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A Mixed Tree-Cotree Gauge for the Reduced Basis Approximation of Maxwell's Eigenvalue Problem

Anna Ziegler, Sebastian Schöps

Abstract

Model order reduction methods are a powerful tool to drastically reduce the computational effort of problems which need to be evaluated repeatedly, i.e., when computing the same system for various parameter values. When applying a reduced basis approximation algorithm to the Maxwell eigenvalue problem, we encounter spurious solutions in the reduced system which hence need to be removed during the basis construction. In this paper, we discuss two tree-cotree gauge-based methods for the removal of the spurious eigenmodes.

A Mixed Tree-Cotree Gauge for the Reduced Basis Approximation of Maxwell's Eigenvalue Problem

Abstract

Model order reduction methods are a powerful tool to drastically reduce the computational effort of problems which need to be evaluated repeatedly, i.e., when computing the same system for various parameter values. When applying a reduced basis approximation algorithm to the Maxwell eigenvalue problem, we encounter spurious solutions in the reduced system which hence need to be removed during the basis construction. In this paper, we discuss two tree-cotree gauge-based methods for the removal of the spurious eigenmodes.
Paper Structure (10 sections, 10 equations, 3 figures, 1 table)

This paper contains 10 sections, 10 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Reduced Basis Approximation and Spurious Modes
  4. Tree-Cotree Gauge
  5. Mixed Gauged Variant
  6. Initialization of the reduced basis
  7. Greedy enrichment of the reduced basis
  8. Computational Considerations and Numerical Results
  9. Conclusion
  10. Acknowledgement

Figures (3)

  • Figure 1: Flowchart of the mixed gauged algorithm. The colors indicate the computational space. Yellow corresponds to the sparse IGA space, red to dense matrices of the cotree space and green to the reduced space. The two phases are indicated with the gray background boxes.
  • Figure 2: 3-cell and 1-cell TESLA cavity at $t = 0$ (left, middle) morphed to the Pillbox cavity at $t=1$ (right). For $t \in (0,1)$, the system matrices are obtained via matrix interpolation and do not correspond to physical geometries.
  • Figure 3: Average error of the first five approximated eigenvalues in the one-cell TESLA cavity with 50 initial basis vectors.