Jeffrey Willert

Short note: Leveraging Anderson Acceleration for improved convergence of iterative solutions to transport systems

In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computational results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.

Hybrid Deterministic/Monte Carlo Neutronics

In this paper we describe a hybrid deterministic/Monte Carlo algorithm for neutron transport simulation. The algorithm is based on nonlinear accelerators for source iteration, using Monte Carlo methods for the purely absorbing high-order problem and a Jacobian-free Newton--Krylov iteration for the low-order problem. We couple the Monte Carlo solution with the low-order problem using filtering to smooth the flux and current from the Monte Carlo solver and an analytic Jacobian-vector product to avoid numerical differentiation of the Monte Carlo results. We use a continuous energy deposition tally for the Monte Carlo simulation. We conclude the paper with numerical results which illustrate the effectiveness of the new algorithm.

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton-Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

NEWTON'S METHOD FOR MONTE CARLO-BASED RESIDUALS ∗

We analyze the behavior of inexact Newton methods for problems where the nonlinear residual, Jacobian, and Jacobian-vector products are the outputs of Monte Carlo simulations. We propose algorithms which account for the randomness in the iteration, develop theory for the behavior of these algorithms, and illustrate the results with an example from neutronics.

Residual Monte Carlo high-order solver for Moment-Based Accelerated Thermal Radiative Transfer equations

In this article we explore the possibility of replacing Standard Monte Carlo (SMC) transport sweeps within a Moment-Based Accelerated Thermal Radiative Transfer (TRT) algorithm with a Residual Monte Carlo (RMC) formulation. Previous Moment-Based Accelerated TRT implementations have encountered trouble when stochastic noise from SMC transport sweeps accumulates over several iterations and pollutes the low-order system. With RMC we hope to significantly lower the build-up of statistical error at a much lower cost. First, we display encouraging results for a zero-dimensional test problem. Then, we demonstrate that we can achieve a lower degree of error in two one-dimensional test problems by employing an RMC transport sweep with multiple orders of magnitude fewer particles per sweep. We find that by reformulating the high-order problem, we can compute more accurate solutions at a fraction of the cost.

Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

Abstract High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Furthermore, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

Using Anderson Acceleration to Accelerate the Convergence of Neutron Transport Calculations with Anisotropic Scattering

Abstract In two recent publications, it was demonstrated that the nonlinear diffusion acceleration (NDA) algorithm, a moment-based accelerator, could be modified to accelerate the solution to neutron transport calculations with anisotropic scattering. It was demonstrated, however, that as the scattering became less isotropic, the performance of the algorithm degraded. Furthermore, it has been shown that Anderson acceleration (AA) could be used to speed up neutron transport and plasma physics calculations. In this paper, we combine these ideas to demonstrate that AA can be used to remedy the degraded performance of NDA when scattering is anisotropic. We describe each of the methods in detail and demonstrate the results on a series of fixed-source calculations and a pair of k-eigenvalue calculations.

A Hybrid Deterministic/Monte Carlo Method for Solving the k-Eigenvalue Problem with a Comparison to Analog Monte Carlo Solutions

In this article we present a hybrid deterministic/Monte Carlo algorithm for computing the dominant eigenvalue/eigenvector pair for the neutron transport k-eigenvalue problem in multiple space dimensions. We begin by deriving the Nonlinear Diffusion Acceleration method (Knoll, Park, and Newman, 2011; Park, Knoll, and Newman, 2012) for the k-eigenvalue problem. We demonstrate that we can adapt the algorithm to utilize a Monte Carlo simulation in place of a deterministic transport sweep. We then show that the new hybrid method can be used to solve a two-group, two dimensional eigenvalue problem. The hybrid method is competitive with analog Monte Carlo in terms of number of particle flights required to compute the eigenvalue; however it produces a much less noisy eigenvector and fission source distribution. Furthermore, we show that we can reduce the error induced by the discretization of the low-order system by appropriate refinement of the mesh.

Scalable Hybrid Deterministic/Monte Carlo Neutronics Simulations in Two Space Dimensions

In this paper we discuss a parallel hybrid deterministic/Monte Carlo (MC) method for the solution of the neutron transport equation in two space dimensions. The algorithm uses an NDA formulation of the transport equation, with a MC solver for the high-order equation. The scalability arises from the concentration of work in the MC phase of the algorithm, while the overall run-time is a consequence of the deterministic phase.