# American Institute of Mathematical Sciences

June  2020, 14(3): 437-461. doi: 10.3934/ipi.2020021

## Regularization of inverse problems via box constrained minimization

 Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria

* corresponding author

Received  May 2019 Revised  December 2019 Published  March 2020

Fund Project: This work was supported by the Austrian Science Fund FWF under the grants P28008, I2271, and P30054 as well as partially by the Karl Popper Kolleg "Modeling-Simulation-Optimization", funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF)

In the present paper we consider minimization based formulations of inverse problems $(x, \Phi)\in \mbox{argmin}\left\{{ \mathcal{J}(x, \Phi;y)}:{(x, \Phi)\in M_{ad}(y)}\right\}$ for the specific but highly relevant case that the admissible set $M_{ad}^\delta(y^\delta)$ is defined by pointwise bounds, which is the case, e.g., if $L^\infty$ constraints on the parameter are imposed in the sense of Ivanov regularization, and the $L^\infty$ noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems.

Discretization of $(x, \Phi)$ with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.

Citation: Philipp Hungerländer, Barbara Kaltenbacher, Franz Rendl. Regularization of inverse problems via box constrained minimization. Inverse Problems & Imaging, 2020, 14 (3) : 437-461. doi: 10.3934/ipi.2020021
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##### References:
Left: exact coefficient ${b_f}_{ex} = c_{ex} = a_{ex}$. Right: locations of spots for testing weak * $L^\infty$ convergence
Left: reconstructed coefficient $c_k$; Middle: active set for lower bound; Right: active set for upper bound; For $k = 1, 2, 3, 4$ (top to bottom) and $\delta = 0.1$
Left: reconstructed coefficient $c_k$; Middle: active set for lower bound; Right: active set for upper bound; For $k = 1, 4, 8, 12$ (top to bottom) and $\delta = 0.01$
Linear system solves (stars) and average size of linear systems (diamonds); Left: warm start; Right: cold start; Top: N = 32; Bottom: N = 64
Test 2: left: exact coefficient $c_{ex}$; $\underline{{c}} = -9$, $\overline{{c}} = 6$; right: locations of spots for testing weak * $L^\infty$ convergence
Test 2: Left: reconstructed coefficient $c_k$; Middle: active set for lower bound; Right: active set for upper bound; For $k = 1, 4, 8, 12$ (top to bottom) and $\delta = 0.01$
Test 3: left: exact coefficient $c_{ex}$; $\underline{{c}} = -10$, $\overline{{c}} = 0$; right: locations of spots for testing weak * $L^\infty$ convergence
Test 3: Left: reconstructed coefficient $c_k$; Middle: active set for lower bound; Right: active set for upper bound; For $k = 1, 4, 8, 12$ (top to bottom) and $\delta = 0.01$
Generation of a primal feasible pair $(\mathcal B, \mathcal D)$
 $\mathcal B \leftarrow \mathcal B_s, \ \mathcal D \leftarrow \mathcal D_s$. while $(\mathcal B, \mathcal D)$ not primal feasible: $y= \text{KKT}(\mathcal B, \mathcal D), \ \mathcal B \leftarrow \mathcal B \cup \{i \in \mathcal N \setminus \mathcal B : y_{i} \geq u_{i} \}, \ \mathcal D \leftarrow \mathcal D \cup \{i \in \mathcal N \setminus \mathcal D : y_{i} \leq \ell_{i} \}$.
 $\mathcal B \leftarrow \mathcal B_s, \ \mathcal D \leftarrow \mathcal D_s$. while $(\mathcal B, \mathcal D)$ not primal feasible: $y= \text{KKT}(\mathcal B, \mathcal D), \ \mathcal B \leftarrow \mathcal B \cup \{i \in \mathcal N \setminus \mathcal B : y_{i} \geq u_{i} \}, \ \mathcal D \leftarrow \mathcal D \cup \{i \in \mathcal N \setminus \mathcal D : y_{i} \leq \ell_{i} \}$.
Algorithmic description of the feasible active set method from [10]
 Feasible active set method for solving (34) Input: $Q \succ 0$, $\ell, \ u, \ q \in \mathbb R^n, \ \ell< u$. $\mathcal A, \ \mathcal C \subseteq \mathcal N, \ \mathcal A \cap \mathcal C = \emptyset$, $(\mathcal A, \mathcal C)$ primal feasible. Output: $(\mathcal A, \mathcal C)$ optimal for (34). $[x, \alpha, \gamma] =$KKT$(\mathcal A, \mathcal C)$ while $(\mathcal A, \mathcal C)$ not optimal for (34) $\mathcal B_{s} \leftarrow \{i \in \mathcal A: \alpha_{i} \geq 0 \}; \mathcal B \leftarrow \mathcal B_{s}$. $\mathcal D_{s} \leftarrow \{i \in \mathcal C: \gamma_{i} \geq 0 \}; \mathcal D \leftarrow \mathcal D_{s}$. $y=$KKT$(\mathcal B, \mathcal D)$. while $(\mathcal B, \mathcal D)$ not primal feasible $\mathcal B \leftarrow \mathcal B \cup \{ i \in \overline{\mathcal B}: y_{i} \geq u_{i} \}$. $\mathcal D \leftarrow \mathcal D \cup \{ i \in \overline{\mathcal D}: y_{i} \leq \ell_{i} \}$. $y=$KKT$(\mathcal B, \mathcal D)$. endwhile Case 1: $J(y)< J(x)$ $\mathcal A \leftarrow \mathcal B, \ \mathcal C \leftarrow \mathcal D$. Case 2: $J(y) \geq J(x)$ and $|\mathcal A|+|\mathcal C|=1$. Let $(\mathcal A_{opt}, \mathcal C_{opt})$ be the optimal pair for (34) with the upper respectively lower bound on $\mathcal A \cup \mathcal C =\{ j \}$ removed. $(\mathcal A_{opt}, \mathcal C_{opt})$ is optimal for (34), hence stop. Case 3: $J(y) \geq J(x)$ and $|\mathcal A| + |\mathcal C|> 1$ Choose $\mathcal A_0 \subseteq \mathcal A$, $\mathcal C_0 \subseteq \mathcal C$ with $\mathcal A_0 \cup \mathcal C_0 \not = \emptyset$ such that $x$ is feasible but not optimal for (36), for details see [10]. Let $(\mathcal B_{0}, \mathcal D_0)$ be the optimal pair for (36). $\mathcal A \leftarrow \mathcal A_0 \cup \mathcal B_{0}, \ \mathcal C \leftarrow \mathcal C_0 \cup \mathcal D_0$. $[\alpha, \gamma]=$KKT$(\mathcal A, \mathcal C)$ endwhile
 Feasible active set method for solving (34) Input: $Q \succ 0$, $\ell, \ u, \ q \in \mathbb R^n, \ \ell< u$. $\mathcal A, \ \mathcal C \subseteq \mathcal N, \ \mathcal A \cap \mathcal C = \emptyset$, $(\mathcal A, \mathcal C)$ primal feasible. Output: $(\mathcal A, \mathcal C)$ optimal for (34). $[x, \alpha, \gamma] =$KKT$(\mathcal A, \mathcal C)$ while $(\mathcal A, \mathcal C)$ not optimal for (34) $\mathcal B_{s} \leftarrow \{i \in \mathcal A: \alpha_{i} \geq 0 \}; \mathcal B \leftarrow \mathcal B_{s}$. $\mathcal D_{s} \leftarrow \{i \in \mathcal C: \gamma_{i} \geq 0 \}; \mathcal D \leftarrow \mathcal D_{s}$. $y=$KKT$(\mathcal B, \mathcal D)$. while $(\mathcal B, \mathcal D)$ not primal feasible $\mathcal B \leftarrow \mathcal B \cup \{ i \in \overline{\mathcal B}: y_{i} \geq u_{i} \}$. $\mathcal D \leftarrow \mathcal D \cup \{ i \in \overline{\mathcal D}: y_{i} \leq \ell_{i} \}$. $y=$KKT$(\mathcal B, \mathcal D)$. endwhile Case 1: $J(y)< J(x)$ $\mathcal A \leftarrow \mathcal B, \ \mathcal C \leftarrow \mathcal D$. Case 2: $J(y) \geq J(x)$ and $|\mathcal A|+|\mathcal C|=1$. Let $(\mathcal A_{opt}, \mathcal C_{opt})$ be the optimal pair for (34) with the upper respectively lower bound on $\mathcal A \cup \mathcal C =\{ j \}$ removed. $(\mathcal A_{opt}, \mathcal C_{opt})$ is optimal for (34), hence stop. Case 3: $J(y) \geq J(x)$ and $|\mathcal A| + |\mathcal C|> 1$ Choose $\mathcal A_0 \subseteq \mathcal A$, $\mathcal C_0 \subseteq \mathcal C$ with $\mathcal A_0 \cup \mathcal C_0 \not = \emptyset$ such that $x$ is feasible but not optimal for (36), for details see [10]. Let $(\mathcal B_{0}, \mathcal D_0)$ be the optimal pair for (36). $\mathcal A \leftarrow \mathcal A_0 \cup \mathcal B_{0}, \ \mathcal C \leftarrow \mathcal C_0 \cup \mathcal D_0$. $[\alpha, \gamma]=$KKT$(\mathcal A, \mathcal C)$ endwhile
Algorithmic description of the infeasible active set method from [11].
 Ineasible active set method for solving (34) Input: $Q \succ 0$, $\ell, \ u, \ q \in \mathbb R^n, \ \ell< u$. $\mathcal A, \ \mathcal C \subseteq \mathcal N, \ \mathcal A \cap \mathcal C = \emptyset$, $(x, \alpha, \gamma)$ Output: $(\mathcal A, \mathcal C)$, $(x, \alpha, \gamma)$ optimal for (34). ${w} =\max\{\min\{x, u\}, \ell\}$ while $(\mathcal A, \mathcal C)$ not optimal for (34) $\mathcal B \leftarrow \{i \not\in \mathcal A: x_{i} \geq u_i \}\cup \{i \in \mathcal A: \alpha_{i} \geq 0 \}$. $\mathcal D \leftarrow \{i \not\in \mathcal C: x_{i} \leq l_i \}\cup\{i \in \mathcal C: \gamma_{i} \geq 0 \}$. $[y, \beta, \delta]=$KKT$(\mathcal B, \mathcal D)$; $v =\max\{\min\{y, u\}, \ell\}$. Case 1: $J(v)< J({w})$ Continue with $[Q, \ell, u, q, \mathcal{B}, \mathcal{D}, (y, \beta, \delta)]$. Case 2: $J(v) \geq J({w})$. Determine $\mathcal{A}^+$, $\mathcal{C}^+$: If $\ell_iu_i\}\cup\{i:y_i<\ell_i\}} J(\max\{\min\{{z}(\lambda_j), u\}, \ell\})$ where $\lambda_{j}:=\left\{\begin{array}{ll} \frac{y_{j}-u_{j}}{y_{j}-w_{j}} & \text { for } j \in\left\{i: y_{i}>u_{i}\right\} \\ \frac{\ell_{j}-y_{j}}{w_{j}-y_{j}} & \text { for } j \in\left\{i: y_{i} < \ell_{i}\right\} \end{array} ; \quad z(\lambda)=\lambda w+(1-\lambda) y\right.$; $x^+={z}(\lambda_*)$; $-\alpha^+_{\tilde{\mathcal{A}}^+} = q_{\tilde{\mathcal{A}}^+} + Q_{\tilde{\mathcal{A}}^+, \mathcal N} x^+$; $\gamma^+_{\mathcal C^+} = q_{\mathcal C^+} + Q_{\mathcal C^+, \mathcal N} x^+$, where $\tilde{\mathcal{A}}^+=\mathcal{A}^+\cup(\mathcal{N}\setminus\mathcal{C}^+)$; continue with $[Q, \ell, u, q, \mathcal{A}^+, \mathcal{C}^+, (x^+, \alpha^+, \gamma^+)]$. endwhile
 Ineasible active set method for solving (34) Input: $Q \succ 0$, $\ell, \ u, \ q \in \mathbb R^n, \ \ell< u$. $\mathcal A, \ \mathcal C \subseteq \mathcal N, \ \mathcal A \cap \mathcal C = \emptyset$, $(x, \alpha, \gamma)$ Output: $(\mathcal A, \mathcal C)$, $(x, \alpha, \gamma)$ optimal for (34). ${w} =\max\{\min\{x, u\}, \ell\}$ while $(\mathcal A, \mathcal C)$ not optimal for (34) $\mathcal B \leftarrow \{i \not\in \mathcal A: x_{i} \geq u_i \}\cup \{i \in \mathcal A: \alpha_{i} \geq 0 \}$. $\mathcal D \leftarrow \{i \not\in \mathcal C: x_{i} \leq l_i \}\cup\{i \in \mathcal C: \gamma_{i} \geq 0 \}$. $[y, \beta, \delta]=$KKT$(\mathcal B, \mathcal D)$; $v =\max\{\min\{y, u\}, \ell\}$. Case 1: $J(v)< J({w})$ Continue with $[Q, \ell, u, q, \mathcal{B}, \mathcal{D}, (y, \beta, \delta)]$. Case 2: $J(v) \geq J({w})$. Determine $\mathcal{A}^+$, $\mathcal{C}^+$: If $\ell_iu_i\}\cup\{i:y_i<\ell_i\}} J(\max\{\min\{{z}(\lambda_j), u\}, \ell\})$ where $\lambda_{j}:=\left\{\begin{array}{ll} \frac{y_{j}-u_{j}}{y_{j}-w_{j}} & \text { for } j \in\left\{i: y_{i}>u_{i}\right\} \\ \frac{\ell_{j}-y_{j}}{w_{j}-y_{j}} & \text { for } j \in\left\{i: y_{i} < \ell_{i}\right\} \end{array} ; \quad z(\lambda)=\lambda w+(1-\lambda) y\right.$; $x^+={z}(\lambda_*)$; $-\alpha^+_{\tilde{\mathcal{A}}^+} = q_{\tilde{\mathcal{A}}^+} + Q_{\tilde{\mathcal{A}}^+, \mathcal N} x^+$; $\gamma^+_{\mathcal C^+} = q_{\mathcal C^+} + Q_{\mathcal C^+, \mathcal N} x^+$, where $\tilde{\mathcal{A}}^+=\mathcal{A}^+\cup(\mathcal{N}\setminus\mathcal{C}^+)$; continue with $[Q, \ell, u, q, \mathcal{A}^+, \mathcal{C}^+, (x^+, \alpha^+, \gamma^+)]$. endwhile
Comparison of different solvers for QPs with box constraints
 source N=32 (n=2178) N=64 (n=8450) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 1 1 1 1 1 1 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 6.6901e-06 6.6154e-06 6.6154e-06 3.2887e-05 3.2532e-05 3.2744e-05 $\mbox{err}_{spot_1}$ 2.4023e-06 0 0 6.7780e-07 0 0 $\mbox{err}_{spot_2}$ 7.3665e-06 0 0 0.0462 0 0 $\mbox{err}_{spot_3}$ 1.0890e-05 0 0 1.3992 1.3917 1.3917 $\mbox{err}_{L^1(\Omega)}$ 0.0462 0.0465 0.0465 0.0832 0.0834 0.0834 CPU 1.68 1.16 1.09 29.62 112.97 22.85 potential N=32 (n=3137) N=64 (n=12417) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 4 4 6 3 3 3 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 6.2155e-06 6.0569e-06 8.0316e-07 1.6318e-04 1.6286e-04 1.6286e-04 $\mbox{err}_{spot_1}$ 6.9611e-10 0 0 1.2184e-10 0 0 $\mbox{err}_{spot_2}$ 1.7502e-06 0 0 0.7183 0.7145 0.7145 $\mbox{err}_{spot_3}$ 1.5392 1.6093 1.4363 3.2625 3.2629 3.2629 $\mbox{err}_{L^1(\Omega)}$ 0.1051 0.1044 0.0938 0.1460 0.1444 0.1444 CPU 21.88 24.44 18.34 432.06 368.67 276.17 diffusion N=32 (n=3137) N=64 (n=12417) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 4 4 4 8 8 8 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 0.0931 0.0931 0.0931 0.0543 0.0543 0.0543 $\mbox{err}_{spot_1}$ 3.7303e-14 0 0 3.6415e-14 0 0 $\mbox{err}_{spot_2}$ 4.4418 4.4418 4.4418 7.8278e-05 0 0 $\mbox{err}_{spot_3}$ 0.3200 0.3199 0.3199 4.3077e-13 0 0 $\mbox{err}_{L^1(\Omega)}$ 0.3259 0.3259 0.3259 0.3799 0.3799 0.3799 CPU 32.01 6.30 5.25 1193.00 532.57 463.89
 source N=32 (n=2178) N=64 (n=8450) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 1 1 1 1 1 1 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 6.6901e-06 6.6154e-06 6.6154e-06 3.2887e-05 3.2532e-05 3.2744e-05 $\mbox{err}_{spot_1}$ 2.4023e-06 0 0 6.7780e-07 0 0 $\mbox{err}_{spot_2}$ 7.3665e-06 0 0 0.0462 0 0 $\mbox{err}_{spot_3}$ 1.0890e-05 0 0 1.3992 1.3917 1.3917 $\mbox{err}_{L^1(\Omega)}$ 0.0462 0.0465 0.0465 0.0832 0.0834 0.0834 CPU 1.68 1.16 1.09 29.62 112.97 22.85 potential N=32 (n=3137) N=64 (n=12417) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 4 4 6 3 3 3 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 6.2155e-06 6.0569e-06 8.0316e-07 1.6318e-04 1.6286e-04 1.6286e-04 $\mbox{err}_{spot_1}$ 6.9611e-10 0 0 1.2184e-10 0 0 $\mbox{err}_{spot_2}$ 1.7502e-06 0 0 0.7183 0.7145 0.7145 $\mbox{err}_{spot_3}$ 1.5392 1.6093 1.4363 3.2625 3.2629 3.2629 $\mbox{err}_{L^1(\Omega)}$ 0.1051 0.1044 0.0938 0.1460 0.1444 0.1444 CPU 21.88 24.44 18.34 432.06 368.67 276.17 diffusion N=32 (n=3137) N=64 (n=12417) $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $\mathtt{quadprog}$ $\mathtt{Infeas\_{AS}}$ $\mathtt{Feas\_AS}$ $k$ 4 4 4 8 8 8 $\frac{J(x_k^\delta, u_k^\delta)}{J(x_0, u_0)}$ 0.0931 0.0931 0.0931 0.0543 0.0543 0.0543 $\mbox{err}_{spot_1}$ 3.7303e-14 0 0 3.6415e-14 0 0 $\mbox{err}_{spot_2}$ 4.4418 4.4418 4.4418 7.8278e-05 0 0 $\mbox{err}_{spot_3}$ 0.3200 0.3199 0.3199 4.3077e-13 0 0 $\mbox{err}_{L^1(\Omega)}$ 0.3259 0.3259 0.3259 0.3799 0.3799 0.3799 CPU 32.01 6.30 5.25 1193.00 532.57 463.89
Convergence as $\delta\to0$: averaged errors of five test runs with uniform noise
 source potential diffusion $\delta$ 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 $\mbox{err}_{spot_1}$ 0 0.2000 0.2000 0 0 0 0 0 0 $\mbox{err}_{spot_2}$ 0 2.7488 4.0702 0 0.7960 4.8689 0 8.4141 9.8436 $\mbox{err}_{spot_3}$ 0 0.5678 1.9445 1.0840 2.1512 2.5862 0.6572 0 0 $\mbox{err}_{L^1(\Omega)}$ 0.0472 0.5288 0.5721 0.1472 0.2136 0.3671 0.7200 0.3783 0.3745
 source potential diffusion $\delta$ 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 $\mbox{err}_{spot_1}$ 0 0.2000 0.2000 0 0 0 0 0 0 $\mbox{err}_{spot_2}$ 0 2.7488 4.0702 0 0.7960 4.8689 0 8.4141 9.8436 $\mbox{err}_{spot_3}$ 0 0.5678 1.9445 1.0840 2.1512 2.5862 0.6572 0 0 $\mbox{err}_{L^1(\Omega)}$ 0.0472 0.5288 0.5721 0.1472 0.2136 0.3671 0.7200 0.3783 0.3745
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