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Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

Margherita Guido, Daniel Kressner, Paolo Ricci

TL;DR

An acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints, is presented, showing that the time needed for solving the linear systems is significantly reduced.

Abstract

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

TL;DR

An acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints, is presented, showing that the time needed for solving the linear systems is significantly reduced.

Abstract

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.
Paper Structure (12 sections, 5 theorems, 41 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 12 sections, 5 theorems, 41 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm
  3. Proper Orthogonal Decomposition
  4. Randomized range finder
  5. Convergence Analysis
  6. Compressibility of the solution time history
  7. Quality of prediction without compression
  8. Proof of Theorem \ref{['theorem:main']}
  9. Optimality of the prediction with compression
  10. Numerical results: Test Case
  11. Numerical results: Plasma Simulations
  12. Conclusions

Key Result

Theorem 2

Under Assumption assume:analyticity, the $k$th largest singular value $\sigma_k$ of the history matrix $X(\boldsymbol{t})$ from def:X satisfies

Figures (4)

  • Figure 1: GMRES iterations per timestep when solving equation \ref{['eq:diff1']} with different initial guesses.
  • Figure 2: Computational time per timestep corresponding to Figure \ref{['fig:1_1']} and Figure \ref{['fig:1_2']}. The average speedup per iteration of the randomized method with respect to the baseline is a factor 9 for $\Delta t = 10^{-5}$ and a factor 10 for $\Delta t = 10^{-3}$.
  • Figure 3: GBS computational domain. The toroidal direction is along $\varphi$, the radial direction is along $R$, and the vertical direction is along $Z$. The domain consists of $N_{\varphi}$ rectangular poloidal planes, each discretized on a $N_{R} \times N_{Z}$ Cartesian grid.
  • Figure 4: Performance of the algorithm applied to the solution of Poisson equation in GBS simulations. The time for the RAND algorithm is on average approximately one fourth of the time for the baseline.

Theorems & Definitions (7)

  • Theorem 2: Kressner2011
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • proof
  • Corollary 6
  • proof