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Inverse Problems in Science and Engineering Volume 29, 2021 - Issue 13

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Research Article

Sequential estimation of the time-dependent heat transfer coefficient using the method of fundamental solutions and particle filters

W. B. da Silvaa Federal University of Espí-rito Santo, Vitória, BrazilCorrespondence[email protected]

https://orcid.org/0000-0003-2242-7825 View further author information

J. C. S. Dutraa Federal University of Espí-rito Santo, Vitória, Brazil

https://orcid.org/0000-0001-6784-4150 View further author information

C. E. P. Kopperschimidta Federal University of Espí-rito Santo, Vitória, BrazilView further author information

D. Lesnicb Department of Applied Mathematics, University of Leeds, Leeds, UKView further author information

R. G. Aykroydc Department of Statistics, University of Leeds, Leeds, UK

https://orcid.org/0000-0003-3700-0816 View further author information

Pages 3322-3341 | Received 21 Nov 2019, Accepted 17 Oct 2021, Published online: 07 Nov 2021

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https://doi.org/10.1080/17415977.2021.1998040
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Figure 1. The analytical and MFS numerical solutions for: (a) $T (x, 0.5)$ , (b) $T (1, t)$ , (c) $T (0, t)$ and (d) $E (t)$ , obtained when solving the direct problem for Example 1. Corresponding to the results for $T (x, 0.5)$ , $T (1, t)$ , $T (0, t)$ and $E (t)$ , the maximum pointwise relative errors between the analytical and numerical MFS solutions are $0.5 %$ , $0.2 %$ , $0.3 %$ and $0.01 %$ , respectively.

Figure 2. Estimated $ρ (t)$ and the filtered boundary temperature measurements (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) contaminated with $p = 1 %$ noise, obtained using the particle filter with $N_{part} = 200$ particles for (a) and (c) $σ_{ρ} = 0.2$ ; (b) and (d) $σ_{ρ} = 0.4$ , for Example 1.

Figure 2. Estimated ρ(t) and the filtered boundary temperature measurements (Equation19(19) Ya(tk)=Y(tk)+ϵk,k=1,…,M,(19) ) contaminated with p=1% noise, obtained using the particle filter with Npart=200 particles for (a) and (c) σρ=0.2; (b) and (d) σρ=0.4, for Example 1.

Table 1. Results for various $σ_{ρ}$ for $p = 1 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) for Example 1.

Display Table

Table 2. Results for $p = 5 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) with $σ_{ρ} = 0.2$ for Example 1.

Display Table

Table 3. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 1.

Display Table

Figure 3. Estimated $ρ (t)$ and the filtered measurements (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) contaminated with $p = 1 %$ noise, obtained using the particle filter with $σ_{ρ} = 0.2$ and $N_{part} = 200$ particles, for Example 1.

Figure 3. Estimated ρ(t) and the filtered measurements (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ) contaminated with p=1% noise, obtained using the particle filter with σρ=0.2 and Npart=200 particles, for Example 1.

Figure 4. Estimated $ρ (t)$ from the measurements (a) (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) or (b) (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) contaminated with $p = 5 %$ noise, as filtered in (c) and (d), respectively, obtained using the particle filter with $σ_{ρ} = 0.2$ and $N_{part} = 200$ particles, for Example 1.

Figure 4. Estimated ρ(t) from the measurements (a) (Equation19(19) Ya(tk)=Y(tk)+ϵk,k=1,…,M,(19) ) or (b) (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ) contaminated with p=5% noise, as filtered in (c) and (d), respectively, obtained using the particle filter with σρ=0.2 and Npart=200 particles, for Example 1.

Figure 5. The variation of the MWCI over time for (a) $p = 1 %$ and (b) $p = 5 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) or Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ), obtained using the particle filter with $σ_{ρ} = 0.2$ and $N_{part} = 200$ particles, for Example 1.

Figure 5. The variation of the MWCI over time for (a) p=1% and (b) p=5% noise in Equation (Equation19(19) Ya(tk)=Y(tk)+ϵk,k=1,…,M,(19) ) or Equation (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ), obtained using the particle filter with σρ=0.2 and Npart=200 particles, for Example 1.

Figure 6. The analytical and MFS numerical solutions for: (a) $T (1, t)$ , (b) $T (0, t)$ , (c) $E (t)$ , obtained when solving the direct problem for Example 2. Corresponding to the results for $T (1, t)$ , $T (0, t)$ and $E (t)$ , the maximum pointwise relative errors between the analytical and numerical MFS solutions are $4 %$ , $2 %$ and $0.2 %$ , respectively.

Figure 7. (a) Estimated $ρ (t)$ from the measurements (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with (a) $p = 1 %$ and (b) $p = 5 %$ noise, as filtered in (c) and (d), respectively, along with (e) the variation of the MWCI over time, obtained using the particle filter with $σ_{ρ} = 0.2$ and $N_{part} = 200$ particles, for Example 2.

Figure 7. (a) Estimated ρ(t) from the measurements (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ) with (a) p=1% and (b) p=5% noise, as filtered in (c) and (d), respectively, along with (e) the variation of the MWCI over time, obtained using the particle filter with σρ=0.2 and Npart=200 particles, for Example 2.

Table 4. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 2.

Display Table

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Sequential estimation of the time-dependent heat transfer coefficient using the method of fundamental solutions and particle filters

Table 1. Results for various $σ_{ρ}$ for $p = 1 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) for Example 1.

Table 2. Results for $p = 5 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) with $σ_{ρ} = 0.2$ for Example 1.

Table 3. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 1.

Table 4. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 2.

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Sequential estimation of the time-dependent heat transfer coefficient using the method of fundamental solutions and particle filters

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Table 1. Results for various σρ for p=1% noise in Equation (Equation19(19) Ya(tk)=Y(tk)+ϵk,k=1,…,M,(19) ) for Example 1.

Table 2. Results for p=5% noise in Equation (Equation19(19) Ya(tk)=Y(tk)+ϵk,k=1,…,M,(19) ) with σρ=0.2 for Example 1.

Table 3. Results for p∈{1,5}% noise in Equation (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ) with σρ=0.2 for Example 1.

Table 4. Results for p∈{1,5}% noise in Equation (Equation20(20) Ea(tk)=E(tk)+ϵk,k=1,…,M,(20) ) with σρ=0.2 for Example 2.

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Table 1. Results for various $σ_{ρ}$ for $p = 1 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) for Example 1.

Table 2. Results for $p = 5 %$ noise in Equation (Equation19(19) $\begin{aligned} Y_{a} (t_{k}) = Y (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (19) ) with $σ_{ρ} = 0.2$ for Example 1.

Table 3. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 1.

Table 4. Results for $p \in {1, 5} %$ noise in Equation (Equation20(20) $\begin{aligned} E_{a} (t_{k}) = E (t_{k}) + ϵ_{k}, k = 1, \dots, M, \end{aligned}$ (20) ) with $σ_{ρ} = 0.2$ for Example 2.