel Controllability Methods For the Helmholtz Equation

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

Parallel Controllability Methods For the Helmholtz Equation

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

Parallel Controllability Methods For the Helmholtz Equation

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

On Controllability Methods for the Helmholtz Equation

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 (Bristeau et al., 1994) 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 boundary-value 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

Fully scalable solver for frequency-domain visco-elastic wave equations in 3D heterogeneous media: A controllability approach

International audienceWe develop a controllability strategy for the computation of frequency-domain solutions of the 3D visco-elastic wave equation, in the perspective of seismic imaging applications. We generalize the controllability results for such equations beyond the sound-soft scattering (obstacle) problem. We detail the conjugate gradient implementation and show how an inner elliptic problem needs to be solved to compute the Riesz representative of the gradient at each iteration. We select a spectral-element spatial discretization and a fourth-order Runge-Kutta time discretization. We implement the controllability method in the framework of the SEM46 full waveform modeling and inversion software, to inherit for its excellent scalability which relies on an efficient domain decomposition algorithm. We perform a series of numerical experiments to validate the approach and illustrate its scalability up to more than fifteen hundred cores. In this case, with an elapsed time of less than 50 minutes, we solve a problem on a cubic domain containing up to 160 wavelengths in each direction, involving more than 1.7 billion unknowns

Solving Frequency-Domain Elastic Wave Equations via Parallel Controllability Methods

International audienceConventional 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

Parallel Controllability Methods For the Helmholtz Equation

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