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Reconstruction algorithms for source term recovery from dynamical samples in catalyst models

Akram Aldroubi, Le Gong, Ilya Krishtal, Brendan Miller, Sumati Thareja

Abstract

This paper investigates the problem of recovering source terms in abstract initial value problems (IVP) commonly used to model various scientific phenomena in physics, chemistry, economics, and other fields. We consider source terms of the form $F=h+η$, where $η$ is a Lipschitz continuous background source. The primary objective is to estimate the unknown parameters of non-instantaneous sources $h(t)=\sum\limits_{j=0}^M h_je^{-ρ_j(t-t_j)}χ_{[t_j,\infty)}(t)$, such as the decay rates, initial intensities and activation times. We present two novel recovery algorithms that employ distinct sampling methods of the solution of the IVP. Algorithm 1 combines discrete and weighted average measurements, whereas Algorithm 2 uses a different variant of weighted average measurements. We analyze the performance of these algorithms, providing upper bounds on the recovery errors of the model parameters. Our focus is on the structure of the dynamical samples used by the algorithms and on the error guarantees they yield.

Reconstruction algorithms for source term recovery from dynamical samples in catalyst models

Abstract

This paper investigates the problem of recovering source terms in abstract initial value problems (IVP) commonly used to model various scientific phenomena in physics, chemistry, economics, and other fields. We consider source terms of the form , where is a Lipschitz continuous background source. The primary objective is to estimate the unknown parameters of non-instantaneous sources , such as the decay rates, initial intensities and activation times. We present two novel recovery algorithms that employ distinct sampling methods of the solution of the IVP. Algorithm 1 combines discrete and weighted average measurements, whereas Algorithm 2 uses a different variant of weighted average measurements. We analyze the performance of these algorithms, providing upper bounds on the recovery errors of the model parameters. Our focus is on the structure of the dynamical samples used by the algorithms and on the error guarantees they yield.
Paper Structure (22 sections, 13 theorems, 132 equations, 6 figures)

This paper contains 22 sections, 13 theorems, 132 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Paper Organization
  3. IVP toolkit
  4. Reconstruction algorithms
  5. Description of Algorithm 1.
  6. Description of Algorithm 2.
  7. Derivation of algorithms and their guaranties
  8. Approximation Bounds for the Source Recovery by Algorithm 1
  9. Details of Algorithm 1
  10. Proof of Theorem \ref{['thm1']}
  11. Derivation of \ref{['Esthj']} in Theorem \ref{['thm1']}
  12. Recovery of Decay Rate.
  13. Approximation Bounds for the Source Recovery by Algorithm 2
  14. Details of Algorithm 2: Prony-Laplace Method
  15. Proof of Theorem \ref{['case1_theorem']}
  16. ...and 7 more sections

Key Result

Theorem 3.1

Assume $u_0\in D(A).$ Let $\beta$ be the sampling time step and, given a parameter $D>0$, assume a minimal distance $t_{j+1}-t_j\ge D+4\beta$ between any two consecutive times $t_{j+1}$ and $t_j$ of catalyst intake. Assume also that $\sup_j\|h_j\|\le H,$$\sup_{g \in \mathcal{G},}\|g\|\le R$, and $\r where $Q(g,\beta)=K\widetilde{Q}(g,\beta)$ is the threshold in Algorithm 1 given by Thresh and $v_2

Figures (6)

  • Figure 1: Simulation results for Algorithm 1. The results for $h_i$ lie in the $i$-th column. Pink plus signs stand for the ground truth $\langle h_i, g_j\rangle$, blue stars stand for the output $\mathfrak f_i(g_j)$ when $\beta=0.01$ and green stars stand for the output $\mathfrak f_i(g_j)$ when $\beta=0.005.$
  • Figure 2: Simulation results for Algorithm 2. The results for $h_i$ lie in the $i$-th column. Pink plus signs stand for the ground truth $\langle h_i, g_j\rangle$, blue stars stand for the output $\mathfrak f_i(g_j)$ when $\beta=0.01$.
  • Figure 3: The error estimates of decay rate $\rho_j\ vs.\ N$ for Algorithm 1 in the simulation and the ideal model.
  • Figure 4: The error estimate of $\langle h_i,g_2\rangle\ vs.\ \beta$: $L=10^{-2},$$\sigma=10^{-3}$. The background sources are $\eta(x,t)=xe^{-Lt}$ and $\eta(x,t)=x\sin(Lt)$ for the first and second columns, respectively. Algorithm 1 is given in the top row, Algorithm 2 on the bottom.
  • Figure 5: The error estimate of $\langle h_i,g_2\rangle\ vs.\ L$: $\beta = 0.01,$$\sigma=10^{-3}$. The background sources are $\eta(x,t)=xe^{-Lt}$ and $\eta(x,t)=x\sin(Lt)$ for the first and second columns, respectively. Algorithm 1 is given in the top row, Algorithm 2 on the bottom.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 16 more