Tensor Approach to Optimal Control problems with Fractional Elliptic - PowerPoint PPT Presentation

Tensor Approach to Optimal Control problems with Fractional Elliptic Operator Volker Schulz Gennadij Heidel Britta Schmitt www.alop.uni-trier.de Boris Khoromskij / Venera Khoromskaia (MPI Leipzig)

Applications causing recent interest Application fields of fractional operators: viscoelastics biophysics nonlocal electrostatics anomalous diffusion heat equation in plasmonic nanostructure networks/composite materials ... 2 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

An optimal control problem Given a function y Ω ∈ L 2 ( Ω ) on Ω := (0 , 1) d , we consider the optimization problem � � ( y ( x ) − y Ω ( x )) 2 d x + γ u 2 ( x ) d x min y , u J ( y , u ) := 2 Ω Ω s. t. − ∆ y = β u y , u ∈ H 1 0 ( Ω ) 3 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

An optimal control problem Given a function y Ω ∈ L 2 ( Ω ) on Ω := (0 , 1) d , we consider the optimization problem � � ( y ( x ) − y Ω ( x )) 2 d x + γ u 2 ( x ) d x min y , u J ( y , u ) := 2 Ω Ω s. t. A α y = β u where A α is the spectral fractional Laplacian operator for some α ∈ (0 , 1). 3 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The KKT system ⇒ ( β A − α + γ       β A α ) u = y Ω A α 0 id y y Ω ⇒ p = γ  = 0 γ id − β id u 0 β u      A α − β id 0 0 ⇒ y = β A − α u p Thus, we find the following necessary optimality conditions: � β A − α + γ β A α � − 1 y Ω u = for the control u , and � β 2 A 2 α � − 1 y Ω I + γ y = β A − α u = for the state y . 4 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The KKT system ⇒ ( β A − α + γ       β A α ) u = y Ω A α 0 id y y Ω ⇒ p = γ  = 0 γ id − β id u 0 β u      A α − β id 0 0 ⇒ y = β A − α u p Thus, we find the following necessary optimality conditions: � β A − α + γ β A α � − 1 y Ω u = for the control u , and � β 2 A 2 α � − 1 y Ω I + γ y = β A − α u = for the state y . 4 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The KKT system ⇒ ( β A − α + γ       β A α ) u = y Ω A α 0 id y y Ω ⇒ p = γ  = 0 γ id − β id u 0 β u      A α − β id 0 0 ⇒ y = β A − α u p Thus, we find the following necessary optimality conditions: � β A − α + γ β A α � − 1 y Ω u = for the control u , and � β 2 A 2 α � − 1 y Ω I + γ y = β A − α u = for the state y . 4 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The KKT system ⇒ ( β A − α + γ       β A α ) u = y Ω A α 0 id y y Ω  = ⇒ p = γ 0 γ id − β id u 0 β u      A α − β id 0 0 ⇒ y = β A − α u p Thus, we find the following necessary optimality conditions: � β A − α + γ β A α � − 1 u = y Ω � �� � G 1 for the control u , and � β 2 A 2 α � − 1 I + γ y = β A − α u = y Ω ���� � �� � G 3 G 2 for the state y . 5 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The spectral fractional Laplacian Let Ω ∈ ❘ d be a bounded Lipschitz domain, and let λ k and e k be the eigenvalues and the corresponding eigenfunctions of the Laplacian, i. e. − ∆ e k = λ k e k in Ω, e k = 0 on ∂Ω, and the functions e k are an orthonormal basis of L 2 ( Ω ). Then, for α ∈ [0 , 1] and a function g ∈ H 1 0 ( Ω ) ∞ � g = a k e k , k =1 we consider the operator ∞ � A α g = a k λ α k e k . k =1 6 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The Riesz fractional Laplacian For α ∈ (0 , 1), the fractional Laplacian ( − ∆) α of a function g : ❘ d → ❘ at a point x ∈ ❘ d is defined by � g ( x ) − g ( y ) ( − ∆) α g ( x ) := C d ,α � x − y � d +2 α d y . ❘ d coincides with A α on ❘ d , cf. details in: Lischke et al. (2018, arXiv:1801.09767) leads to multilevel Toeplitz structures on tensor grids (Ch. Vollmann, V. Schulz, CVS 2019) 7 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

The Riesz fractional Laplacian For α ∈ (0 , 1), the fractional Laplacian ( − ∆) α of a function g : ❘ d → ❘ at a point x ∈ ❘ d is defined by � g ( x ) − g ( y ) ( − ∆) α g ( x ) := C d ,α � x − y � d +2 α d y . ❘ d coincides with A α on ❘ d , cf. details in: Lischke et al. (2018, arXiv:1801.09767) leads to multilevel Toeplitz structures on tensor grids (Ch. Vollmann, V. Schulz, CVS 2019) 7 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

General nonlocal operator � L φ g ( x ) = g ( x ) φ ( x , y ) − g ( y ) φ ( y , x ) dy Ω nonlocal calculus developed by Max Gunzburger et. al. unstructured discretization and shape optimization discussed in Ch. Vollmann: Nonlocal Models with Truncated Interaction Kernels– Analysis, Finite Element Methods and Shape Optimization , PhD dissertation Trier University, 2019 V. Schulz, Ch. Vollmann: Shape optimization for interface identification in nonlocal models , arXiv:1909.08884, 2019 → more details in 2nd RICAM workshop in two weeks... 8 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

Example of nonlocal shape numerics 9 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

A tale of two fractional Laplacians On a bounded domain, the operators are different. Theorem, Servadei/Valdinoci (2014) The operators A α and ( − ∆) α are not the same, since they have different eigenvalues and eigenfunctions (with respect to Dirichlet boundary conditions). In particular, the first eigenvalues of ( − ∆) α is strictly less than that of A α the eigenfunctions of ( − ∆) α are only H¨ older continuous up to the boundary, in contrast with those of A α , which are as smooth up to the boundary as the boundary allows. Lischke et al. (2018, arXiv:1801.09767): Numerical tests for the error between A α and ( − ∆) α . 10 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

A tale of two fractional Laplacians On a bounded domain, the operators are different. Theorem, Servadei/Valdinoci (2014) The operators A α and ( − ∆) α are not the same, since they have different eigenvalues and eigenfunctions (with respect to Dirichlet boundary conditions). In particular, the first eigenvalues of ( − ∆) α is strictly less than that of A α the eigenfunctions of ( − ∆) α are only H¨ older continuous up to the boundary, in contrast with those of A α , which are as smooth up to the boundary as the boundary allows. Lischke et al. (2018, arXiv:1801.09767): Numerical tests for the error between A α and ( − ∆) α . 10 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

Yet another fractional operator R L β := − R D β 1 x 1 − R D β 2 x 2 , β 1 , β 2 ∈ (1 , 2) R D β i x i : 1D Riemann-Liouville derivative This operator is considered in the related publications: S. Dolgov, J. W. Pearson, D. V. Savostyanov, M. Stoll: Fast tensor product solvers for optimization problems with fractional differential equations as constraints , Applied Mathematics and Computation, 2016 T. Breiten, V. Simoncini, M. Stoll: Low-rank solvers for fractional differential equations , ETNA 2016 S. Pougkakiotis, J. W. Pearson, S. Leveque, J. Gondzio: Fast Solution Methods for Convex Fractional Differential Equation Optimization , arXiv:1907.13428, 2019 Note: ( − ∆) α � = A α � = R L (2 α, 2 α ) 11 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

Yet another fractional operator R L β := − R D β 1 x 1 − R D β 2 x 2 , β 1 , β 2 ∈ (1 , 2) R D β i x i : 1D Riemann-Liouville derivative This operator is considered in the related publications: S. Dolgov, J. W. Pearson, D. V. Savostyanov, M. Stoll: Fast tensor product solvers for optimization problems with fractional differential equations as constraints , Applied Mathematics and Computation, 2016 T. Breiten, V. Simoncini, M. Stoll: Low-rank solvers for fractional differential equations , ETNA 2016 S. Pougkakiotis, J. W. Pearson, S. Leveque, J. Gondzio: Fast Solution Methods for Convex Fractional Differential Equation Optimization , arXiv:1907.13428, 2019 Note: ( − ∆) α � = A α � = R L (2 α, 2 α ) 11 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019

Separation of variables and the Laplacian For a function with separated variables, the Laplacian can be applied in one dimension: Let g : (0 , 1) 2 → ❘ , g ( x 1 , x 2 ) = g 1 ( x 1 ) g 2 ( x 2 ) . Then − ∆ g ( x 1 , x 2 ) = − g ′′ 1 ( x 1 ) g 2 ( x 2 ) − g 1 ( x 1 ) g ′′ 2 ( x 2 ) . The case S � g ( j ) 1 ( x 1 ) g ( j ) g ( x 1 , x 2 ) = 2 ( x 2 ) j =1 follows immediately. 12 Volker Schulz, Tensor approach to optimal control problems, October 16, 2019