Photoacoustic inversion formulas using mixed data on finite time intervals

We study the inverse source problem in photoacoustic tomography (PAT) for mixed data, which denote a weighted linear combination of the acoustic pressure and its normal derivative on an observation surface. We consider in particular the case where the data are only available on finite time intervals, which accounts for real-world usage of PAT where data are only feasible within a certain time interval. Extending our previous work, we derive explicit formulas up to a smoothing integral on convex domains with a smooth boundary, yielding exact reconstruction for circular or elliptical domains. We also present numerical reconstructions of our new exact inversion formulas on finite time intervals and compare them with the reconstructions of our previous formulas for unlimited time wave measurements

A simple preconditioned domain decomposition method for electromagnetic scattering problems

We present a domain decomposition method (DDM) devoted to the iterative solution of time-harmonic electromagnetic scattering problems, involving large and resonant cavities. This DDM uses the electric field integral equation (EFIE) for the solution of Maxwell problems in both interior and exterior subdomains, and we propose a simple preconditioner for the global method, based on the single layer operator restricted to the fictitious interface between the two subdomains.Comment: 23 page

Boundary integral equation based numerical solutions of helmholtz transmission problems for composite scatters

In this dissertation, an in-depth comparison between boundary integral equation solvers and Domain Decomposition Methods (DDM) for frequency domain Helmholtz transmission problems in composite two-dimensional media is presented. Composite media are characterized by piece-wise constant material properties (i.e., index of refraction) and thus, they exhibit interfaces of material discontinuity and multiple junctions. Whenever possible to use, boundary integral methods for solution of Helmholtz boundary value problems are computationally advantageous. Indeed, in addition to the dimensional reduction and straightforward enforcement of the radiation conditions that these methods enjoy, they do not suffer from the pollution effect present in volumetric discretization. The reformulation of Helmholtz transmission problems in composite media in terms of boundary integral equations via multi-traces constitutes one of the recent success stories in the boundary integral equation community. Multi-trace formulations (MTF) incorporate local Dirichlet and Neumann traces on subdomains within Green’s identities and use restriction and extension by zero operators to enforce the intradomain continuity of the fields and fluxes. Through usage of subdomain Calderon projectors, the transmission problem is cast into a linear system form whose unknowns are local Dirichlet and Neumann traces (two such traces per interface of material discontinuity) and whose operator matrix consists of diagonal block boundary integral operators associated with the subdomains and extension/projections off diagonal blocks. This particular form of the matrix operator associated with MTF is amenable to operator preconditioning via Calderon projectors. DDM rely on subdomain solutions that are matched via transmission conditions on the subdomain interfaces that are equivalent to the physical continuity of fields and traces. By choosing the appropriate transmission conditions, the convergence of DDM for frequency domain scattering problems can be accelerated. Traditionally, the intradomain transmission conditions were chosen to be the classical outgoing Robin/impedance boundary conditions. When the ensuing DDM linear system is solved via Krylov subspace methods, the convergence of DDM with classical Robin transmission conditions is slow and adversely affected by the number of subdomains. Heuristically, this behavior is explained by the fact that Robin boundary conditions are first order approximations of transparent boundary conditions, and thus there is significant information that is reflected back into a given subdomain from adjacent subdomains. Clearly, using more sophisticated transparent boundary conditions facilitates the information exchange between subdomains. For instance, Dirichlet-to-Neumann (DtN) operators of adjacent domains or suitable approximations of these can be used in the form of generalized Robin boundary conditions to increase the rate of the convergence of iterative solvers of DDM linear systems. The approximations of DtN operators that are expressed in terms of Helmholtz hypersingular operators (e.g., the normal derivative of the double layer operator) are used in this dissertation. The incorporation of these in a DDM framework is subtle, and an effective method is proposed to blend these transmission operators in the presence of multiple junctions. Conceptually, the information exchange between subdomains is realized through certain Robin-to-Robin (RtR) operators, which how to compute robustly via integral equation formulations is shown. All of the Helmholtz boundary integral operators that feature in Calderon’s calculus are discretized via Nystr¨om methods that rely on sigmoid transforms, trigonometric interpolation, and singular kernel splitting. Sigmoid transforms are means to polynomially accumulate discretization points toward corners without compromising the discretization density in smooth boundary portions. A wide variety of numerical results is presented in this dissertation that illustrate the merits of each of the two approaches (MTF and DDM) for the solution of transmission problems in composite domains

A differential semblance algorithm for the inverse problem of reflection seismology

AbstractThis paper presents an analysis of stability and convergence for a special case of differential semblance optimization (DSO). This approach to model estimation for reflection seismology is a variant of the output least squares inversion of seismograms, enjoying analytical and numerical properties superior to those of more straightforward versions. We study a specialization of DSO appropriate to the inversion of convolutional-approximation planewave seismograms over layered constant-density acoustic media. We prove that the differential semblance variational principle is locally convex in suitable model classes for a range of data noise. Moreover, the structure of the convexity estimates suggest a family of quasi-Newton algorithms. We describe an implementation of one of these algorithms, and present some numerical results

Nystrom methods for high-order CQ solutions of the wave equation in two dimensions

An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate to eventually explore numerical homogenization to replace a chaff cloud by a dispersive lossy dielectric that produces the same scattering. To this end, a variety of accuracy tests are presented that showcase the high-order in time convergence (up to and including fifth order) that the Nystrom CQ discretizations are capable of delivering for a variety of two dimensional single and multiple scatterers. Particular emphasis is given to Lipschitz boundaries and open arcs with both Dirichlet and Neumann boundary conditions

Benchmarking preconditioned boundary integral formulations for acoustics.

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves