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In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph…
In this article we continue our investigation of the iterative regularization method for optimization problems based on Bregman distances. The optimization problems are subject to pointwise inequality constraints in $L^2(\Omega)$. We…
Many key problems in machine learning and data science are routinely modeled as optimization problems and solved via optimization algorithms. With the increase of the volume of data and the size and complexity of the statistical models used…
We consider a regularization problem whose objective function consists of a convex fidelity term and a regularization term determined by the $\ell_1$ norm composed with a linear transform. Empirical results show that the regularization with…
We deal with a boundary detection problem arising in nondestructive testing of materials. The problem consists in recovering an unknown portion of the boundary, where a Robin condition is satisfied, with the use of a Cauchy data pair…
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix…
A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The problem: given a propositional logic program P and a non-negative integer k, decide whether P has a stable…
We consider semilinear parabolic optimal control problems subject to Neumann boundary conditions, control constraints, and an infinite time horizon. The control constraints are pointwise in time, but they can be pointwise or integral in the…
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on…
The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type…
Uncertainty quantification is vital for decision-making and risk assessment in machine learning. Mean-variance regression models, which predict both a mean and residual noise for each data point, provide a simple approach to uncertainty…
In this paper, we describe the hyper-parameter search problem in the field of machine learning and present a heuristic approach in an attempt to tackle it. In most learning algorithms, a set of hyper-parameters must be determined before…
The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to…
Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant.…
This paper is dedicated to the stability analysis of the optimal solutions of a control problem associated with a semilinear elliptic equation. The linear differential operator of the equation is neither monotone nor coercive due to the…
For linear inverse problem with Gaussian random noise we show that Tikhonov regularization algorithm is minimax in the class of linear estimators and is asymptotically minimax in the sense of sharp asymptotic in the class of all estimators.…
In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares…
Data-driven control strategies for dynamical systems with unknown parameters are popular in theory and applications. An essential problem is to prevent stochastic linear systems becoming destabilized, due to the uncertainty of the…
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this…
The Least Absolute Shrinkage and Selection Operator (LASSO) has gained attention in a wide class of continuous parametric estimation problems with promising results. It has been a subject of research for more than a decade. Due to the…