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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.

We study the problem of recovering a scatterer object boundary by measuring the acoustic far field using Bayesian inference. This is the inverse acoustic scattering problem, and Bayesian inference is used to quantify the uncertainty on the unknowns (e.g., boundary shape and position). Aiming at sampling efficiently from the arising posterior probability distribution, we introduce a probability transition kernel (sampler) that is invariant under affine transformations of space. The sampling is carried out over a cloud of control points used to interpolate candidate boundary solutions. We demonstrate the performance of our method through a classical problem.

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The adoption of machine learning frameworks in areas beyond computer science have been facilitated by the development of user-friendly software tools that do not require an advanced understanding of computer programming. In this paper, we present a new package (sparse kernel methods, SKM) software developed in R language for implementing six (generalized boosted machines, generalized linear models, support vector machines, random forest, Bayesian regression models and deep neural networks) of the most popular supervised machine learning algorithms with the optional use of sparse kernels. The SKM focuses on user simplicity, as it does not try to include all the available machine learning algorithms, but rather the most important aspects of these six algorithms in an easy-to-understand format. Another relevant contribution of this package is a function for the computation of seven different kernels. These are Linear, Polynomial, Sigmoid, Gaussian, Exponential, Arc-Cosine 1 and Arc-Cosine L (with L = 2, 3, … ) and their sparse versions, which allow users to create kernel machines without modifying the statistical machine learning algorithm. It is important to point out that the main contribution of our package resides in the functionality for the computation of the sparse version of seven basic kernels, which is indispensable for reducing computational resources to implement kernel machine learning methods without a significant loss in prediction performance. Performance of the SKM is evaluated in a genome-based prediction framework using both a maize and wheat data set. As such, the use of this package is not restricted to genome prediction problems, and can be used in many different applications.