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To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have…
It is shown that a simple Dirichlet process mixture of multivariate normals offers Bayesian density estimation with adaptive posterior convergence rates. Toward this, a novel sieve for non-parametric mixture densities is explored, and its…
The inverse design of optical metasurfaces is a rapidly emerging field that has already shown great promise in miniaturizing conventional optics as well as developing completely new optical functionalities. Such a design process relies on…
The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However,…
We construct an adaptive wavelet estimator that attains minimax near-optimal rates in a wide range of Besov balls. The convergence rates are affected only by the weakest dependence amongst the channels, and take into account both noise…
In this paper, we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise.We propose two new wavelet estimators in this general context.…
We present a multi-objective Bayesian optimisation algorithm that allows the user to express preference-order constraints on the objectives of the type "objective A is more important than objective B". These preferences are defined based on…
Implicit neural representations have recently demonstrated promising potential in arbitrary-scale Super-Resolution (SR) of images. Most existing methods predict the pixel in the SR image based on the queried coordinate and ensemble nearby…
The Bayesian approach to Inverse Problems relies predominantly on Markov Chain Monte Carlo methods for posterior inference. The typical nonlinear concentration of posterior measure observed in many such Inverse Problems presents severe…
A novel block prior is proposed for adaptive Bayesian estimation. The prior does not depend on the smoothness of the function or the sample size. It puts sufficient prior mass near the true signal and automatically concentrates on its…
The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires…
In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability…
We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the $\ell_0$ norm under a tight…
Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances…
When dealing with Bayesian inference the choice of the prior often remains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the…
We revisit the problem of model-based object recognition for intensity images and attempt to address some of the shortcomings of existing Bayesian methods, such as unsuitable priors and the treatment of residuals with a non-robust error…
Due to their conjugate posteriors, Gaussian process priors are attractive for estimating the drift of stochastic differential equations with continuous time observations. However, their performance strongly depends on the choice of the…
Compressive sensing is an impressive approach for fast MRI. It aims at reconstructing MR image using only a few under-sampled data in k-space, enhancing the efficiency of the data acquisition. In this study, we propose to learn priors based…
The paper is concerned with the sparse approximation of functions having hybrid regularity borrowed from the theory of solutions to electronic Schr\"odinger equations due to Yserentant [43]. We use hyperbolic wavelets to introduce…
In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the…