An adaptive dynamical low-rank optimizer for solving kinetic parameter identification inverse problems

TL;DR

This paper introduces an adaptive dynamical low-rank scheme for efficiently solving inverse problems in kinetic equations, significantly reducing computational and memory costs while maintaining accuracy.

Contribution

It develops a novel DLRA-based method with adaptive basis updates and gradient refinement for kinetic parameter identification.

Findings

Reduces computational cost compared to full solvers

Maintains high accuracy in reconstructing scattering parameters

Demonstrates efficiency across various initial conditions

Abstract

The numerical solution of parameter identification inverse problems for kinetic equations can exhibit high computational and memory costs. In this paper, we propose a dynamical low-rank scheme for the reconstruction of the scattering parameter in the radiative transfer equation from a number of macroscopic time-independent measurements. We first work through the PDE constrained optimization procedure in a continuous setting and derive the adjoint equations using a Lagrangian reformulation. For the scattering coefficient, a periodic B-spline approximation is introduced and a gradient descent step for updating its coefficients is formulated. After the discretization, a dynamical low-rank approximation (DLRA) is applied. We make use of the rank-adaptive basis update & Galerkin integrator and a line search approach for the adaptive refinement of the gradient descent step size and the DLRA tolerance. We show that the proposed scheme significantly reduces both memory and computational cost. Numerical results computed with different initial conditions validate the accuracy and efficiency of the proposed DLRA scheme compared to solutions computed with a full solver.