Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differentiable Singular Value Decomposition

Rohit Kanchi, Sicheng He

Abstract

Singular value decomposition is widely used in modal analysis, such as proper orthogonal decomposition and resolvent analysis, to extract key features from complex problems. SVD derivatives need to be computed efficiently to enable the large scale design optimization. However, for a general complex matrix, no method can accurately compute this derivative to machine precision and remain scalable with respect to the number of design variables without requiring the all of the singular variables. We propose two algorithms to efficiently compute this derivative based on the adjoint method and reverse automatic differentiation and RAD-based singular value derivative formula. Differentiation results for each method proposed were compared with FD results for one square and one tall rectangular matrix example and matched with the FD results to about 5 to 7 digits. Finally, we demonstrate the scalability of the proposed method by calculating the derivatives of singular values with respect to the snapshot matrix derived from the POD of a large dataset for a laminar-turbulent transitional flow over a flat plate, sourced from the John Hopkins turbulence database.

Differentiable Singular Value Decomposition

Abstract

Singular value decomposition is widely used in modal analysis, such as proper orthogonal decomposition and resolvent analysis, to extract key features from complex problems. SVD derivatives need to be computed efficiently to enable the large scale design optimization. However, for a general complex matrix, no method can accurately compute this derivative to machine precision and remain scalable with respect to the number of design variables without requiring the all of the singular variables. We propose two algorithms to efficiently compute this derivative based on the adjoint method and reverse automatic differentiation and RAD-based singular value derivative formula. Differentiation results for each method proposed were compared with FD results for one square and one tall rectangular matrix example and matched with the FD results to about 5 to 7 digits. Finally, we demonstrate the scalability of the proposed method by calculating the derivatives of singular values with respect to the snapshot matrix derived from the POD of a large dataset for a laminar-turbulent transitional flow over a flat plate, sourced from the John Hopkins turbulence database.
Paper Structure (37 sections, 124 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 37 sections, 124 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Gram matrix method (GMM)
  4. Symmetric embedding matrix method
  5. Discussion
  6. Derivative computation
  7. Adjoint method
  8. Adjoint equation of GMM
  9. Adjoint equation of LGMM
  10. Adjoint equation of RGMM
  11. Adjoint equation of SEMM
  12. RAD method for singular value derivative
  13. Summary
  14. Numerical results
  15. Square matrix derivative computation
  16. ...and 22 more sections

Figures (6)

  • Figure 1: Computational efficiency comparison of real-valued proposed methods, FD and RAD. $p$ represents the index for slope as a reference and $p=1$ corresponds to our proposed methods, $p=2$ is for RAD and $p=3$ is for FD in the plot. The smaller slope and lower wall time, yields better scalability.
  • Figure 2: POD modes of the JHTDB transitional boundary layer dataset. Vortical structures are $Q$ iso-surfaces ($Q$ = $0.001$)
  • Figure 3: $x$-direction singular value derivatives of the POD modes of JHTDB transitional boundary layer dataset. Vortical structures are $Q$ iso-surfaces ($Q$ = $0.001$)
  • Figure 4: Key vortical structures in the transitional region of the flow over flat plate visualized by ($Q$ = $0.001$) iso-surfaces colored by velocity-magnitude
  • Figure 5: The common research model 315 bar wing-truss Shahabsafa2018a
  • ...and 1 more figures