Gonzalo R. Feijoo

The variational multiscale method—a paradigm for computational mechanics

Abstract We present a general treatment of the variational multiscale method in the context of an abstract Dirichlet problem. We show how the exact theory represents a paradigm for subgrid-scale models and a posteriori error estimation. We examine hierarchical p -methods and bubbles in order to understand and, ultimately, approximate the ‘fine-scale Greens function’ which appears in the theory. We review relationships between residual-free bubbles, element Greens functions and stabilized methods. These suggest the applicability of the methodology to physically interesting problems in fluid mechanics, acoustics and electromagnetics.

Solution of inverse problems in elasticity imaging using the adjoint method

We consider the problem of determining the shear modulus of a linear-elastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

An application of shape optimization in the solution of inverse acoustic scattering problems

We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference between the measured and expected signals. The measured and expected signals are assumed to satisfy a boundary-value problem given by the Helmholtz equation with the Sommerfeld condition imposed at infinity. Gradient-based algorithms are used to solve this optimization problem. At every step of the algorithm the derivative of the cost function with respect to the parameters that describe the shape of the object is calculated. We develop an efficient method based on the adjoint equations to calculate the derivative and show how this method is implemented in a finite element setting. The predominant cost of each step of the algorithm is equal to one forward solution and one adjoint solution and therefore is independent of the number of parameters used to describe the shape of the object. Numerical examples showing the efficacy of the proposed methodology are presented.

Shape sensitivity calculations for exterior acoustics problems

The purpose of this paper is to present a method to calculate the derivative of a functional that depends on the shape of an object. This functional depends on the solution of a linear acoustic problem posed in an unbounded domain. We rewrite this problem in terms of another one posed in a bounded domain using the Dirichlet‐to‐Neumann (DtN) map or the modified DtN map. Using a classical method in shape sensitivity analysis, called the adjoint method, we are able to calculate the derivative of the functional using the solution of an auxiliary problem. This method is particularly efficient because the cost of calculating the derivatives is independent of the number of parameters used to approximate the shape of the domain. The resulting variational problems are discretized using the finite‐element method and solved using an efficient Krylov‐subspace iterative scheme. Numerical examples that illustrate the efficacy of our approach are presented.

Lanczos iterated time-reversal

A new iterative time-reversal algorithm capable of identifying and focusing on multiple scatterers in a relatively small number of iterations is developed. It is recognized that the traditional iterated time-reversal method is based on utilizing power iterations to determine the dominant eigenpairs of the time-reversal operator. The convergence properties of these iterations are known to be suboptimal. Motivated by this, a new method based on Lanczos iterations is developed. In several illustrative examples it is demonstrated that for the same number of transmitted and received signals, the Lanczos iterations based approach is substantially more accurate.

An acoustic finite‐element model to study sonar interactions with marine mammals.

A computer model based on the finite‐element method (FEM) is being developed to study the interaction of sonar signals with marine mammals. This model solves the Helmholtz equation in a computational box that includes the animal and the surrounding medium, water. The FEM code has been validated with analytical solutions for scattering of a plane wave by a fluid sphere over a range of parameters and frequencies of interest. The same FEM code has been applied to a 142‐cm‐long specimen of the common dolphin (Delphinus delphis); internal pressure and displacement fields have been computed. The animal is represented in the computer model by a set of tissue groups whose acoustic properties, density and sound speed, are taken from the literature. The geometry of each tissue group was constructed from segmented computerized tomography images. Results are presented for harmonic signals in the 1‐10 kHz frequency range. [Work supported by NOPP through ONR award N000140710992.]

A new imaging method in diffraction tomography based on the topological derivative

The development and implementation of a new method in diffraction tomography, which is based on the optimization of a topology, is discussed. The method relies on the definition of a function, called the topological derivative, that has support in the image and at every point quantifies the sensitivity (or derivative) of the scattered field to the introduction of an infinitesimal scatterer at that point. If the scatterers are rigid, the expression for the topological derivative is calculated analytically. As a result, the proposed scheme is not iterative. Furthermore, no assumptions such as the Born or Rylov approximations are made. It is shown through several numerical experiments that the image produced by this function is capable of reconstructing both the position and the shape of scatterers in the domain with excellent agreement.

Prediction of elastic material properties via an adjoint formulation

The prediction of elastic material properties via an adjoint formulation in this talk an efficient computational formulation to determine the acoustic properties of an elastic material, given its response to a time‐harmonic excitation, is presented. Such problems arise in biomedical imaging using techniques such as Magnetic Resonance Elastography. The problem is posed as an inverse problem and its solution is computed using a mathematical programming algorithm (from mathematical programming). The functional to be minimized is the norm of the difference of the ‘‘measured’’ and a ‘‘trial’’ velocity field. At every iteration the trial velocity field is determined by solving the Helmholtz equation. The derivatives of the functional with respect to material properties, which form the input to the optimization algorithm, are computed by solving an adjoint problem. Using this formulation, the cost of computing derivatives is independent of the number of parameters used to represent the material properties, and t...

Krylov methods in inverse scattering and imaging

Inverse scattering requires the measurement and inversion of a compact operator. The compactness of the operator implies that its range is low dimensional (i.e., sparse.) This implies the theoretical possibility of measuring the full operator with relatively few measurements and inverting it on a sparse basis. One issue, however, is that the basis on which the operator is sparse is unknown a priori. We show that Krylov methods can be used to simultaneously identify an efficient basis for the measurements and facilitate the inversion for imaging purposes. In particular, we show how imaging via Multiple SIgnal Classification (MUSIC) and by Krischs factorization method can be efficiently implemented in a Krylov space context. This method allows us to make most efficient use of all available acoustic sensors with few measurements and with minimal mutual interference.

Krylov methods in time‐reversal imaging by multiple‐signal‐classification.

Multiple signal classification (MUSIC) has been used to form images and identify sound sources since 1986 [R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., AP‐34, 276–280 (1986)]. In active sonar imaging of point targets, the MUSIC method can be used to estimate the range of the time‐reversal operator. In this context, the method is related to the decomposition of the time‐reversal operator method. Typical implementations of these methods utilize measurements of the entire time‐reversal operator, and require computations of its eigenvalues and eigenvectors. By contrast, we show that Krylov iterative methods can be used to perform MUSIC imaging with relatively few acoustic excitations. Furthermore, by using the Lanczos technique, no eigenvalues or eigenvectors need be computed. Rather, an orthonormal basis for the range space of the time‐reversal operator can be constructed directly from the received data. Most of the necessary computing is performed...