arXiv subjects

Michal Béreš

Publications and source records attributed to Michal Béreš.

5 recordsLinked to original sources

arXiv2026

Delayed acceptance sampling with Hamiltonian proposal subchains for random field materials inference

This paper focuses on accelerating Markov chain Monte Carlo sampling in Bayesian inverse problems in which forward model evaluations dominate the computational cost. It builds on several established ingredients previously used in related scenarios: delayed acceptance, neural network surrogate models, Hamiltonian proposals, and proposal subchains. The main framework is the delayed-acceptance Metropolis-Hastings algorithm of Christen and Fox (2005). The first-stage proposal distribution is constructed from a subchain of Hamiltonian trajectories targeting the surrogate posterior. For each fixed surrogate model, the Hamiltonian subchain and delayed-acceptance correction define a kernel invariant with respect to the exact posterior. In the present work, the surrogate is updated only during a burn-in phase, after which the production run uses a fixed surrogate model. The sampling framework is implemented in Python using parallel processes. Several chains are generated in parallel and share a single surrogate model trained during burn-in on all collected data. The forward model is treated as a black box; therefore, the application area is broad. However, the main motivation is efficient solution of geotechnical inverse problems with material properties represented by Gaussian random fields. In this study, the sampling framework is applied to a geotechnical inverse problem in which hydraulic conductivity and porosity are modeled as non-stationary Gaussian random fields approximated using truncated Karhunen-Loeve expansions. Based on a precomputation, the truncation dimensions are chosen separately for hydraulic conductivity and porosity. The forward model outputs are pore pressure values at control points and selected observation times. These are compared with in situ pore pressure measurements collected over one year during the Tunnel Sealing Experiment in an underground laboratory in Canada.

stat.CO↗

arXiv2025

Legendre Polynomials and Their Use for Karhunen-Loève Expansion

This paper makes two main contributions. First, we present a pedagogical review of the derivation of the three-term recurrence relation for Legendre polynomials, without relying on the classical Legendre differential equation, Rodrigues' formula, or generating functions. This exposition is designed to be accessible to undergraduate students. Second, we develop a computational framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The framework leverages Legendre polynomials and their associated Gaussian quadrature, and it remains efficient even in higher spatial dimensions. A covariance kernel is first approximated by a non-negative mixture of squared-exponentials, obtained via a Newton-optimized fit with a theoretically informed initialization. The resulting separable kernel enables a Legendre-Galerkin discretization in the form of a Kronecker product over single dimensions, with submatrices that exhibit even/odd parity structure. For assembly, we introduce a Duffy-type transformation followed by quadrature. These structural properties significantly reduce both memory usage and arithmetic cost compared to naive approaches. All algorithms and numerical experiments are provided in an open-source repository that reproduces every figure and table in this work.

cs.CE↗

arXiv2025

Convex optimization problems inspired by geotechnical stability analysis

This paper is motivated by the limit load, limit analysis and shear strength reduction methods, which are commonly employed in geotechnical stability analysis or similar applications. The aim is to make these methods more approachable by introducing a unified framework based on abstract convex optimization and its parametric studies. We establish suitable assumptions on the abstract problems that capture the selected features of these methods and facilitate rigorous theoretical investigation. Further, we propose continuation techniques tailored to the resulting parametric problem formulations and show that the developed abstract framework could also be useful outside the domain of geotechnical stability analysis. The main results are illustrated with analytical and numerical examples. The numerical example deals with a 3D slope stability problem.

math.NA↗

arXiv2024

Minimization of Nonlinear Energies in Python Using FEM and Automatic Differentiation Tools

This contribution examines the capabilities of the Python ecosystem to solve nonlinear energy minimization problems, with a particular focus on transitioning from traditional MATLAB methods to Python's advanced computational tools, such as automatic differentiation. We demonstrate Python's streamlined approach to minimizing nonlinear energies by analyzing three problem benchmarks - the p-Laplacian, the Ginzburg-Landau model, and the Neo-Hookean hyperelasticity. This approach merely requires the provision of the energy functional itself, making it a simple and efficient way to solve this category of problems. The results show that the implementation is about ten times faster than the MATLAB implementation for large-scale problems. Our findings highlight Python's efficiency and ease of use in scientific computing, establishing it as a preferable choice for implementing sophisticated mathematical models and accelerating the development of numerical simulations.

cs.MS↗

arXiv2024

On the parallel solution of hydro-mechanical problems with fracture networks and contact conditions

The paper presents a numerical method for simulating flow and mechanics in fractured rock. The governing equations that couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the contact conditions prevent fractures from self-penetration. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of dualization and an optimal quadratic programming algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for tunnel excavation with hundreds of fractures in 3D. The paper's novelty consists in a combination of three crucial ingredients: (i) application of discrete fracture-matrix approach to poroelasticity, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems, and (iii) efficient solution of its mechanical quadratic programming part with a large number of fractures in mutual contact by means of own solvers implemented into an in-house software library.

math.NA↗