arXiv subjects

Caroline Lasser

Publications and source records attributed to Caroline Lasser.

At least 19 recordsLinked to original sources

arXiv2026

Residual-Based Time Discretization on Nonlinear Approximation Manifolds: Analysis and Gaussian Applications

We study time-discrete parametric approximations of evolution equations in Hilbert spaces based on residual minimization. The solution is represented by a parametrized ansatz belonging to a low-dimensional nonlinear manifold, and time stepping is performed by minimizing suitably defined residuals at each step. Two natural residual formulations are considered: discretization followed by parametrization of the evolution equation, and discretization of the Dirac--Frenkel variational principle governing the parameter dynamics. A unified error analysis is developed for both approaches within the family of $\zeta$-methods. The resulting bounds separate the effects of time discretization from those of residual minimization and yield first- and second-order convergence under Lipschitz, one-sided Lipschitz, and dissipativity assumptions. For the variational formulation, additional stability conditions involving the conditioning of the parametrization map arise naturally. The framework is applied to Gaussian approximation manifolds, for which residual norms and gradients admit explicit closed-form expressions when polynomial operators are involved. This enables efficient implementation without spatial discretization. Numerical experiments for time-dependent Schr\"odinger equations illustrate the theoretical convergence rates and the influence of residual accuracy on conservation properties.

math.NA↗

arXiv2026

Meshfree versus grid-based Schr\"odinger solvers for modeling the interactions between free-electron wave packets and light

The interaction of free-electron wave packets with electromagnetic fields provides a powerful route toward coherent electron control, enabling the generation of energy combs, momentum-state superpositions, and aberration-engineered electron beams. Existing theoretical descriptions, however, often rely on eikonal or no-recoil approximations. Here, we present a mesh-free numerical framework that directly solves the time-dependent single-particle Schr\"odinger equation for arbitrary electromagnetic potentials. Comparison with a benchmark mesh-based Schr\"odinger solver reveals excellent quantitative agreement. By eliminating the need for spatial meshing, our method offers an efficient and scalable route for simulating electron wave packet dynamics in complex time-dependent and static electromagnetic environments, while the simulation time is significantly improved by up to 800 times faster. These capabilities establish a versatile computational tool for quantum electron optics and free-electron-light interactions beyond eikonal approximations.

physics.optics↗

arXiv2026

Structure-Preserving Integration for Magnetic Gaussian Wave Packet Dynamics

We develop structure-preserving time integration schemes for Gaussian wave packet dynamics associated with the magnetic Schr\"odinger equation. The variational Dirac--Frenkel formulation yields a finite-dimensional Hamiltonian system for the wave packet parameters, where the presence of a magnetic vector potential leads to a non-separable structure and a modified symplectic geometry. By introducing kinetic momenta through a minimal substitution, we reformulate the averaged dynamics as a Poisson system that closely parallels the classical equations of charged particle motion. This representation enables the construction of Boris-type integrators adapted to the variational setting. In addition, we propose explicit high-order symplectic schemes based on splitting methods and partitioned Runge--Kutta integrators. The proposed methods conserve the quadratic invariants characterizing the Hagedorn parametrization, preserve linear and angular momentum under symmetry assumptions, and exhibit near-conservation of the averaged Hamiltonian over long time intervals. Rigorous error estimates are derived for both the wave packet parameters and observable quantities, with bounds uniform in the semiclassical parameter. Numerical experiments demonstrate the favorable long-time behavior and structure preservation of the integrators.

math.NA↗

arXiv2026

An equivalence of moment closure and nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow

We establish the equivalence between a classical moment closure and a nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow in the linearized Hookean spring chain setting. The variational formulation is based on the Dirac-Frankel principle applied to a Gaussian approximation manifold endowed with the Fisher-Rao information metric. We show that the invariance of this manifold under the linear configurational dynamics yields an exact evolution for the macroscopic conformation tensor, recovering the classical diffusive Oldroyd-B closure. While the equivalence only holds in the linearized setting, the associated variational framework provides an abstract error representation. Thus it can serve in future work as a starting point for the systematic construction of reduced approximation schemes for polymeric flows with nonlinear forcing laws.

math.SP↗

arXiv2025

Vibronic spectra at nonzero temperatures from Herman-Kluk coherence thermofield dynamics

We combine the semiclassical Herman--Kluk approximation with the coherence thermofield dynamics in order to evaluate vibrationally resolved electronic spectra at nonzero temperatures. In coherence thermofield dynamics, the dipole time correlation function is rewritten exactly as a wavepacket autocorrelation function, and the corresponding wavepacket is a solution to a zero-temperature time-dependent Schr\"odinger equation on an augmented configuration space of doubled dimension. We derive the Herman--Kluk representation for the thermofield wavepacket autocorrelation function and demonstrate how it can be computed from individual trajectories. To analyze this method, we compare spectra of Morse potentials of increasing anharmonicity evaluated at various temperatures with a numerically exact approach, with the Herman--Kluk coherence thermofield dynamics, and with the single-trajectory thawed Gaussian coherence thermofield dynamics. At low anharmonicity, both approximate methods yield accurate spectra. However, in a Morse potential with higher anharmonicity, the thawed Gaussian thermofield dynamics, based on the local harmonic approximation, fails to capture emerging hot bands, whereas the Herman--Kluk thermofield approach successfully reproduces them.

physics.chem-ph↗

arXiv2025

Time-integration of Gaussian variational approximation for the magnetic Schr\"odinger equation

In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schr\"odinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.

math.NA↗

arXiv2024

Sampling strategies for expectation values within the Herman--Kluk approximation

When computing quantum-mechanical observables, the ``curse of dimensionality'' limits the naive approach that uses the quantum-mechanical wavefunction. The semiclassical Herman--Kluk propagator mitigates this curse by employing a grid-free ansatz to evaluate the expectation values of these observables. Here, we investigate quadrature techniques for this high-dimensional and highly oscillatory propagator. In particular, we analyze Monte Carlo quadratures using three different initial sampling approaches. The first two, based either on the Husimi density or its square root, are independent of the observable whereas the third approach, which is new, incorporates the observable in the sampling to minimize the variance of the Monte Carlo integrand at the initial time. We prove sufficient conditions for the convergence of the Monte Carlo estimators and provide convergence error estimates. The analytical results are validated by numerical experiments in various dimensions on a harmonic oscillator and on a Henon-Heiles potential with an increasing degree of anharmonicity.

math.NA↗

arXiv2024

Regularized dynamical parametric approximation

This paper studies the numerical approximation of evolution equations by nonlinear parametrizations $u(t)=\Phi(\param(t))$ with time-dependent parameters $\param(t)$, which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and neural networks. In all these cases, the parametrization is typically irregular: the derivative $\Phi'(\param)$ can have arbitrarily small singular values and may have varying rank. We derive approximation results for a regularized approach in the time-continuous case as well as in time-discretized cases. For the latter, there is a nontrivial interplay between the regularization parameter and the time stepsize, depending also on the defect size and local bounds of the second derivative of the parametrization map $\Phi$. When this is appropriately taken into account, the approach can be successfully applied in irregular situations, even though it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm. The paper also includes two theoretical case studies: regularized parametric steepest descent for optimization and regularized parametric Strang splitting for the time-dependent Schr\"odinger equation. Numerical experiments with sums of Gaussians for approximating quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results.

math.NA↗

arXiv2024

FSSH-2: Fewest Switches Surface Hopping with robust switching probability

This study introduces the FSSH-2 scheme, a redefined and numerically stable adiabatic Fewest Switches Surface Hopping (FSSH) method for mixed quantum-classical dynamics. It reformulates the standard FSSH hopping probability without non-adiabatic coupling vectors and allows for numerical time integration with larger step sizes. The advantages of FSSH-2 are demonstrated by numerical experiments for five different model systems in one and two spatial dimensions with up to three electronic states.

physics.comp-ph↗

arXiv2023

Variational Gaussian approximation for the magnetic Schrödinger equation

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove $L^2$-error bounds and observable error bounds for the approximating Gaussian wave packet.

math.NA↗

arXiv2023

The Gaussian Wave Packet Transform via Quadrature Rules

We analyse the Gaussian wave packet transform. Based on the Fourier inversion formula and a partition of unity, which is formed by a collection of Gaussian basis functions, a new representation of square-integrable functions is presented. Including a rigorous error analysis, the variants of the wave packet transform are then derived by a discretisation of the Fourier integral via different quadrature rules. Based on Gauss--Hermite quadrature, we introduce a new representation of Gaussian wave packets in which the number of basis functions is significantly reduced. Numerical experiments in 1D illustrate the theoretical results.

math.NA↗

arXiv2023

Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schr\"odinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schr\"odinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter.

math.AP↗

arXiv2023

No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schrödinger equation

Linear combinations of complex gaussian functions, where the linear and nonlinear parameters are allowed to vary, are shown to provide an extremely flexible and effective approach for solving the time-dependent Schrödinger equation in one spatial dimension. The use of flexible basis sets has been proven notoriously hard within the systematics of the Dirac--Frenkel variational principle. In this work we present an alternative time-propagation scheme that de-emphasizes optimal parameter evolution but directly targets residual minimization via the method of Rothe's method, also called the method of vertical time layers. We test the scheme using a simple model system mimicking an atom subjected to an extreme laser pulse. Such a pulse produces complicated ionization dynamics of the system. The scheme is shown to perform very well on this model and notably does not rely on a computational grid. Only a handful of gaussian functions are needed to achieve an accuracy on par with a high-resolution, grid-based solver. This paves the way for accurate and affordable solution of the time-dependent Schrödinger equation for atoms and molecules within and beyond the Born--Oppenheimer approximation.

quant-ph↗

arXiv2022

Sampling strategies for the Herman-Kluk propagator of the wavefunction

When the semiclassical Herman-Kluk propagator is used for evaluating quantum-mechanical observables or time-correlation functions, the initial conditions for the guiding trajectories are typically sampled from the Husimi density. Here, we employ this propagator to evolve the wavefunction itself. We investigate two grid-free strategies for the initial sampling of the Herman-Kluk propagator applied to the wavefunction and validate the resulting time-dependent wavefunctions evolved in harmonic and anharmonic potentials. In particular, we consider Monte Carlo quadratures based either on the initial Husimi density or on its square root as possible and most natural sampling densities. We prove analytical convergence error estimates and validate them with numerical experiments on the harmonic oscillator and on a series of Morse potentials with increasing anharmonicity. In all cases, sampling from the square root of Husimi density leads to faster convergence of the wavefunction.

quant-ph↗

arXiv2022

Laser-induced dynamic alignment of the HD molecule without the Born-Oppenheimer approximation

Laser-induced molecular alignment is well understood within the framework of the Born-Oppenheimer (BO) approximation Without the BO approximation, however, the concept of molecular structure is lost, making alignment hard to define precisely. In this work, we demonstrate the emergence of alignment from the first-ever non-BO quantum dynamics simulations, using the HD molecule exposed to ultrashort laser pulses as a few-body test case We extract the degree of alignment from the non-BO wave function by means of an operator expressed in terms of pseudo-proton coordinates that mimics the BO-based definition of alignment The only essential approximation, in addition to the semiclassical electric-dipole approximation for the matter-field interaction, is the choice of time-independent explicitly correlated Gaussian basis functions. We use a variational, electric-field-dependent basis-set construction procedure, which allows us to keep the basis-set dimension low whilst capturing the main effects of electric polarization on the nuclear and electronic degrees of freedom. The basis-set construction procedure is validated by comparing with virtually exact grid-based simulations for two one-dimensional model systems: laser-driven electron dynamics in a soft attractive Coulomb potential and nuclear rovibrational dynamics in a Morse potential.

physics.chem-ph↗

arXiv2022

An Error Bound for the Time-Sliced Thawed Gaussian Propagation Method

We study the time-sliced thawed Gaussian propagation method, which was recently proposed for solving the time-dependent Schrödinger equation. We introduce a triplet of quadrature-based analysis, synthesis and re-initialization operators to give a rigorous mathematical formulation of the method. Further, we derive combined error bounds for the discretization of the wave packet transform and the time-propagation of the thawed Gaussian basis functions. Numerical experiments in 1D illustrate the theoretical results.

math.NA↗

arXiv2022

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{ö}dinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.

math.AP↗

arXiv2021

Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work [I. Burghardt, R. Carles, C. Fermanian Kammerer, B. Lasorne, C. Lasser, J. Phys. A. 54, 414002 (2021)]. Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The timedependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.

physics.chem-ph↗