arXiv subjects

Boyan Lazarov

Publications and source records attributed to Boyan Lazarov.

7 recordsLinked to original sources

arXiv2026

Implicit Lagrangian Hydrodynamics with High-Order Finite Elements

We present an implicit time integration capability for high-order curvilinear finite element Lagrangian hydrodynamics. Starting from an existing explicit formulation, the implicit treatment builds directly on the original discretization and operator structure and does not alter the underlying spatial formulation or physics model. We demonstrate our implementation using the Laghos miniapp, which is built on the MFEM finite element library. To support gradient-based nonlinear solution methods, we compute Jacobian actions automatically using MFEM's $\partial$FEM interface together with Enzyme-based automatic differentiation, and apply the resulting Jacobian in a matrix-free or fully-assembled manner within a Newton-Krylov solver. To ensure robust and differentiable nonlinear solves in the presence of shocks, we introduce a smooth artificial viscosity treatment based on smooth approximations of non-differentiable pointwise operations. The differentiable artificial viscosity presently lacks a limiter to ensure high-order scaling away from shocks, but is sufficient for illustrating the benefits of implicit Lagrangian hydrodynamics. The behavior and performance of the implicit method are demonstrated on several standard benchmark problems. We verify high-order convergence on the smooth Taylor-Green vortex in the absence of artificial viscosity, show correct strong-shock behavior on the Sedov blast problem, and obtain significant improvements in accuracy-per-time-to-solution on the Triple Point problem where explicit stability constraints become increasingly severe for high-order discretizations.

physics.comp-ph↗

arXiv2026

Exponential Adaptive Smoothing and Importance Sampling for Optimization of the Conditional Value-at-Risk

We present a novel method for solving conditional value-at-risk (CVaR) optimization problems based on the dual representation of CVaR, which is defined as the worst-case expectation over a risk envelope. The method is based on the Bregman proximal point algorithm and alternates between stochastic primal and dual stages. Every (inner) primal stage involves a subproblem solved by sampling from a probability distribution updated at each dual stage (outer iteration). The likelihood ratio of the dual probability distributions relative to the distribution underlying the original problem converges to the risk identifier of the solution's CVaR. Thus, the dual distribution provides the algorithm with a built-in importance sampling mechanism that draws from the tail of the underlying distribution. Because only samples in the tail influence the CVaR, and samples outside the tail are drawn with decreasing probability, the algorithm delivers exceptional performance over other stochastic approximation methods. We prove the convergence of the algorithm for convex objective functions. Our numerical experiments target representative problems in financial mathematics and machine learning, focusing on portfolio optimization and support-vector machines, respectively.

math.OC↗

arXiv2025

PDE-Constrained High-Order Mesh Optimization

We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regularization. We adopt the Target-Matrix Optimization Paradigm to control geometric properties across the mesh, independent of the PDE of interest. To incorporate the accuracy of the PDE solution, we introduce error measures that control the finite element discretization error. The implicit dependence of these error measures on the mesh nodal positions is accurately captured by adjoint sensitivity analysis. Additionally, a convolution-based gradient regularization strategy is used to ensure stable and effective adaptation of high-order meshes. We demonstrate that the proposed framework can improve mesh quality and reduce the error by up to 10 times for the solution of Poisson and linear elasto-static problems. The approach is general with respect to the dimensionality, the order of the mesh, the types of mesh elements, and can be applied to any PDE that admits well-defined adjoint operators.

math.NA↗

arXiv2025

Scalable Analysis and Design Using Automatic Differentiation

This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to localize the application of AD at the integration point level by combining it with the so-called Finite Element Operator Decomposition. The proposed methods are computationally effective, scalable, automatic, and non-intrusive, making them ideal for existing serial and parallel solvers and complex multiphysics applications. The performance is demonstrated on large-scale steady-state non-linear scalar problems. The chosen testbed, the MFEM library, is free and open-source finite element discretization library with proven scalability to thousands of parallel processes and state-of-the-art high-order discretization techniques.

math.NA↗

arXiv2024

Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization

This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on applications in computational mechanics and design optimization. We leverage Matérn-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models and the topology of optimized designs. We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications. The flexibility and efficiency of SPDE-based GRF generation empower us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model geometric uncertainties of reconstructed submanifolds, such as the surfaces of cerebral aneurysms. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.

cs.CE↗

arXiv2024

High-performance finite elements with MFEM

The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability across leadership-class supercomputing facilities, including exascale supercomputers, as well as new capabilities and functionality, enabling a wider range of applications. Much of this work was undertaken as part of the Department of Energy's Exascale Computing Project (ECP) in collaboration with the Center for Efficient Exascale Discretizations (CEED).

cs.MS↗

arXiv2023

An Adaptive Sampling Augmented Lagrangian Method for Stochastic Optimization with Deterministic Constraints

The primary goal of this paper is to provide an efficient solution algorithm based on the augmented Lagrangian framework for optimization problems with a stochastic objective function and deterministic constraints. Our main contribution is combining the augmented Lagrangian framework with adaptive sampling, resulting in an efficient optimization methodology validated with practical examples. To achieve the presented efficiency, we consider inexact solutions for the augmented Lagrangian subproblems, and through an adaptive sampling mechanism, we control the variance in the gradient estimates. Furthermore, we analyze the theoretical performance of the proposed scheme by showing equivalence to a gradient descent algorithm on a Moreau envelope function, and we prove sublinear convergence for convex objectives and linear convergence for strongly convex objectives with affine equality constraints. The worst-case sample complexity of the resulting algorithm, for an arbitrary choice of penalty parameter in the augmented Lagrangian function, is $O(ε^{-3-δ})$, where $ε> 0$ is the expected error of the solution and $δ> 0$ is a user-defined parameter. If the penalty parameter is chosen to be $O(ε^{-1})$, we demonstrate that the result can be improved to $O(ε^{-2})$, which is competitive with the other methods employed in the literature. Moreover, if the objective function is strongly convex with affine equality constraints, we obtain $O(ε^{-1}\log(1/ε))$ complexity. Finally, we empirically verify the performance of our adaptive sampling augmented Lagrangian framework in machine learning optimization and engineering design problems, including topology optimization of a heat sink with environmental uncertainty.

math.OC↗