Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Abstract

The goals and contributions of this paper are twofold. It provides a new computational tool for data driven Koopman spectral analysis by taking up the formidable challenge to develop a numerically robust algorithm by following the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly -these are defined as the inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to curb ill-conditioning is the discrete Fourier transform of the snapshots; in the new representation, the Vandermonde matrix is transformed into a generalized Cauchy matrix, which then allows accurate computation by specially tailored algorithms of numerical linear algebra. The second goal is to shed light on the connection between the formulas for optimal reconstruction weights when reconstructing snapshots using subsets of the computed Koopman modes. It is shown how using a certain weaker form of generalized inverses leads to explicit reconstruction formulas that match the abstract results from Koopman spectral theory, in particular the Generalized Laplace Analysis.AMS subject classifications: 15A12, 15A23, 65F35, 65L05, 65M20, 65M22, 93A15, 93A30, 93B18, 93B40, 93B60, 93C05, 93C10, 93C15, 93C20, 93C57In this paper, we revisit the natural formulation of finite dimensional Koopman spectral analysis in terms of Krylov bases and the Frobenius companion matrix. In this formulation, reviewed in §2.1 below, a spectral approximation of A is obtained by the Rayleigh-Ritz extraction using the primitive Krylov basis X m = (f 1 , . . . , f m ). This means that the Rayleigh quotient of A is the Frobenius companion matrix, whose eigenvector matrix is the inverse of the Vandermonde matrix V m , parametrized by its eigenvalues λ i . The DMD (Koopman) modes, that is, the Ritz vectors z i , are then computed as. . , m. Unfortunately, Vandermonde matrices can have extremely high conditions numbers, so a straightforward implementation of this algebraically elegant scheme can lead to inaccurate results in finite precision computation. Most other DMD-variants bypass this issue by computing an orthonormal basis from the snaphshots using a truncated singular value decomposition [34,33,38]. We note that the original formulation of DMD was based on the SVD ([34], reviewed in Algorithm 2.1 in this paper), whereas the first connection between DMD and Koopman operator theory was formulated in terms of the companion matrix [32]. In Rowley et al. [32], though, there was no consideration of the deep numerical issues related to working with Vandermonde matrices which has likely contributed to the prevalence of the SVD-based variants of DMD. There is, however, a certain intrinsic elegance in the decomposition of the snapshots in terms of the spectral structure of the companion matrix in addition to a stronger connection to Generalized Laplace Analysis (GLA) theory [27,26] of Koopman operator theory than the SVD-based DMD variants have.Following this natural formulation, we present a DMD alg...

An algorithm for two‐variable rational interpolation suitable for matrix manipulations with the evaluation–interpolation method

“…As shown by Gautschi and Inglese, 9 the condition number of Vandermonde-like matrices grows exponentially with n. However, algorithms that can accurately perform linear algebra operations with ill-conditioned Vandermonde matrices have been developed (see e.g. [10][11][12][13][14] ). All the algorithms in References 10-13 concerning Vandermonde matrices are based on the fact that, after performing a Discrete Fourier Transform (DFT) on a Vandermonde matrix, the latter is transformed into a Cauchy-like matrix.…”

“…Therefore, the ability of the ASR algorithm to decide the necessary reductions in order to arrive to a correct solution of the system that corresponds to the perturbed matrix, mainly relies to the ability of the reciprocal condition number estimator rcond that is called by ASR (see Appendices A3 and A4) to behave in a stable manner when applied to the submatrices of (13). In ASR the estimator rcond is used in two phases: (a) to arrive to a uniquely solvable system and (b) to further reduce this system, leaving out the zero-elements of the solution and obtaining a much smaller-sized system.…”

“…That is, some reciprocal condition numbers became large earlier than expected and the algorithm stopped at the reduced size N = 16 instead of N = 10. So, questions arose on what made the difference between the perturbed rows of matrix (13), which forced the reciprocal condition numbers to increase on one occasion, while remaining close to zero on the other. Extending this reasoning, if some interpolation points lead to better algorithmic stability than other, then there should be a criterion different than randomness when choosing which matrix row to discard at each step of the ASR.…”

“…For Cauchy-like matrices, Demmel 10 has developed a high-accuracy Gaussian elimination with Complete Pivoting (GECP) algorithm (algorithm 3 of Reference 10), which, by use of a recursive formula, calculates Schur complements both accurately and efficiently. Drmač et al 13 have developed a MATLAB code that performs a pivoted LDU decomposition on a Cauchy-like matrix. The latter is the outcome of a DFT transform on a Vandermonde matrix.…”

Mean Subtraction and Mode Selection in Dynamic Mode Decomposition

“…A setback with this approach is the deluge of DMD eigenvalues to contend with. A common choice is to neglect those corresponding to DMD modes with very small norm [29,33,2,18,11,10]. We justify this choice by establishing a 1-1 correspondence between DMD modes of negligible norm and DMD eigenvalues that are numerical artifacts.…”