arXiv subjects

Denis Ridzal

Publications and source records attributed to Denis Ridzal.

6 recordsLinked to original sources

arXiv2026

Structure-Preserving Optimal Control of Maxwell's Equations with Applications to Source Cloaking

We develop a structure-preserving solution framework for the optimal control of the time-dependent Maxwell's equations. Building on a well-posedness theory for a weak form of the forward problem, we first analyze a forward solver that couples N\'ed\'elec and Raviart--Thomas finite elements with Crank--Nicolson time stepping. The solver preserves the de~Rham structure, enforces a discrete Gauss law, exactly satisfies a per-time-step energy balance, and converges to the weak solution under low regularity assumptions on the problem data, which are dictated by the optimal control setting. To control the Maxwell system, we add the curl of a space-time current density as a source to Amp\'ere's law. The curl form yields charge conservation without auxiliary constraints. We prove the well-posedness and continuity of the control-to-state map, derive the adjoint system and a gradient representation for a tracking-type objective functional, and formulate a discrete optimization scheme that inherits structure preservation from the forward solver. Our discrete stationarity conditions are consistent with their continuous counterparts, and the discrete optimal controls converge, with mesh and time refinements, to the continuous optima. We demonstrate the merits of our optimal control formulation and the theoretical developments by numerically solving a series of source-cloaking model problems.

math.OC↗

arXiv2025

Trilinos: Enabling Scientific Computing Across Diverse Hardware Architectures at Scale

Trilinos is a community-developed, open-source software framework that facilitates building large-scale, complex, multiscale, multiphysics simulation code bases for scientific and engineering problems. Since the Trilinos framework has undergone substantial changes to support new applications and new hardware architectures, this document is an update to ``An Overview of the Trilinos project'' by Heroux et al. (ACM Transactions on Mathematical Software, 31(3):397-423, 2005). It describes the design of Trilinos, introduces its new organization in product areas, and highlights established and new features available in Trilinos. Particular focus is put on the modernized software stack based on the Kokkos ecosystem to deliver performance portability across heterogeneous hardware architectures. This paper also outlines the organization of the Trilinos community and the contribution model to help onboard interested users and contributors.

cs.MS↗

arXiv2024

Scalable optimal control for inequality-constrained discretizations of scalar conservation laws

Optimization-based (OB) alternatives to traditional flux limiters couch preservation of properties such as local bounds and maximum principles into optimization problems, which impose these properties through inequality constraints. In this paper, we propose a new potential-target OB approach that enforces these properties using an optimal control formulation, in which the control is the source term expressed through flux potentials. The resulting OB formulation combines superb accuracy with excellent local conservation properties, but complicates the development of scalable iterative solvers, which is greatly influenced by the choice of semi-norms for the objective function. We use this fact to design scalable iterative solvers based on matrix-free trust-region Newton methods with projections onto convex sets. These solvers leverage inexpensive multigrid V-cycles while satisfying all constraints to machine precision. Numerical experiments reveal that the convergence behavior of the solvers can be greatly improved by a simple scaling of the inequality constraints. We demonstrate excellent performance in applications to linear test problems, such as $L^2$ projection and solid body rotation, and to the Cahn-Hilliard equation.

math.OC↗

arXiv2024

R-Adaptive Mesh Optimization to Enhance Finite Element Basis Compression

Modern computing systems are capable of exascale calculations, which are revolutionizing the development and application of high-fidelity numerical models in computational science and engineering. While these systems continue to grow in processing power, the available system memory has not increased commensurately, and electrical power consumption continues to grow. A predominant approach to limit the memory usage in large-scale applications is to exploit the abundant processing power and continually recompute many low-level simulation quantities, rather than storing them. However, this approach can adversely impact the throughput of the simulation and diminish the benefits of modern computing architectures. We present two novel contributions to reduce the memory burden while maintaining performance in simulations based on finite element discretizations. The first contribution develops dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh. These schemes are shown to reduce memory requirements by more than 99 percent on meshes with large numbers of nearly identical mesh cells. For applications where this structure does not exist, our second contribution leverages a recently developed augmented Lagrangian sequential quadratic programming algorithm to enable r-adaptive mesh optimization, with the goal of enhancing redundancies in the mesh. Numerical results demonstrate the effectiveness of the proposed methods to detect, exploit and enhance mesh structure on examples inspired by large-scale applications.

math.OC↗

arXiv2024

A Parallel-in-Time Multigrid Preconditioner for Optimal Control

We develop a parallel-in-time multigrid preconditioner for augmented systems. These saddle-point systems are foundational to numerical optimization. Our preconditioner, when paired with a suitable optimization method, accelerates the solution of optimal control problems. We construct the preconditioner by introducing virtual interface variables that enable time-domain decomposition. After permuting the resulting augmented system into block tridiagonal form, we develop a geometric multigrid scheme with a block Jacobi smoother, which parallelizes trivially in time. As the coarse grid solver we use GMRES preconditioned with a symmetric Gauss-Seidel iteration. We use the multigrid scheme to precondition a flexible GMRES [1] iteration for the solution of the augmented system. We combine our preconditioner with the matrix-free sequential quadratic programming (SQP) algorithm [2] to solve optimal control problems involving the van der Pol oscillator and the viscous Burgers' equation. We find that the preconditioner is remarkably effective when the problems are suitably scaled.

math.OC↗

arXiv2001

SU(1,1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1,1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1,1) random polynomial of degree $N$ are concentrated in a narrow annulus of the order of $N^{-1}$ around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit $N\to\infty$. Our results are supported through various numerical simulations. We then extend results of Hannay \cite{H} and Bleher, Shiffman, Zelditch \cite{BSZ2} to derive different formulae for correlations between zeros of the SU(1,1) random analytic functions, by applying the generalized Kac-Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1,1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros $|z_j|>1$ of the SU(1,1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.

math-ph↗