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Sparse Discrete Empirical Interpolation Method: State Estimation from Few Sensors

Mohammad Farazmand

TL;DR

S-DEIM leverages a kernel vector which has been neglected in previous DEIM-based methods, and it is proved that, under certain conditions, data assimilated S-DEIM converges exponentially fast towards the true state.

Abstract

Discrete empirical interpolation method (DEIM) estimates a function from its incomplete pointwise measurements. Unfortunately, DEIM suffers large interpolation errors when few measurements are available. Here, we introduce Sparse DEIM (S-DEIM) for accurately estimating a function even when very few measurements are available. To this end, S-DEIM leverages a kernel vector which has been neglected in previous DEIM-based methods. We derive theoretical error estimates for S-DEIM, showing its relatively small error when an optimal kernel vector is used. When the function is generated by a continuous-time dynamical system, we propose a data assimilation algorithm which approximates the optimal kernel vector using observational time series. We prove that, under certain conditions, data assimilated S-DEIM converges exponentially fast towards the true state. We demonstrate the efficacy of our method on two numerical examples.

Sparse Discrete Empirical Interpolation Method: State Estimation from Few Sensors

TL;DR

S-DEIM leverages a kernel vector which has been neglected in previous DEIM-based methods, and it is proved that, under certain conditions, data assimilated S-DEIM converges exponentially fast towards the true state.

Abstract

Discrete empirical interpolation method (DEIM) estimates a function from its incomplete pointwise measurements. Unfortunately, DEIM suffers large interpolation errors when few measurements are available. Here, we introduce Sparse DEIM (S-DEIM) for accurately estimating a function even when very few measurements are available. To this end, S-DEIM leverages a kernel vector which has been neglected in previous DEIM-based methods. We derive theoretical error estimates for S-DEIM, showing its relatively small error when an optimal kernel vector is used. When the function is generated by a continuous-time dynamical system, we propose a data assimilation algorithm which approximates the optimal kernel vector using observational time series. We prove that, under certain conditions, data assimilated S-DEIM converges exponentially fast towards the true state. We demonstrate the efficacy of our method on two numerical examples.
Paper Structure (16 sections, 12 theorems, 63 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 12 theorems, 63 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline
  4. Preliminaries and set-up
  5. Finite-dimensional discretization
  6. Vanilla DEIM
  7. Equal number of sensors and modes
  8. Different number of sensors and modes
  9. Sparse DEIM
  10. Withholding measurements would not work
  11. Data assimilation with S-DEIM
  12. Convergence of DAS-DEIM
  13. Numerical results
  14. Lorenz63 system
  15. Lorenz96 system
  16. ...and 1 more sections

Key Result

Lemma 1

\newlabellem:vDEIM_err0 Assume that the number of sensors and the number of modes are equal, $n=m$, and that the matrix $S^\top \Phi\in\mathbb R^{n\times n}$ is invertible. The vanilla DEIM reconstruction eq:vDEIM_rec satisfies where $\mathcal{E}_{m}(\mathbf u)=\|\mathbf u-\hat{\mathbf u}\|$ is the truncation error, $\hat{\mathbf u} = \Phi\Phi^\top\mathbf u$ is the orthogonal reconstruction of $

Figures (5)

  • Figure 1: Mappings between the high-resolution space $\mathbb R^N$, the measurement space $\mathbb R^n$, and the coefficient space $\mathbb R^m$. The matrix $\mathbb P_+$ is shorthand for $\mathbb P_{\mathcal{R}[(S^\top\Phi)^+]}$, i.e., orthogonal projection onto the range of $(S^\top\Phi)^+$. Note that the mappings in this figure do not necessarily commute.
  • Figure 1: Orthogonal decomposition of $\mathcal{R}[\Phi]$ with fewer sensors than modes, $n<m$. Dashed black lines represent orthogonal projections, whereas the dashed orange line represents an oblique projection.
  • Figure 1: Vanilla DEIM and S-DEIM state estimation for Lorenz63. (a) State space showing the true trajectory (blue), vanilla Q-DEIM (straight black line), and DAS-DEIM (red). The red circle marks the initial DAS-DEIM estimation. (b) Relative errors as a function of time.
  • Figure 2: State estimation for Lorenz96 system. (a) True solution, DAS-DEIM, and vanilla Q-DEIM reconstructions. (b) Close-up view of the observational data from $n=1$ sensor. Dashed blue shows the true observation data and solid red line marks the corresponding noisy observations. (c) Relative error of vanilla Q-DEIM compared to DAS-DEIM.
  • Figure 3: The matrix norm $\|(S_n^\top\Phi_m)^+\|_2$ for the Lorenz96 system with $n=1$ sensors and increasing number of modes $m$.

Theorems & Definitions (30)

  • Definition 1: Orthogonal reconstruction
  • Lemma 1: Vanilla DEIM error estimate Sorensen2010
  • Lemma 2: Ref. Drmac2018
  • Remark 1: See Ref. Drmac2018
  • Definition 2: Sparse DEIM reconstruction
  • Lemma 3
  • Proof 1
  • Corollary 1
  • Proof 2
  • Definition 3: Kernel matrix
  • ...and 20 more