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Mathematical Considerations on Randomized Orthgonal Decomposition Method for Developing Twin Data Models

Diana A. Bistrian

Abstract

This paper introduces the approach of Randomized Orthogonal Decomposition (ROD) for producing twin data models in order to overcome the drawbacks of existing reduced order modelling techniques. When compared to Fourier empirical decomposition, ROD provides orthonormal shape modes that maximize their projection on the data space, which is a significant benefit. A shock wave event described by the viscous Burgers equation model is used to illustrate and evaluate the novel method. The new twin data model is thoroughly evaluated using certain criteria of numerical accuracy and computational performance.

Mathematical Considerations on Randomized Orthgonal Decomposition Method for Developing Twin Data Models

Abstract

This paper introduces the approach of Randomized Orthogonal Decomposition (ROD) for producing twin data models in order to overcome the drawbacks of existing reduced order modelling techniques. When compared to Fourier empirical decomposition, ROD provides orthonormal shape modes that maximize their projection on the data space, which is a significant benefit. A shock wave event described by the viscous Burgers equation model is used to illustrate and evaluate the novel method. The new twin data model is thoroughly evaluated using certain criteria of numerical accuracy and computational performance.
Paper Structure (5 sections, 5 theorems, 52 equations, 3 figures, 2 tables)

This paper contains 5 sections, 5 theorems, 52 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Randomized Orthogonal Decomposition
  3. Mathematical model and the exact solution
  4. The Twin Data Model. Qualitative analysis
  5. Conclusions

Key Result

Proposition 1

(Fourier Empirical Orthogonal Decomposition) Let be a real-valued data matrix of rank $r \le \min \left( {{N_x},{N_t} + 1} \right)$, whose columns ${u_j} \in {\mathbb{R}^{{N_x}}}$, $j = 1,...,{N_t} + 1$ are data snapshots. The Singular Value Decomposition (SVD) yields the factorization of the matrix $V$, where $\Psi = \left[ {{\psi _1},...,{\psi _{{N_x}}}} \right] \in {\mathbb{R}^{{N_x} \times

Figures (3)

  • Figure 1: The exact solution of the viscid Burgers equation model (\ref{['burgers']}), with initial condition (\ref{['test1']})
  • Figure 2: The objectives of the optimization problem (\ref{['optimprob']}) and the Pareto front solution obtained by genetic algorithm
  • Figure 3: a.The twin data model as the solution of ROD algorithm; b.The modal amplitudes; c.The corresponding leading shape modes

Theorems & Definitions (7)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2