# 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
##### References:

show all references

##### 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
 [1] Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053 [2] Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1 [3] Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631 [4] Xiaoliang Cheng, Rongfang Gong, Weimin Han. A new Kohn-Vogelius type formulation for inverse source problems. Inverse Problems & Imaging, 2015, 9 (4) : 1051-1067. doi: 10.3934/ipi.2015.9.1051 [5] Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042 [6] Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19 [7] Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749 [8] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [9] David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control & Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391 [10] Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347 [11] Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 [12] Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 [13] Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059 [14] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [15] Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399 [16] Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 [17] Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 [18] Monica Motta, Caterina Sartori. Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 513-535. doi: 10.3934/dcds.2008.21.513 [19] Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51 [20] Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems & Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185

2019 Impact Factor: 1.373

## Metrics

• HTML views (146)
• Cited by (0)

• on AIMS