arXiv subjects

Aleš Wodecki

Publications and source records attributed to Aleš Wodecki.

7 recordsLinked to original sources

arXiv2026

Penalised and constrained geodesics in geometric control theory

In many problems in optimal control, one seeks to minimise an objective function subject to constraints on the velocity of the system. Imposing these constraints directly -- the ``hard-constrained'' approach -- is often analytically and computationally challenging. A natural alternative is to penalise violations of the constraints, solving a sequence of ``soft-constrained'' problems indexed by a penalty parameter $q$, and hoping that solutions converge to solutions of the hard-constrained problem as $q \to \infty$. We show that this approach is justified when applied to a broad class of geometric control problems on a Riemannian manifold $(M,g)$. We first consider the case where there are no autonomous dynamics, and so the control problem reduces to the problem of finding a curve of minimal length or energy between two points, subject to a nonholonomic velocity constraint/penalty determined by the choice of a bracket-generating subbundle $D$ of $TM$. We show that any sequence of solutions to the soft-constrained problem has an accumulation point which is a solution to the hard-constrained problem. Subsequently, we show how to transform a broad class of optimal control problems to the problem of finding a geodesic, by trivialising the inherent dynamics of the system using a change of coordinates inspired by the interaction picture transformation in quantum mechanics.

math.OC↗

arXiv2026

On Moment-Based Recovery of Measures with Atomic and Continuous Parts

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

math.OC↗

arXiv2026

Geometric quantum control and the random Schrödinger equation

Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very fundamental, but hard to analyze (evolution by general dynamical map) and others perhaps too simplistic to represent physical reality. In this paper, we present an intermediate approach, introducing the random Schrödinger equation, with a noise term given by a time-varying random Hermitian matrix as a means to model noisy quantum systems. We derive bounds on the error of the synthesized unitary in terms of bounds on the norm of the noise, and show that for certain noise processes these bounds are tight. We then show that in certain situations, minimizing the error is equivalent to finding a geodesic on SU(n) with respect to a Riemannian metric encoding the coupling between the control pulse and the noise process, thus connecting our work to the complexity geometry pioneered by Michael Nielsen.

quant-ph↗

arXiv2025

On Taylor-like Estimates for Chebyshev Series Approximations of $e^x$

Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and expansions derived based on families of $L^{2}-$orthogonal polynomials on bounded intervals. Traditionally, however, the properties of these series are studied in an isolated manner and most results on the $L^{2}-$based approximations are restricted to the interval of approximation. In many applications, theoretical guarantees require global estimates. Since these often follow trivially from Taylor's remainder theorem, Taylor's series is often favored in these applications. Its suboptimal numerical performance on compact intervals, however, ultimately results in such numerical algorithms being suboptimal. We show that "Taylor-like" bounds valid outside of the interval of approximation may be derived for the important case of $e^{x}$ and Chebyshev polynomials on $[-1,1]$ and provide links to substantial numerical applications.

math.GM↗

arXiv2022

Numerical Optimization of the Dirichlet Boundary Condition in the Phase Field Model with an Application to Pure Substance Solidification

As opposed to the distributed control of parabolic PDE's, very few contributions currently exist pertaining to the Dirichlet boundary condition control for parabolic PDE's. This motivates our interest in the Dirichlet boundary condition control for the phase field model describing the solidification of a pure substance from a supercooled melt. In particular, our aim is to control the time evolution of the temperature field on the boundary of the computational domain in order to achieve the prescribed shape of the crystal at the given time. To obtain efficient means of computing the gradient of the cost functional, we derive the adjoint problem formally. The gradient is then used to perform gradient descent. The viability of the proposed optimization method is supported by several numerical experiments performed in one and two spatial dimensions. Among other things, these experiments show that a linear reaction term in the phase field equation proves to be insufficient in certain scenarios and so an alternative reaction term is considered to improve the models behavior.

math.OC↗

arXiv2021

Focusing the Latent Heat Release in 3D Phase Field Simulations of Dendritic Crystal Growth

We investigate a family of phase field models for simulating dendritic growth of a pure supercooled substance. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for spatial distribution of latent heat release during solidification. In this context, several existing forms of the reaction term are analyzed. Inspired by the known conclusions of matched asymptotic analysis, we propose new variants that are simple enough to allow mathematical and numerical analysis and robust enough to be applicable to solidification under very large supercooling. The resulting models are tested in a number of numerical simulations focusing on mesh-dependence and model parameter settings. Despite the phase interface thickness being relatively large to make numerical computations feasible, the obtained results exhibit a good quantitative agreement with experimental data from rapid solidification of nickel melts.

physics.comp-ph↗

arXiv2020

Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification

We present a convergence result for the finite volume method applied to a particular phase field problem suitable for simulation of pure substance solidification. The model consists of the heat equation and the phase field equation with a general form of the reaction term which encompasses a variety of existing models governing dendrite growth and elementary interface tracking problems. We apply the well known compact embedding techniques in the context of the finite volume method on admissible unstructured polyhedral meshes. We develop the necessary interpolation theory and derive an a priori estimate to obtain boundedness of the key terms. Based on this estimate, we conclude the convergence of all of the terms in the equation system.

math.NA↗