Wenjia Jing

Mathematical and statistical methods for multistatic imaging

Mathematical and Probabilistic Tools.- Small Volume Expansions and Concept of Generalized Polarization Tensors.- Multistatic Configuration.- Localization and Detection Algorithms.- Dictionary Matching and Tracking Algorithms.- Imaging of Extended Targets.- Invisibility.- Numerical Implementations and Results.- References.- Index.

Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity ∗

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

Resolution and stability analysis in acousto-electric imaging

We consider the optimization approach to the acousto-electric imaging problem. Assuming that the electric conductivity distribution is a small perturbation of a constant, we investigate the least-squares estimate analytically using (multiple) Fourier series, and confirm the widely observed fact that acousto-electric imaging has high resolution and is statistically stable. We also analyze the case of partial data and the case of limited-view data, in which some singularities of the conductivity can still be imaged.

Corrector Theory for MsFEM and HMM in Random Media

We analyze the random fluctuations of several multiscale algorithms, such as the multiscale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multiscale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one-dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multiscale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size h, we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length e→0. We next ...

Corrector theory for elliptic equations with long-range correlated random potential

We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochas- tic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Greens function, we show that either the deterministic or the random part of the corrector dominates.

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

This paper analyzes the random fluctuations obtained by a heterogeneous multiscale first-order finite element method applied to solve elliptic equations with a random potential. Several multi-scale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a second-order elliptic equations with random potential in two dimensions of space. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained in [8] for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence computationally less intensive, algorithms.

Nonlinear Optimization Algorithms

In this chapter we consider the nonlinear optimization problem for reconstructing the shape of an extended target from multistatic data. Because of the nonlinearity of the problem, iterative algorithms have to be introduced.

3. Generalized Polarization Tensors

3.1. Definition 3.2. Uniqueness Result 3.3. Symmetry and Positivity of GPT’s 3.4. Bounds for the Polarization Tensor of Polya-Szego 3.5. Estimates of the Weighted Volume and the Center of Mass 3.6. Polarization Tensors of Multiple Inclusions 3.6.1. Definition 3.6.2. Properties 3.6.3. Representation by Equivalent Ellipses

Limiting distribution of elliptic homogenization error with periodic diffusion and random potential

We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that in the short range correlation setting, the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by the homogenized Greens function, the homogenized solution and the statistics of the random potential. Similar results hold for random potentials that are functions of long range correlated Gaussian random fields. These results generalize previous ones in the setting with slowly varying diffusion coefficients, and the current setting with fast oscillations in the differential operator requires new methods to prove compactness of the probability distributions of the random fluctuation.

Passive Array Correlation-Based Imaging in a Random Waveguide

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.