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We address a prototype inverse scattering problem in the interface of applied mathematics, statistics, and scientific computing. We pose the acoustic inverse scattering problem in a Bayesian inference perspective and simulate from the posterior distribution using MCMC. The PDE forward map is implemented using high performance computing methods. We implement a standard Bayesian model selection method to estimate an effective number of Fourier coefficients that may be retrieved from noisy data within a standard formulation.

Inverse problems are typically ill-posed and analytical solutions are seldom available. Approaches to inverse problems in the interface of applied mathematics, statistics, and scientific computing represent a setting with a myriad of tools for robust solutions, for a number of reasons including its sound theoretical setting for uncertainty quantification [

In this paper we consider a nonlinear acoustic scattering inverse problem as a case study. The rationale is that the well posedness of the direct problem and robust numerical methods has been comprehensively studied [

The propagation of acoustic waves in a homogeneous isotropic medium with constant speed of sound is governed by the linear wave equation:

Given an incident wave

In this work we address the problem of estimating the boundary

In this section we describe the forward mapping evaluation using the integral equations method and the layer acoustic potentials approach as in [

The fundamental solution of the Helmholtz equation (

The direct problem is formulated in the form of a

The three potentials (single layer, double layer, and combined potential) reproduce the same far field pattern given a boundary

The far field pattern for the combined potential is given by

The integral (

As a test case we consider the two-dimensional kite-shaped domain shown in Figure

Synthetic example: a kite-shaped object. Numerical results for forward mapping evaluations are obtained by generating synthetic far field pattern measurements for this smooth, nonconvex boundary.

The combined potential equation (

Matrix

Consequently, in order to have an efficient numerical machinery to evaluate the direct problem we have implemented a parallel version of the general conjugate residuals method (GCR) to solve the linear system (

Elapsed times of serial and parallel numerical implementations for a single forward mapping evaluation using the combined potential.

The far field pattern value

Relative errors of the forward mapping evaluation on direction

First let us remember the Bayes rule:

We consider the inverse problem defined by the nonlinear forward mapping (

We construct a prior model based on a parameterization for

The posterior distribution does not have to our knowledge a closed form. Consequently, we resort to

The t-walk algorithm evaluates the energy function (minus log of the nonnormalized posterior). Consider the following:

The output simulations of the t-walk are vectors of Fourier coefficients

We establish the prior model as follows. Suppose that

Due to smoothness of

We pose a normal distribution for each pair

The coefficient

It is not clear how to set the constant

Prior distribution samples for coefficients on our boundary representation. The sample is drawn from a Gaussian distribution with mean

Assume that the measurements of the far field pattern have additive Gaussian noise

In order to avoid committing an “inverse crime” [

We present results of the MCMC simulations, varying the number of Fourier coefficients in Section

For our experiments we discretize the boundary of the kite-shaped object with 256 points. The far field pattern was computed on 16 evenly spaced points. We use 16 incident waves and we set the coupling parameter

Due to space constrains, we present only marginal posterior probability distributions of cosine terms for

Marginal posterior distributions for

We present in Figure

Probability Regions from the MCMC simulations (gray). The boundaries defined from mean of marginal posterior distribution are shown in blue. The corresponding MAP boundaries are shown in green. The results are compared to the kite object in dashed red line.

Also, in Figure

We refer to

We define a collection of

We use the super-model approach; that is, a new model is defined

The posterior probability for model

The ratio of two normalizing constants is defined as follows:

A consistent estimator of the ratio is

We observe from (

The importance sampling density

The kernel density estimator has the form

We present in Figure

Minus logarithm of estimated normalizing constants for Bayesian model comparisons. Best model refers to the highest normalizing constant (

Based on the Bayes factors, the best model corresponds to

Fourier series is a classical representation for smooth periodic closed curves. Furthermore, we have used a well understood and high order numerical method for forward mapping evaluation. However, these two elements by themselves are not enough to have confidence regarding the solution of the inverse problem. The quality of the solution depends also on the number of coefficients of the Fourier series used for the representation.

Our effective dimension discussion relies on MCMC methods to provide a quantitative strategy to determine the number of parameters that can be retrieved given a data set.

In this work we address the acoustic inverse scattering problem with a classical Fourier-based representation of the solution. We pose the inverse problem as a Bayesian inference problem and use the output of a MCMC method (namely, the t-walk) for our effective dimension results. For the corresponding direct problem we have used the classical layer potential approach, which was solved in a fast and reliable manner with a robust numerical method and parallel computing. Using Fourier series to represent solutions allows for a straightforward formulation that incorporates the smoothness of the solutions into the prior distribution. On the other hand, the finite Fourier representation is numerically unstable. Although other approaches to represent the scattering obstacle are applicable (e.g., wavelet basis which correspond to Besov priors), a fundamental question remains: How much information can be retrieved, within the representation, from a noisy data set?

The main contribution of this paper is the effective dimension method, which is a quantitative method to estimate and quantify the uncertainty of the estimable parameters given a noisy data set. Given a parametric representation of the solution of the inverse problem, the effective dimension method is implemented via Bayesian model selection where the normalizing constant for each model is approximated using the MCMC output. Of note, the effective dimension method is applicable regardless of the parametric representation of the solution.

The t-walk (for “traverse” or “thoughtful” walk) is a MCMC sampler for arbitrary continuous distributions that require no tuning. The t-walk maintains two independent points in the sample space and all moves are based on four proposals (walk, traverse, hop, and blow) that are accepted with a standard Metropolis-Hastings acceptance probability on the product space. These moves produce an efficient sampling algorithm that is invariant to scale and approximately invariant to affine transformations of the state space.

For an objective function (e.g., posterior distribution)

The numerical implementation of the algorithm only requires the user to define three functions.

The algorithm is available for download from Andres Christen’s personal web page

Although MCMC methods are by definition serial procedures, the high computational cost can be reduced by performing each evaluation of the objective function in a parallel computing scheme when possible. In this appendix we present a way to solve the direct problem for combined potential and Nÿstrom method.

We recall the linear system matrix for combined potential and Nÿstrom method:

The evaluation of each entry of the matrix involves the numerical evaluation of Bessel functions which is computationally costly. On the other hand, each entry of the matrix is independent. Then the matrix setup is recommended to be performed in a parallel scheme in order to reduce the computing time. That is, given a number of knots

In order to solve the linear system (

The most demanding part is computing the vector

When using multiple incident waves, the term

In our experiments, this parallel computing implementation has a performance enhancement of about 7x speedup over the original. An implementation with a more effective method than this parallel version of GCR is left as future work.

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors would like to thank Editor Fatih Yaman and anonymous referees for their constructive and detailed remarks which have helped to improve this paper in a major way. This research was supported by Grant GTO-2011-C04-168676 Guanajuato-CONACyT.