[Paper Review] Nonlinear Geometric Optics Based Multiscale Stochastic Galerkin Methods for Highly Oscillatory Transport Equations with Random Inputs

Nicolas Crouseilles, Shi Jin+2 more|arXiv (Cornell University)|Apr 4, 2017

Probabilistic and Robust Engineering Design13 references3 citations

TL;DR

This paper develops a generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method for highly oscillatory transport equations with random inputs, leveraging nonlinear geometric optics (NGO) to achieve frequency-independent accuracy. By introducing a phase-based time variable that eliminates oscillations in random space, the method captures pointwise solution statistics without resolving high-frequency oscillations in mesh size, time step, or polynomial chaos degree.

ABSTRACT

We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method, developed in \cite{NGO} for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics {\em pointwisely} even though none of the numerical parameters resolve the high frequencies of the solution.

Motivation & Objective

• Address the challenge of efficiently computing uncertainty propagation in highly oscillatory transport equations arising in semiclassical quantum dynamics with random potential inputs.
• Overcome the computational infeasibility of standard gPC-SG methods when applied to oscillatory problems, which require resolution of small wave lengths.
• Develop a stochastic Galerkin method that maintains accuracy even when mesh size, time step, and polynomial chaos degree are independent of the oscillation frequency.
• Extend the nonlinear geometric optics (NGO) framework to stochastic settings, ensuring asymptotic preservation and pointwise solution accuracy.
• Validate the method on both scalar and surface-hopping models with random band gaps, demonstrating robustness and efficiency in uncertainty quantification.

Proposed method

• Adapt the nonlinear geometric optics (NGO) framework to stochastic problems by introducing a phase-based time variable that removes oscillations in the random space.
• Apply generalized polynomial chaos (gPC) approximation to represent random inputs, followed by stochastic Galerkin (SG) projection to derive a system of deterministic PDEs in the random-physical space.
• Decompose the solution into oscillatory and non-oscillatory components using Fourier analysis in the phase variable τ, enabling stable time integration via backward Euler in the Fourier domain.
• Use spectral collocation in the random variable space to compute derivatives of gPC coefficients, which enter the system matrices for the oscillatory part.
• Construct a time-implicit scheme for the oscillatory component using matrix inversion in the Fourier domain, ensuring stability regardless of ε.
• For the surface-hopping model, derive a gPC-SG-D scheme that handles random band gaps through matrix projections involving ψm(z)ψl(z) integrals with respect to the weight function π(z).

Experimental results

Research questions

• RQ1 Can the nonlinear geometric optics (NGO) method be extended to stochastic, highly oscillatory transport equations with random coefficients?
• RQ2 Does the introduction of a phase-based time variable eliminate frequency-dependent oscillations in the random space, enabling frequency-independent discretization?
• RQ3 Can a gPC-SG method be constructed that captures pointwise solution statistics without resolving the small wave length in mesh, time step, or polynomial chaos order?
• RQ4 How does the method perform on the semiclassical surface-hopping model with random band gaps compared to the scalar model?
• RQ5 What is the impact of random uncertainty on the regularity of higher-order derivatives in the random space, and how can it be mitigated?

Key findings

• The standard gPC-SG method for oscillatory problems requires a wave frequency-dependent gPC order, making it computationally prohibitive for high-frequency regimes.
• Introducing a phase-based time variable results in a non-oscillatory random space dependence, enabling the use of frequency-independent mesh size, time step, and polynomial chaos degree.
• The proposed NGO-based gPC-SG method achieves accurate pointwise solution statistics even when no numerical parameter resolves the high-frequency oscillations.
• Numerical examples confirm that the method captures the correct solution statistics for both scalar and surface-hopping models with random band gaps.
• The matrix inversion in the Fourier domain for the oscillatory part remains stable and invertible due to symmetric, real-eigenvalue structure, ensuring numerical robustness.
• The method is asymptotically preserving and maintains accuracy across a range of ε values, demonstrating its robustness in the highly oscillatory limit.