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Linear inverse problems are ubiquitous. Often the measurements do not follow a Gaussian distribution. Additionally, a model matrix with a large condition number can complicate the problem further by making it ill-posed. In this case, the performance of popular estimators may deteriorate significantly. We have developed a new estimator that is both nearly optimal in the presence of Gaussian errors while being also robust against outliers. Furthermore, it obtains meaningful estimates when the problem is ill-posed through the inclusion of $\ell_1$ and $\ell_2$ regularizations. The computation of our estimate involves minimizing a non-convex objective function. Hence, we are not guaranteed to find the global minimum in a reasonable amount of time. Thus, we propose two algorithms that converge to a good local minimum in a reasonable (and adjustable) amount of time, as an approximation of the global minimum. We also analyze how the introduction of the regularization term affects the statistical properties of our estimator. We confirm high robustness against outliers and asymptotic efficiency for Gaussian distributions by deriving measures of robustness such as the influence function, sensitivity curve, bias, asymptotic variance, and mean square error. We verify the theoretical results using numerical experiments and show that the proposed estimator outperforms recently proposed methods, especially for increasing amounts of outlier contamination. Python code for all of the algorithms are available online in the spirit of reproducible research.

Linear inverse problems are ubiquitous. Often the measurements do not follow a Gaussian distribution. Additionally, a model matrix with a large condition number can complicate the problem further by making it ill-posed. In this case, the performance of popular estimators may deteriorate significantly. We have developed a new estimator that is both nearly optimal in the presence of Gaussian errors while being also robust against outliers. Furthermore, it obtains meaningful estimates when the problem is ill-posed through the inclusion of $\ell_1$ and $\ell_2$ regularizations. The computation of our estimate involves minimizing a non-convex objective function. Hence, we are not guaranteed to find the global minimum in a reasonable amount of time. Thus, we propose two algorithms that converge to a good local minimum in a reasonable (and adjustable) amount of time, as an approximation of the global minimum. We also analyze how the introduction of the regularization term affects the statistical properties of our estimator. We confirm high robustness against outliers and asymptotic efficiency for Gaussian distributions by deriving measures of robustness such as the influence function, sensitivity curve, bias, asymptotic variance, and mean square error. We verify the theoretical results using numerical experiments and show that the proposed estimator outperforms recently proposed methods, especially for increasing amounts of outlier contamination. Python code for all of the algorithms are available online in the spirit of reproducible research.

Many problems were studied during the six year period of this research, 1 September 1977-30 September 1983 under the two titles: Smoothing, Regularization, and Ill-Posed Inverse Problems and Robust and Convex Estimation of Functions of Several Variables and Associated Design Problems, Sixteen Technical Reports and twenty papers and discussions appeared in the open literature. Briefly, the problems studied fall in three classes: Problems density estimation, problems in experimental design, and problems in the estimation of functions of one of several variables given noisy observations on functionals and various types of side and or prior information.

This final report summarizes research activities in applied mathematics, including control of distributed systems, optimization, an inverse problems. The report summarizes results published in 23 journal articles. Results include a new method for computing solutions to ill-posed problems via optimal filtering and new results for the analysis and control of switching systems.

This monograph is devoted to statements of multidimensional inverse problems, in particular to methods of their investigation. Questions of the uniqueness of solution, solvability and stability are studied. Methods to construct a solution are given and, in certain cases, inversion formulas are given as well. Concrete applications of the theory developed here are also given. Where possible, the author has stopped to consider the method of investigation of the problems, thereby sometimes losing generality and quantity of the problems, which can be examined by such a method. The book should be of interest to researchers in the field of applied mathematics, geophysics and mathematical biology.

We study a standard method of regularization by projections of the linear inverse problem $Y=Af+\epsilon$, where $\epsilon$ is a white Gaussian noise, and $A$ is a known compact operator with singular values converging to zero with polynomial decay. The unknown function $f$ is recovered by a projection method using the singular value decomposition of $A$. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.