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Projection-based reduced-order models (PROMs) have demonstrated accuracy, reliability, and robustness in approximating high-dimensional, differential equation-based computational models across many applications. For this reason, it has been…

A classical reduced order model for dynamical problems involves spatial reduction of the problem size. However, temporal reduction accompanied by the spatial reduction can further reduce the problem size without losing accuracy much, which…

Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation space. When solving the inverse…

This study develops a framework for a class of constant modulus (CM) optimization problems, which covers binary constraints, discrete phase constraints, semi-orthogonal matrix constraints, non-negative semi-orthogonal matrix constraints,…

We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem under inhomogeneous Robin boundary conditions. The model is motivated by diffusive logistic equations in spatially heterogeneous environments,…

Machine learning techniques for the solution of inverse problems have become an attractive approach in the last decade, while their theoretical foundations are still in their infancy. In this chapter we want to pursue the study of…

We study the inverse problem of parameter identification in non-coercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map by using the first-order and…

Classical benchmark problems utilize multiple transformation techniques to increase optimization difficulty, e.g., shift for anti centering effect and rotation for anti dimension sensitivity. Despite testing the transformation invariance,…

Shape optimization problems for interior eigenvalues is a very challenging task since already the computation of interior eigenvalues for a given shape is far from trivial. For example, a concrete maximizer with respect to shapes of fixed…

In this paper, we provide a mathematical framework for improving generalization in a class of learning problems which is related to point estimations for modeling of high-dimensional nonlinear functions. In particular, we consider a…

This work presents an approach to the inverse design of scattering systems by modifying the transmission matrix using reinforcement learning. We utilize Proximal Policy Optimization to navigate the highly non-convex landscape of the object…

Many inverse problems in nuclear fusion and high-energy astrophysics research, such as the optimization of tokamak reactor geometries or the inference of black hole parameters from interferometric images, necessitate high-dimensional…

Algorithm designers typically assume that the input data is correct, and then proceed to find "optimal" or "sub-optimal" solutions using this input data. However this assumption of correct data does not always hold in practice, especially…

We address the aircraft conflict resolution problem under trajectory prediction uncertainty. We consider that aircraft velocity vectors may be perturbed due to weather effects, such as wind, or measurement errors. Such perturbations may…

Chance constraints are frequently used to limit the probability of constraint violations in real-world optimization problems where the constraints involve stochastic components. We study chance-constrained submodular optimization problems,…

Consider the problem of finding an optimal value of some objective functional subject to constraints over numerical domain. This type of problem arises frequently in practical engineering tasks. Nowdays almost all general methods for…

In this work we collect and compare to each other many different numerical methods for regularized regression problem and for the problem of projection on a hyperplane. Such problems arise, for example, as a subproblem of demand matrix…

The inverse problem method is tested for a class of mean field statistical mechanics models representing a mixture of particles of different species. The robustness of the inversion is investigated for different values of the physical…

Nonlinear parametric inverse problems appear in many applications. Here, we focus on diffuse optical tomography (DOT) in medical imaging to recover unknown images of interest, such as cancerous tissue in a given medium, using a mathematical…

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However…