arXiv subjects

Anders Szepessy

Publications and source records attributed to Anders Szepessy.

18 recordsLinked to original sources

arXiv2026

Pontryagin-Based Solver with Smoothed Hamiltonian, Adaptive $Δt$, and PA-Bundle Refinement

We present a Pontryagin-based numerical solver for deterministic optimal control problems in Bolza form. The solver regularizes the generally nonsmooth Hamiltonian using a log-sum-exp smoothing of a concave piecewise-affine bundle surrogate, yielding a smoothed Hamiltonian $H_δ$ that is $C^\infty$ and concave in the costate and $C^{1,1}$ in the state. The resulting two-point boundary value problem is discretized by a symplectic Euler scheme and solved by damped Newton iteration on the full-space nonlinear system in the discrete state and costate variables. A unified adaptive outer loop jointly controls the time-step distribution $Δt$, the number of planes in the bundle surrogate, and the smoothing parameter $δ$. The time-adaptation strategy follows the error-density framework of Karlsson, Larsson, Sandberg, and Tempone (2015). Related approaches include Pontryagin shooting, Hamilton--Jacobi--Bellman methods, occupation-measure relaxations, max-plus approximations, and direct collocation. Numerical examples, including nonsmooth, singular, hypersensitive, and quadratic-programming-oracle benchmarks, demonstrate the roles of time refinement, bundle enrichment, and smoothing continuation. A linear--quadratic regulator with a closed-form Riccati solution is used as a calibration check.

math.OC↗

arXiv2026

Convergence for adaptive resampling of random Fourier features

The machine learning random Fourier feature method for data in high dimension is computationally and theoretically attractive since the optimization is based on a convex standard least squares problem and independent sampling of Fourier frequencies. The challenge is to sample the Fourier frequencies well. This work proves convergence of a data adaptive method based on resampling the frequencies asymptotically optimally, as the number of nodes and amount of data tend to infinity. Numerical results based on resampling and adaptive random walk steps together with approximations of the least squares problem by conjugate gradient iterations confirm the analysis for regression and classification problems.

math.NA↗

arXiv2024

Convergence rates for random feature neural network approximation in molecular dynamics

Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error $\mathcal{O}\big((K^{-1}+J^{-1/2})^{\frac{1}{2}}\big)$, for networks with $K$ nodes using $J$ data points, provided the Hessians of the potential and the observables are bounded. The loss function is based on the least squares error of the potential and regularizations, with the data points sampled from the Gibbs density. The proof uses an elementary new derivation of the generalization error for random feature networks that does not apply the Rademacher or related complexities.

math.NA↗

arXiv2023

Path integral molecular dynamics approximations of quantum canonical observables

Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components based on Brownian bridges connecting permuted electron coordinates. The computational work for $n$ electrons is $\mathcal O(n^3)$, which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a computational error indicator. It remains to rigorously explain the surprisingly high accuracy.

math.NA↗

arXiv2023

Canonical mean-field molecular dynamics derived from quantum mechanics

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be $\mathcal O(M^{-1})$, provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and $M$ is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain $\mathcal O(M^{-1})$ accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace $h:={\rm Tr}(H e^{-βH})/{\rm Tr}(e^{-βH})$ with respect to the electron degrees of freedom and $H$ is the Weyl symbol corresponding to a quantum many body Hamiltonian $\widehat{H}$. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy $\mathcal O (M^{-1}+ tε^2)$, for correlation time $t$ where $ε^2$ is related to the variance of mean value approximation $h$. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.

math.NA↗

arXiv2021

Smaller generalization error derived for a deep residual neural network compared to shallow networks

Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \mathrm{Re}\sum_{k=1}^K\bar b_{\ell k}e^{\mathrm{i}ω_{\ell k}\bar z_\ell}+ \mathrm{Re}\sum_{k=1}^K\bar c_{\ell k}e^{\mathrm{i}ω'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(ω_{\ell k},ω'_{\ell k})$ of the random Fourier features $e^{\mathrm{i}ω_{\ell k}\bar z_\ell}$ and $e^{\mathrm{i}ω'_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(KL)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $KL$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network. Promising performance of the proposed new algorithm is demonstrated in computational experiments.

math.NA↗

arXiv2020

Adaptive random Fourier features with Metropolis sampling

The supervised learning problem to determine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hatβ_k e^{\mathrm{i}ω_k\cdot x}$ with one hidden layer is studied as a random Fourier features algorithm. The Fourier features, i.e., the frequencies $ω_k\in\mathbb{R}^d$, are sampled using an adaptive Metropolis sampler. The Metropolis test accepts proposal frequencies $ω_k'$, having corresponding amplitudes $\hatβ_k'$, with the probability $\min\big\{1, (|\hatβ_k'|/|\hatβ_k|)^γ\big\}$, for a certain positive parameter $γ$, determined by minimizing the approximation error for given computational work. This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hatβ_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods. Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is tested both on synthetic data and a real-world high-dimensional benchmark.

math.NA↗

arXiv2019

Classical Langevin dynamics derived from quantum mechanics

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

math-ph↗

arXiv2019

Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics

It is known that ab initio molecular dynamics based on the electron ground state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature. The proof uses the semiclassical Weyl law to show that canonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The result covers observables that depend on time-correlations. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's law shows that the error estimate holds for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position and the observables are in the diagonal form with respect to the electron eigenstates.

math-ph↗

arXiv2019

The classical limit of quantum observables in the conservation laws of fluid dynamics

In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.

math-ph↗

arXiv2016

Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data

We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.

math.NA↗

arXiv2015

Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces

The difference of the values of observables for the time-independent Schroedinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics and stability analysis of a perturbed eigenvalue problem.

math.NA↗

arXiv2014

An adaptive mass algorithm for Car-Parrinello and Ehrenfest ab initio molecular dynamics

Ehrenfest and Car-Parrinello molecular dynamics are computational alternatives to approximate Born-Oppenheimer molecular dynamics without solving the electron eigenvalue problem at each time-step. A non-trivial issue is to choose the artificial electron mass parameter appearing in the Car-Parrinello method to achieve both good accuracy and high computational efficiency. In this paper, we propose an algorithm, motivated by the Landau-Zener probability, to systematically choose an artificial mass dynamically, which makes the Car-Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data. Numerical experiments for simple model problems show that the time-dependent adaptive artificial mass parameter improves the efficiency of the Car-Parrinello and Ehrenfest molecular dynamics.

physics.chem-ph↗

arXiv2014

An a posteriori error estimate for Symplectic Euler approximation of optimal control problems

This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian system connected with the optimal control problem. The error representation has a leading order term consisting of an error density that is computable from Symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading error density part in the error representation. With the error representation, it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. The performance is illustrated by numerical tests.

math.OC↗

arXiv2012

Monte Carlo Euler approximations of HJM term structure financial models

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

math.NA↗

arXiv2011

How accurate is molecular dynamics?

Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding Hamilton-Jacobi equation, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy microcanonical ensembles.

math-ph↗

arXiv2011

Langevin molecular dynamics derived from Ehrenfest dynamics

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac-Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio, $M$, of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy $o(M^{-1/2})$ on bounded time intervals and by $o(1)$ on unbounded time intervals, which makes the small $\mathcal{O}(M^{-1/2})$ friction and $o(M^{-1/2})$ diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperture, derived by a stability and consistency argument: starting with any equilibrium measure of the Ehrenfest Hamiltonian system, the initial electron distribution is sampled from the equilibrium measure conditioned on the nuclei positions, which after long time leads to the nuclei positions in a Gibbs distribution (i.e. asymptotic stability); by consistency the original equilibrium measure is then a Gibbs measure.The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation-dissipation relation.

math-ph↗

arXiv2010

Stochastic and deterministic molecular dynamics derived from the time-independent Schrödinger equation

Ehrenfest, Born-Oppenheimer, Langevin and Smoluchowski dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system avoiding caustics, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding Hamilton-Jacobi equation, bypasses the usual separation of nuclei and electron wave functions, includes crossing electron eigenvalues, and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems, stochastic electron equilibrium states and numerical simulation in molecular dynamics modeling.

math-ph↗