Numerical Analysis of initial data identification of parabolic - PowerPoint PPT Presentation

Numerical Analysis of initial data identification of parabolic problems Dmitriy Leykekhman University of Connecticut Joint work with Boris Vexler (TU M¨ unchen) and Daniel Walter, (RICAM) New trends in PDE constrained optimization RICAM October 14-18, 2019 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 1 / 39

Outline Model Problem 1 Finite element discretization in space and time 2 Main Results 3 Numerical examples 4 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 2 / 39

Model Problem 1 Finite element discretization in space and time 2 Main Results 3 Numerical examples 4 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 3 / 39

Model Problem Homogeneous Parabolic PDEs ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω , u (0) = u 0 in Ω . I = (0 , T ) Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 4 / 39

Model Problem Homogeneous Parabolic PDEs ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω , u (0) = u 0 in Ω . I = (0 , T ) Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral Goal: Recover initial data u 0 from the final time observation u ( T ) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 4 / 39

Homogeneous Parabolic PDEs Main difficulty: backward parabolic problem is exponentially ill-posed! 1 0.9 0.8 0.7 0.6 x 2 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 5 / 39

Homogeneous Parabolic PDEs Main difficulty: backward parabolic problem is exponentially ill-posed! 1 0.9 0.8 5 0.7 4 0.6 x 2 3 0.5 0.4 2 0.3 1 1 0.2 0 1 0.1 0.8 0.5 0.6 0.4 0 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 x 1 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 5 / 39

Homogeneous Parabolic PDEs Difficulty Backward parabolic is exponentially ill-posed! Easy to see, thus in 1D ( Ω = (0 , 1) ) simple separation of variables u ( x, t ) = V ( t ) W ( x ) gives ∞ c n e − π 2 n 2 t sin ( πnx ) , � u ( x, t ) = n =1 where c n has to be find out from final time condition ∞ c n e − π 2 n 2 T sin ( πnx ) � u ( x, T ) = n =1 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 6 / 39

Homogeneous Parabolic PDEs Difficulty Backward parabolic is exponentially ill-posed! Easy to see, thus in 1D ( Ω = (0 , 1) ) simple separation of variables u ( x, t ) = V ( t ) W ( x ) gives ∞ c n e − π 2 n 2 t sin ( πnx ) , � u ( x, t ) = n =1 where c n has to be find out from final time condition ∞ c n e − π 2 n 2 T sin ( πnx ) � u ( x, T ) = n =1 Additional assumption Let u 0 be sparse, i.e. u 0 = � M j =1 a j δ x j ( x ) , � where δ x j ( x ) are delta functions, i.e. Ω f ( x ) δ x j ( x ) dx = f ( x j ) . D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 6 / 39

Initial value control of parabolic PDEs Optimal control problem Minimize J ( q, u ) = 1 2 � u ( T ) − u d � 2 L 2 (Ω) + α � q � X , q ∈ X, u ∈ V subject to the state equation ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω u (0) = q in Ω . I = (0 , T ) , Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral u d ∈ L 2 (Ω) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 7 / 39

Initial value control of parabolic PDEs Optimal control problem Minimize J ( q, u ) = 1 2 � u ( T ) − u d � 2 L 2 (Ω) + α � q � X , q ∈ X, u ∈ V subject to the state equation ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω u (0) = q in Ω . I = (0 , T ) , Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral u d ∈ L 2 (Ω) X = L 2 (Ω) Setting I → We can not recover accurately q from u ( T ) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 7 / 39

Initial value control of parabolic PDEs Optimal control problem Minimize J ( q, u ) = 1 2 � u ( T ) − u d � 2 L 2 (Ω) + α � q � X , q ∈ X, u ∈ V subject to the state equation ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω u (0) = q in Ω . I = (0 , T ) , Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral u d ∈ L 2 (Ω) X = L 2 (Ω) Setting I → We can not recover accurately q from u ( T ) Setting II X = M (Ω) → q has a sparse structure and we can (hopefully) recover q from u ( T ) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 7 / 39

Look for q in the space of regular Borel measures M (Ω) = ( C 0 (Ω)) ∗ Optimal control problem 2 � u ( T ) − u d � 2 q ∈M (Ω) J ( q, u ) = 1 min L 2 (Ω) + α � q � M , subject to ∂ t u − ∆ u = 0 in I × Ω , u (0) = q in Ω . D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 8 / 39

Look for q in the space of regular Borel measures M (Ω) = ( C 0 (Ω)) ∗ Optimal control problem 2 � u ( T ) − u d � 2 q ∈M (Ω) J ( q, u ) = 1 min L 2 (Ω) + α � q � M , subject to ∂ t u − ∆ u = 0 in I × Ω , u (0) = q in Ω . Equivalent problem q ∈M J ( q ) = � q � M , min subject to ∂ t u − ∆ u = 0 in I × Ω , u (0) = q in Ω and � u ( T ) − u d � L 2 (Ω) ≤ ε D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 8 / 39

Literature Measure valued control for elliptic equations: C. Clason and K. Kunisch, 2011 E. Casas, C. Clason, and K. Kunisch, 2012 K. Pieper and B. Vexler, 2013 E. Casas and K. Kunisch, 2013 Measure valued control for parabolic equations: E. Casas, and K. Kunisch, 2016 E. Casas, C. Clason, and K. Kunisch, 2013 K. Kunisch, K. Pieper, and B. Vexler, 2014 Measure valued control for initial data identification: E. Casas and E. Zuazua, 2013 E. Casas, B. Vexler, and E. Zuazua 2015 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 9 / 39

Literature Measure valued control for elliptic equations: C. Clason and K. Kunisch, 2011 E. Casas, C. Clason, and K. Kunisch, 2012 with FEM analysis K. Pieper and B. Vexler, 2013 with FEM analysis E. Casas and K. Kunisch, 2013 Measure valued control for parabolic equations: E. Casas, and K. Kunisch, 2016 E. Casas, C. Clason, and K. Kunisch, 2013 with FEM analysis K. Kunisch, K. Pieper, and B. Vexler, 2014 with FEM analysis Measure valued control for initial data identification: E. Casas and E. Zuazua, 2013 E. Casas, B. Vexler, and E. Zuazua 2015 D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 9 / 39

Literature Measure valued control for elliptic equations: C. Clason and K. Kunisch, 2011 E. Casas, C. Clason, and K. Kunisch, 2012 with FEM analysis K. Pieper and B. Vexler, 2013 with FEM analysis E. Casas and K. Kunisch, 2013 Measure valued control for parabolic equations: E. Casas, and K. Kunisch, 2016 E. Casas, C. Clason, and K. Kunisch, 2013 with FEM analysis K. Kunisch, K. Pieper, and B. Vexler, 2014 with FEM analysis Measure valued control for initial data identification: E. Casas and E. Zuazua, 2013 E. Casas, B. Vexler, and E. Zuazua 2015 with FEM analysis, no error estimates D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 9 / 39

Existence and uniqueness Optimal control problem 2 � u ( T ) − u d � 2 q ∈M (Ω) J ( q, u ) = 1 min L 2 (Ω) + α � q � M , subject to ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω , u (0) = q in Ω . I = (0 , T ) , Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral u d ∈ L 2 (Ω) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 10 / 39

Existence and uniqueness Optimal control problem 2 � u ( T ) − u d � 2 q ∈M (Ω) J ( q, u ) = 1 min L 2 (Ω) + α � q � M , subject to ∂ t u − ∆ u = 0 in I × Ω , u = 0 on I × ∂ Ω , u (0) = q in Ω . I = (0 , T ) , Ω ⊂ R d ( d = 2 , 3 ) convex and polygonal/polyhedral u d ∈ L 2 (Ω) ⇒ For each q ∈ M (Ω) there is u = S ( q ) with u ∈ L r ( I ; W 1 ,p (Ω)) with r, p ∈ [1; 2) and 2 r + d p > d + 1 and u ( T ) ∈ L 2 (Ω) D. Leykekhman (University of Connecticut) Numerical Analysis of initial data identification of parabolic problems New trends 10 / 39