arXiv subjects

Pavel Bochev

Publications and source records attributed to Pavel Bochev.

At least 19 recordsLinked to original sources

arXiv2026

Non-intrusive data-driven model order reduction for circuits based on Hammerstein architectures

We demonstrate that system identification techniques can provide a basis for effective, non-intrusive model order reduction (MOR) for common circuits that are key building blocks in microelectronics. Our approach is motivated by the practical operation of these circuits and utilizes a canonical Hammerstein architecture. To demonstrate the approach we develop parsimonious Hammerstein models for a nonlinear CMOS differential amplifier and an operational amplifier circuit. We train these models on a combination of direct current (DC) and transient Spice circuit simulation data using a novel sequential strategy to identify their static nonlinear and linear dynamical parts. Simulation results show that the Hammerstein model is an effective surrogate for for these types of circuits that accurately and efficiently reproduces their behavior over a wide range of operating points and input frequencies.

eess.SY↗

arXiv2025

Dynamic flux surrogate-based partitioned methods for interface problems

Loosely coupled partitioned methods for multiphysics problems treat each subproblem as a separate entity and advance them independently in time. In so doing these methods enable code reuse, increase concurrency and provide a convenient framework for plug-and-play multiphysics simulations. However, mathematically loosely coupled schemes are equivalent to a single step of an iterative solution method, which can compromise their accuracy and stability. We present a new data-driven partitioned method for coupled parametric PDEs that can improve upon the accuracy of traditional loosely coupled methods without incurring a performance penalty. To that end, we replace conventional field transfers across the interface by a surrogate for the dynamics of the interface flux exchanged between the subdomains. To develop this surrogate we apply dynamic mode decomposition to a non-standard staggered-in-time state, comprising the interface flux and small solution patches near the interface. The new approach shifts the main computational burden to an offline training phase, whereas application of the surrogate in the online phase amounts to a single matrix-vector multiplication. We provide stability analysis of the surrogate-based partitioned scheme and include numerical results that demonstrate its potential.

cs.CE↗

arXiv2025

An optimization-based coupling of reduced order models with efficient reduced adjoint basis generation approach

Optimization-based coupling (OBC) is an attractive alternative to traditional Lagrange multiplier approaches in multiple modeling and simulation contexts. However, application of OBC to time-dependent problems has been hindered by the computational cost of finding the stationary points of the associated Lagrangian, which requires primal and adjoint solves. This issue can be mitigated by using OBC in conjunction with computationally efficient reduced order models (ROM). To demonstrate the potential of this combination, in this paper we develop an optimization-based ROM-ROM coupling for a transient advection-diffusion transmission problem. We pursue the ``optimize-then-reduce'' path towards solving the minimization problem at each timestep and solve reduced-space adjoint system of equations, where the main challenge in this formulation is the generation of adjoint snapshots and reduced bases for the adjoint systems required by the optimizer. One of the main contributions of the paper is a new technique for efficient adjoint snapshot collection for gradient-based optimizers in the context of optimization-based ROM-ROM couplings. We present numerical studies demonstrating the accuracy of the approach along with comparison between various approaches for selecting a reduced order basis for the adjoint systems, including decay of snapshot energy, average iteration counts, and timings.

cs.CE↗

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↗

arXiv2023

Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases

This paper formulates, analyzes, and demonstrates numerically a method for the partitioned solution of coupled interface problems involving combinations of projection-based reduced order models (ROM) and/or full order methods (FOMs). The method builds on the partitioned scheme developed in [1], which starts from a well-posed formulation of the coupled interface problem and uses its dual Schur complement to obtain an approximation of the interface flux. Explicit time integration of this problem decouples its subdomain equations and enables their independent solution on each subdomain. Extension of this partitioned scheme to coupled ROM-ROM or ROM-FOM problems required formulations with non-singular Schur complements. To obtain these problems, we project a well-posed coupled FOM-FOM problem onto a composite reduced basis comprising separate sets of basis vectors for the interface and interior variables, and use the interface reduced basis as a Lagrange multiplier. Our analysis confirms that the resulting coupled ROM-ROM and ROM-FOM problems have provably non-singular Schur complements, independent of the mesh size and the reduced basis size. In the ROM-FOM case, analysis shows that one can also use the interface FOM space as a Lagrange multiplier. We illustrate the theoretical and computational properties of the partitioned scheme through reproductive and predictive tests for a model advection-diffusion transmission problem.

math.NA↗

arXiv2022

A Novel Partitioned Approach for Reduced Order Model -- Finite Element Model (ROM-FEM) and ROM-ROM Coupling

Partitioned methods allow one to build a simulation capability for coupled problems by reusing existing single-component codes. In so doing, partitioned methods can shorten code development and validation times for multiphysics and multiscale applications. In this work, we consider a scenario in which one or more of the "codes" being coupled are projection-based reduced order models (ROMs), introduced to lower the computational cost associated with a particular component. We simulate this scenario by considering a model interface problem that is discretized independently on two non-overlapping subdomains. We then formulate a partitioned scheme for this problem that allows the coupling between a ROM "code" for one of the subdomains with a finite element model (FEM) or ROM "code" for the other subdomain. The ROM "codes" are constructed by performing proper orthogonal decomposition (POD) on a snapshot ensemble to obtain a low-dimensional reduced order basis, followed by a Galerkin projection onto this basis. The ROM and/or FEM "codes" on each subdomain are then coupled using a Lagrange multiplier representing the interface flux. To partition the resulting monolithic problem, we first eliminate the flux through a dual Schur complement. Application of an explicit time integration scheme to the transformed monolithic problem decouples the subdomain equations, allowing their independent solution for the next time step. We show numerical results that demonstrate the proposed method's efficacy in achieving both ROM-FEM and ROM-ROM coupling.

math.NA↗

arXiv2021

An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions

We develop and analyze an optimization-based method for the coupling of a static peri-dynamic (PD) model and a static classical elasticity model. The approach formulates the coupling as a control problem in which the states are the solutions of the PD and classical equations, the objective is to minimize their mismatch on an overlap of the PD and classical domains, and the controls are virtual volume constraints and boundary conditions applied at the local-nonlocal interface. Our numerical tests performed on three-dimensional geometries illustrate the consistency and accuracy of our method, its numerical convergence, and its applicability to realistic engineering geometries. We demonstrate the coupling strategy as a means to reduce computational expense by confining the nonlocal model to a subdomain of interest, and as a means to transmit local (e.g., traction) boundary conditions applied at a surface to a nonlocal model in the bulk of the domain.

math.NA↗

arXiv2020

Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method

We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. We present the method in the context of Local-to-Nonlocal diffusion coupling. Numerical examples illustrate the theoretical properties of the approach.

math.AP↗

arXiv2020

Learning Compact Physics-Aware Delayed Photocurrent Models Using Dynamic Mode Decomposition

Radiation-induced photocurrent in semiconductor devices can be simulated using complex physics-based models, which are accurate, but computationally expensive. This presents a challenge for implementing device characteristics in high-level circuit simulations where it is computationally infeasible to evaluate detailed models for multiple individual circuit elements. In this work we demonstrate a procedure for learning compact delayed photocurrent models that are efficient enough to implement in large-scale circuit simulations, but remain faithful to the underlying physics. Our approach utilizes Dynamic Mode Decomposition (DMD), a system identification technique for learning reduced order discrete-time dynamical systems from time series data based on singular value decomposition. To obtain physics-aware device models, we simulate the excess carrier density induced by radiation pulses by solving numerically the Ambipolar Diffusion Equation, then use the simulated internal state as training data for the DMD algorithm. Our results show that the significantly reduced order delayed photocurrent models obtained via this method accurately approximate the dynamics of the internal excess carrier density -- which can be used to calculate the induced current at the device boundaries -- while remaining compact enough to incorporate into larger circuit simulations.

physics.comp-ph↗

arXiv2020

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

math.NA↗

arXiv2020

An energy-based coupling approach to nonlocal interface problems

Nonlocal models provide accurate representations of physical phenomena ranging from fracture mechanics to complex subsurface flows, where traditional partial differential equations fail to capture effects caused by long-range forces at the microscale and mesoscale. However, the application of nonlocal models to problems involving interfaces such as multimaterial simulations and fluid-structure interaction, is hampered by the lack of a rigorous nonlocal interface theory needed to support numerical developments. In this paper, we use an energy-based approach to develop a mathematically rigorous nonlocal interface theory which provides a physically consistent extension of the classical perfect interface PDE formulation. Numerical examples validate the proposed framework and demonstrate the scope of our theory.

math.AP↗

arXiv2019

A conservative, consistent, and scalable meshfree mimetic method

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.

math.NA↗

arXiv2018

An Optimally Convergent Coupling Approach for Interface Problems Approximated with Higher-Order Finite Elements

In this paper, we present a new numerical method for determining the numerical solution of interface problems to optimal accuracy with respect to the polynomial order of the Lagrangian finite element space on polytopial meshes. We introduce the notion of a virtual interface, and on this virtual interface we enforce that "extended" interface conditions are satisfied in the sense of a Dirichlet--Neumann coupling. The virtual interface framework serves to bypass geometric variational crimes incurred by the classical finite element method. Further, this approach does not require that the geometric interfaces are spatially matching. Our analysis indicates that this approach is well--posed and optimally convergent in $H^1$. Numerical experiments indicate that optimal $H^1$ and $L^2$ convergence is achieved.

math.NA↗

arXiv2018

Optimally accurate higher-order finite element methods on polytopial approximations of domains with smooth boundaries

Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on such meshes. On the other hand, the simplicity of affine meshes makes them a desirable modeling tool in many applications. In this paper, we develop and analyze higher-order accurate finite element methods that remain stable and optimally accurate on polytopial approximations of domains with smooth boundaries. This is achieved by constraining a judiciously chosen extension of the finite element solution on the polytopial domain to weakly match the prescribed boundary condition on the true geometric boundary. We provide numerical examples that highlight key properties of the new method and that illustrate the optimal $H^1$ and $L^2$-norm convergence rates.

math.NA↗

arXiv2017

A Coupling Approach for Linear Elasticity Problems with Spatially Noncoincident Interfaces

We present a new formulation based on the classical Dirichlet-Neumann formulation for interface coupling problems in linearized elasticity. By using Taylor series expansions, we derive a new set of interface conditions that allow our formulation to pass the linear consistency test. In addition, we propose an iterative method to determine the solution of our formulation. We demonstrate in our numerical results that we may achieve the desired piecewise linear finite element error bounds for both nonoverlapping domain decomposition problems as well as for interface coupling problems where the Lamé parameters of the structures differ.

math.NA↗

arXiv2016

A high-order staggered meshless method for elliptic problems

We present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points $\bm{x}_i$. This graph defines a local primal-dual grid complex with a \emph{virtual} dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a Generalized Moving Least Squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$ and $H^1$ norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.

math.NA↗

arXiv2014

Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Near the defect core the method employs a potential-based atomistic model, which enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.

math.NA↗

arXiv2013

Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

Atomistic-to-Continuum (AtC) coupling methods are a novel means of computing the properties of a discrete crystal structure, such as those containing defects, that combine the accuracy of an atomistic (fully discrete) model with the efficiency of a continuum model. In this note we extend the optimization-based AtC, formulated in arXiv:1304.4976 for linear, one-dimensional problems to multi-dimensional settings and arbitrary interatomic potentials. We conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.

math.NA↗