arXiv subjects

Michael Herty

Publications and source records attributed to Michael Herty.

At least 37 records · Page 2Linked to original sources

arXiv2026

Kinetic simulated annealing optimization with entropy-based cooling rate

We present a modified simulated annealing method with a dynamical choice of the cooling temperature. The latter is determined via a closed-loop control and is proven to yield exponential decay of the entropy of the particle system. The analysis is carried out through kinetic equations for interacting particle systems describing the simulated annealing method in an extended phase space. Decay estimates are derived under the quasi-invariant scaling of the resulting system of Boltzmann-type equations to assess the consistency with their mean-field limit. Numerical results are provided to illustrate and support the theoretical findings.

math.OC↗

arXiv2026

High-dimensional stochastic finite volumes using the tensor train format

A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and tensor trains in a novel way: the dimensions of physical space and time are kept as full tensors, while all stochastic dimensions are compressed together into a tensor train. The resulting hybrid format has one tensor train for each spatial cell and each time step. The MUSCL scheme is adapted to the proposed hybrid format, and its feasibility is demonstrated through several test cases. For the scalar Burgers' equation, we conduct a convergence study and compare the results with those obtained using the full tensor train format with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions.For systems of conservation laws, we focus on the Euler equations. A parameter study and a comparison with the full tensor train format are carried out for the Sod shock tube problem. As a more complex application, we investigate the Shu-Osher problem, which involves intricate wave interactions. The presented method opens new avenues for integrating uncertainty quantification with established numerical schemes for hyperbolic conservation laws.

math.NA↗

arXiv2025

Consensus-based algorithms for stochastic optimization problems

We address an optimization problem where the cost function is the expectation of a random mapping. To tackle the problem two approaches based on the approximation of the objective function by consensus-based particle optimization methods on the search space are developed. The resulting methods are mathematically analyzed using a mean-field approximation and their connection is established. Several numerical experiments show the validity of the proposed algorithms and investigate their rates of convergence.

math.OC↗

arXiv2025

Solving Random Hyperbolic Conservation Laws Using Linear Programming

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows to compute the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.

math.NA↗

arXiv2025

A modified particle filter that reduces weight collapse

Particle filters are a widely used Monte Carlo based data assimilation technique that estimates the probability distribution of a system's state conditioned on observations through a collection of weights and particles. A known problem for particle filters is weight collapse, or degeneracy, where a single weight attains a value of one while all others are close to zero, thereby collapsing the estimated distribution. We address this issue by introducing a novel modification to the particle filter that is simple to implement and inspired by energy-based diversity measures. Our approach adjusts particle weights to minimize a two-body energy potential, promoting balanced weight distributions and mitigating collapse. We demonstrate the performance of this modified particle filter in a series of numerical experiments with linear and nonlinear dynamical models, where we compare with the classical particle filter and ensemble Kalman filters in the nonlinear case. We find that our new approach improves weight distributions compared to the classical particle filter and thereby improve state estimates.

stat.AP↗

arXiv2025

Turnpike property for hierarchical optimal control problems: from particle systems to hydrodynamic equations

This work is concerned with a hierarchical framework of optimal control problems connecting interacting particle systems, the mean field limit equations, and associated hydrodynamic models. By assuming the existence of solutions, we establish the exponential turnpike property for each level of the hierarchy, showing that optimal trajectories remain close to the associated steady states over long time horizons. The results demonstrate that the exponential turnpike behavior persists consistently across scales, providing a unified connection between microscopic, kinetic, and macroscopic optimal control frameworks.

math.OC↗

arXiv2025

Numerical Study of Random Kelvin-Helmholtz Instability

In this paper, we study random dissipative weak solutions of the compressible Euler equations in the Kelvin-Helmholtz (KH) instability. Motivated by the fact that weak entropy solutions are not unique and can be viewed as inviscid limits of Navier-Stokes flows, we take a statistical approach following ideas from turbulence theory. Our aim is to identify solution features that remain consistent across different realizations and mesh resolutions. For this purpose, we compute stable numerical solutions using a stochastic collocation method implemented with the help of a fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme and seventh-order central weighted essentially non-oscillatory (CWENO) interpolation in the random space. The obtained solutions are averaged over several embedded uniform grids, resulting in Cesáro averages, which are studied using stochastic tools. The analysis includes Reynolds stress and energy defects, probability density functions of averaged quantities, and reduced-order representations using proper orthogonal decomposition. The presented numerical experiments illustrate that random KH instabilities can be systematically described using statistical methods, averaging, and reduced-order modeling, providing a robust methodology for capturing the complex and chaotic dynamics of inviscid compressible flows.

math.NA↗

arXiv2025

Numerical Analysis of Stabilization for Random Hyperbolic Systems of Conservation Laws

This paper extends the deterministic Lyapunov-based stabilization framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite volume discretization method from [{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilization of the linear wave equations and linearized shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearized Saint-Venant system with source terms.

math.NA↗

arXiv2025

Control of Conservation Laws in the Nonlocal-to-Local Limit

We analyze a class of control problems where the initial datum acts as a control and the state is given by the entropy solution of (local) conservation laws by a nonlocal-to-local limiting strategy. In particular we characterize the limit up to subsequence of minimizers to nonlocal control problems as minimizer of the corresponding local ones. Moreover, we also prove an analogous result at a discrete level by means of a Eulerian-Lagrangian scheme.

math.OC↗

arXiv2025

Micro-Macro Decomposition of Particle Swarm Optimization Methods

Solving non-convex minimization problems using multi-particle metaheuristic derivative-free optimization methods is still an active area of research. Popular methods are Particle Swarm Optimization (PSO) methods, that iteratively update a population of particles according to dynamics inspired by social interactions between individuals. We present a modification to include constrained minimization problems using exact penalization. Additionally, we utilize the hierarchical structure of PSO to introduce a micro-macro decomposition of the algorithm. The probability density of particles is written as a convex combination of microscopic and macroscopic contributions, and both parts are propagated separately. The decomposition is dynamically updated based on heuristic considerations. Numerical examples compare the results obtained using the algorithm in the microscopic scale, in the macroscopic scale, and, using the new micro-macro decomposition.

math.OC↗

arXiv2025

Fast and robust consensus-based optimization via optimal feedback control

We propose a variant of consensus-based optimization (CBO) algorithms, controlled-CBO, which introduces a feedback control term to improve convergence towards global minimizers of non-convex functions in multiple dimensions. The feedback law is a gradient of a numerical approximation to the Hamilton-Jacobi-Bellman (HJB) equation, which serves as a proxy of the original objective function. Thus, the associated control signal furnishes gradient-like information to facilitate the identification of the global minimum without requiring derivative computation from the objective function itself. The proposed method exhibits significantly improved performance over standard CBO methods in numerical experiments, particularly in scenarios involving a limited number of particles, or where the initial particle ensemble is not well positioned with respect to the global minimum. At the same time, the modification keeps the algorithm amenable to theoretical analysis in the mean-field sense. The superior convergence rates are assessed experimentally.

math.OC↗

arXiv2025

Kinetic variable-sample methods for stochastic optimization problems

We discuss kinetic-based particle optimization methods and variable-sample strategies for problems where the cost function represents the expected value of a random mapping. Kinetic-based optimization methods rely on a consensus mechanism targeting the global minimizer, and they exploit tools of kinetic theory to establish a rigorous framework for proving convergence to that minimizer. Variable-sample strategies replace the expected value by an approximation at each iteration of the optimization algorithm. We combine these approaches and introduce a novel algorithm based on instantaneous collisions governed by a linear Boltzmann-type equation. After proving the convergence of the resulting kinetic method under appropriate parameter constraints, we establish a connection to a recently introduced consensus-based method for solving the random problem in a suitable scaling. Finally, we showcase its enhanced computational efficiency compared to the aforementioned algorithm and validate the consistency of the proposed modeling approaches through several numerical experiments.

math.OC↗

arXiv2025

High-Order Flux Splitting Schemes for the Euler Equations of Gas Dynamics

We develop high-order flux splitting schemes for the one- and two-dimensional Euler equations of gas dynamics. The proposed schemes are high-order extensions of the existing first-order flux splitting schemes introduced in [ E. F. Toro, M. E. Vázquez-Cendón, Comput. \& Fluids, 70 (2012), pp. 1--12], where the Euler equations of gas dynamics are split into two subsystems: the advection and pressure systems. In this paper, we formulate the TV splitting within the semi-discrete framework to extend it to higher orders of accuracy for the first time. The second-order extension is obtained by using piecewise linear interpolant to reconstruct the one-sided point values of the unknowns. The third- and fifth-order schemes are developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework, which is particularly effective in handling multidimensional problems and provides a more straightforward approach to constructing higher-order WENO schemes. These extensions significantly improve the resolution of discontinuities and the accuracy of numerical solutions, as demonstrated by a series of numerical experiments of both the one- and two-dimensional Euler equations of gas dynamics.

math.NA↗

arXiv2025

A multiscale Consensus-Based algorithm for multi-level optimization

A novel multiscale consensus-based optimization (CBO) algorithm for solving bi- and tri-level optimization problems is introduced. Existing CBO techniques are generalized by the proposed method through the employment of multiple interacting populations of particles, each of which is used to optimize one level of the problem. These particle populations are evolved through multiscale-in-time dynamics, which are formulated as a singularly perturbed system of stochastic differential equations. Theoretical convergence analysis for the multiscale CBO model to an averaged effective dynamics as the time-scale separation parameter approaches zero is provided. The resulting algorithm is presented for both bi-level and tri-level optimization problems. The effectiveness of the approach in tackling complex multi-level optimization tasks is demonstrated through numerical experiments on various benchmark functions. Additionally, it is shown that the proposed method performs well on min-max optimization problems, comparing favorably with existing CBO algorithms for saddle point problems.

math.OC↗

arXiv2025

Sparse Reconstruction of Multi-Dimensional Kinetic Distributions

In the present work, we propose a novel method for reconstruction of multi-dimensional kinetic distributions, based on their representation as a mixture of Dirac delta functions. The representation is found as a solution of an optimization problem. Different target functionals are considered, with a focus on sparsity-promoting regularization terms. The proposed algorithm guarantees non-negativity of the distribution by construction, and avoids an exponential dependence of the computational cost on the dimensionality of the problem. Numerical comparisons with other classical methods for reconstruction of kinetic distributions are provided for model problems, and the role of the different parameters governing the optimization problem is studied.

physics.comp-ph↗

arXiv2025

Sparse control in microscopic and mean-field leader-follower models

This work investigates the decay properties of Lyapunov functions in leader-follower systems seen as a sparse control framework. Starting with a microscopic representation, we establish conditions under which the total Lyapunov function, encompassing both leaders and followers, exhibits exponential decay. The analysis is extended to a hybrid setting combining a mean-field description for followers and a microscopic model for leaders. We identify sufficient conditions on control gain and interaction strengths that guarantee stabilization of the linear system towards a target state. The results highlight the influence of sparse control and interaction parameters in achieving coordinated behavior in multi-agent systems.

math.OC↗

arXiv2025

A numerical method for solving the generalized tangent vector of hyperbolic systems

This work is concerned with the computation of the first-order variation for one-dimensional hyperbolic partial differential equations. In the case of shock waves the main challenge is addressed by developing a numerical method to compute the evolution of the generalized tangent vector introduced by Bressan and Marson (1995). Our basic strategy is to combine the conservative numerical schemes and a novel expression of the interface conditions for the tangent vectors along the discontinuity. Based on this, we propose a simple numerical method to compute the tangent vectors for general hyperbolic systems. Numerical results are presented for Burgers' equation and a 2 x 2 hyperbolic system with two genuinely nonlinear fields.

math.NA↗

arXiv2025

Boundary Stabilization with restricted observability

Lyapunov functions are popularly used to investigate the stabilization problem of systems of hyperbolic conservation laws with boundary controls. In real life applications often not every boundary value can be observed. In this work, we show the stabilization under a restricted boundary observability. Thereby, we apply the boundary control directly on the observed (physical) variables. Using well-known stabilization results from the literature, we also discuss examples such as a density flow model or the Saint-Venant equations. This shows that a restricted observation can result in more restrictive control choices or can prevent the system from stabilizing.

math.OC↗