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Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational…
Abbreviated Abstract: The objective of Evolutionary Computation is to solve practical problems (e.g. optimization, data mining) by simulating the mechanisms of natural evolution. This thesis addresses several topics related to adaptation…
In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…
We present a new method of blackbox optimization via gradient approximation with the use of structured random orthogonal matrices, providing more accurate estimators than baselines and with provable theoretical guarantees. We show that this…
We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible…
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the…
Recent advances in LLM-guided evolutionary computation, particularly AlphaEvolve, have demonstrated remarkable success in discovering novel mathematical constructions and solving challenging optimization problems. In this article, we…
We will discuss an extension of the pseudospectral method developed by Wineberg, McGrath, Gabl, and Scott for the numerical integration of the KdV initial value problem. Our generalization of their algorithm can be used to solve initial…
In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced which converges to the solution under some suitable conditions. In the nonlinear case, after a few iterations the terms of…
The performance of evolutionary algorithms can be heavily undermined when constraints limit the feasible areas of the search space. For instance, while Covariance Matrix Adaptation Evolution Strategy is one of the most efficient algorithms…
Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are…
We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the Free-evolution approach, which…
In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and…
We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends…
We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique…
The main topic of this thesis is the analysis of evolution equations reflecting issues in ecology and population dynamics. In mathematical modelling, the impact of environmental elements and the interaction between species is read into the…
We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved…