Mixed-Integer PDE-Constrained Optimization Frontiers in PDE-constrained Optimization Pelin Cay, Bart van Bloemen Waanders, Drew Kouri, Anna Thuenen and Sven Leyffer Lehigh University, Universit¨ at Magdeburg, Argonne National Laboratory, and Sandia National Laboratories June 8, 2016
Outline Introduction 1 Problem Definition and Applications Theoretical and Computational Challenges Early Numerical Results 2 Source Inversion as MIP with PDE Constraints Eliminating the PDE and States Control of Heat Equation 3 Design and Operation of Actuators Sum-Up Rounding Heuristic for Time-Dependent Controls Conclusions 4 2 / 32
Mixed-Integer PDE-Constrained Optimization (MIPDECO) PDE-constrained MIP ... u = u ( t , x , y , z ) ⇒ infinite-dimensional! t is time index; x , y , z are spatial dimensions minimize F ( u , w ) u , w subject to C ( u , w ) = 0 u ∈ U , and w ∈ Z p (integers) , u ( t , x , y , z ): PDE states, controls, & design parameters w discrete or integral variables MIPDECO Warning w = w ( t , x , y , z ) ∈ Z may be infinite-dimensional integers! 3 / 32
Mixed-Integer PDE-Constrained Optimization (MIPDECO) PDE-constrained MIP ... u = u ( t , x , y , z ) ⇒ infinite-dimensional! t is time index; x , y , z are spatial dimensions minimize F ( u , w ) u , w subject to C ( u , w ) = 0 u ∈ U , and w ∈ Z p (integers) , u ( t , x , y , z ): PDE states, controls, & design parameters w discrete or integral variables MIPDECO Warning w = w ( t , x , y , z ) ∈ Z may be infinite-dimensional integers! Oh my God, alien MIPs! 3 / 32
Grand-Challenge Applications of MIPDECO Topology optimization [Sigmund and Maute, 2013] Nuclear plant design: select core types & control flow rates [Committee, 2010] Well-selection for remediation of contaminated sites [Ozdogan, 2004] Design of next-generation solar cells [Reinke et al., 2011] Design of wind-farms [Zhang et al., 2013] Scheduling for disaster recovery: oil-spills [You and Leyffer, 2010] & wildfires [Donovan and Rideout, 2003] Design & control of gas networks, [De Wolf and Smeers, 2000, Martin et al., 2006, Zavala, 2014] ... any more applications! 4 / 32
Mesh-Independent & Mesh-Dependent Integers Definition (Mesh-Independent & Mesh-Dependent Integers) 1 The integer variables are mesh-independent, iff number of integer variables is independent of the mesh. 2 The integer variables are mesh-dependent, iff the number of integer variables depends on the mesh. Mesh-Independent Mesh-Dependent Manageable tree Exploding tree Theory possible Theory??? 5 / 32
Theoretical Challenges of MIPDECO Functional Analysis (mesh-dependent integers) Denis Ridzal: What function space is w ( x , y ) ∈ { 0 , 1 } ? Consistently approximate w ( x , y ) ∈ { 0 , 1 } as h → 0? Conjecture: { w ( x , y ) ∈ { 0 , 1 }} � = L 2 (Ω) ... e.g. binary support of Cantor set not integrable Likely need additional regularity assumptions Coupling between Discretization & Integers Discretization scheme (e.g. upwinding for wave equation) depends on direction of flow (integers). Application: gas network models with flow reversals 6 / 32
Computational Challenges of MIPDECO Approaches for humongous branch-and-bound trees ... e.g. 3D topology optimization with 10 9 binary variables Warm-starts for PDE-constrained optimization (nodes) Guarantees for nonconvex (nonlinear) PDE constraints ... factorable programming approach hopeless for 10 9 vars! + * ^ log 3 x x 2 1 ... f ( x 1 , x 2 ) = x 1 log( x 2 ) + x 3 x 2 2 7 / 32
MIPDECO: Two Cartoons Collide Observation Mixed-Integer and PDE-optimization developed separately ⇒ different assumptions, methodologies, & computational kernels! Mixed-Integer Programming PDE-Optimization Deliver certificate of optimality Obtain good solutions efficiently Branch-and-Cut Newton’s method Factors & rank-one updates Iterative Krylov solvers Potential for Disaster, or Opportunity for Innovation! 8 / 32
Outline Introduction 1 Problem Definition and Applications Theoretical and Computational Challenges Early Numerical Results 2 Source Inversion as MIP with PDE Constraints Eliminating the PDE and States Control of Heat Equation 3 Design and Operation of Actuators Sum-Up Rounding Heuristic for Time-Dependent Controls Conclusions 4 9 / 32
Source Inversion as MIP with PDE Constraints Simple Example: Locate number of sources to match observation ¯ u J = 1 � u ) 2 d Ω minimize ( u − ¯ least-squares fit 2 u , w Ω � subject to − ∆ u = w kl f kl in Ω Poisson equation k , l � w kl ≤ S and w kl ∈ { 0 , 1 } source budget k , l with Dirichlet boundary conditions u = 0 on ∂ Ω. E.g. Gaussian source term, σ > 0, centered at ( x k , y l ) � −� ( x k , y l ) − ( x , y ) � 2 � f kl ( x , y ) := exp , σ 2 Motivated by porous-media flow application to determine number of boreholes, [Ozdogan, 2004, Fipki and Celi, 2008] 10 / 32
Source Inversion as MIP with PDE Constraints Consider 2D example with Ω = [0 , 1] 2 and discretize PDE: 5-point finite-difference stencil; uniform mesh h = 1 / N Denote u i , j ≈ u ( ih , jh ) approximation at grid points N J h = h 2 � u i , j ) 2 minimize ( u i , j − ¯ 2 u , w i , j =0 N subject to 4 u i , j − u i , j − 1 − u i , j +1 − u i − 1 , j − u i +1 , j � = w kl f kl ( ih , jh ) h 2 k , l =1 u 0 , j = u N , j = u i , 0 = u i , N = 0 N � w kl ≤ S and w kl ∈ { 0 , 1 } k , l =1 ... finite-dimensional (convex) MIQP! 11 / 32
Mesh-Independent Source Inversion Potential source locations (blue dots) on 16 × 16 mesh Create target ¯ u using red square sources 12 / 32
Source Inversion as MIP with PDE Constraints Contours of TARGET ubar(x,y) Relative error in States ubar(x,y) - u(x,y) & Source Location 1 1 0.02 0.15 0.9 0.9 0.018 0.8 0.8 0.1 0.016 0.7 0.7 0.014 0.6 0.6 0.05 0.012 0.5 0.5 0.01 0 0.4 0.4 0.008 0.3 0.3 0.006 -0.05 0.2 0.004 0.2 0.1 0.002 0.1 -0.1 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Target (3 sources), reconstructed sources, & error on 32 × 32 mesh 13 / 32
Mesh-Independent Source Inversion: MINLP Solvers Number of Nodes and CPU time for Increasing Mesh Sizes 10 5 10 5 10 4 10 4 CPU Time 10 3 Nodes 10 2 10 3 10 1 BonminOA BonminOA MINLP MINLP Minotaur Minotaur 10 2 10 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Mesh-Size Mesh-Size Number of Nodes independent of mesh size! MINLP & Minotaur: filterSQP runs out of memory for N ≥ 32 BonminOA takes roughly 100 iterations ... quadratic objective 14 / 32
Mesh-Dependent Source Inversion: MINLP Solvers Number of Nodes and CPU time for Increasing Mesh Sizes 10 8 10 7 10 6 10 6 10 5 CPU Time 10 4 Nodes 10 4 10 2 10 3 10 0 BonminOA BonminOA 10 2 BonminBB BonminBB MINLP MINLP Minotaur Minotaur 10 1 10 -2 4 6 8 10 12 14 16 4 6 8 10 12 14 16 Mesh-Size Mesh-Size Number of nodes & CPU time explodes with mesh size! OA <BREAK> after 130,000 seconds ... stress test for solvers! 15 / 32
Mesh-Dependent MIPDECOs are Really Tough MIPDECO trees become humongous ... ... and unmanageable. 16 / 32
MIPDECO Trick # 1: Eliminating the PDE Discretized PDE constraint (Poisson equation) 4 u i , j − u i , j − 1 − u i , j +1 − u i − 1 , j − u i +1 , j � = w kl f kl ( ih , jh ) , ∀ i , j h 2 k , l ⇔ A u = � w kl f kl , where w kl ∈ { 0 , 1 } only appear on RHS! Elimination of PDE and states u ( x , y , z ) � � � w kl f kl ⇔ u = A − 1 = w kl A − 1 f kl A u = w kl f kl k , l k , l k , l Solve n 2 ≪ 2 n PDEs: u ( kl ) := A − 1 f kl k , l w kl u ( kl ) Eliminate u = � 17 / 32
MIPDECO Trick # 1: Eliminating the PDE k , l w kl u ( kl ) in MINLP: Eliminate u = � 2 N J h = h 2 w kl u ( kl ) � � minimize − ¯ u i , j ij 2 w i , j =0 k , l N � subject to w kl ≤ S and w kl ∈ { 0 , 1 } k , l =1 Eliminates the states u ( N 2 variables) Eliminates the PDE constraint ( N 2 constraints) ... generalizes to other PDEs (with integer controls on RHS) Simplified model is quadratic knapsack problem 18 / 32
Elimination of States & PDEs: Source Inversion CPU Time for Increasing Mesh Sizes: Simplified vs. Original Model 10 4 10 3 10 2 CPU Time 10 1 10 0 10 -1 MINLP MINLP-Simple 10 -2 5 10 15 20 25 30 35 Mesh-Size Eliminating PDEs is two orders of magnitude faster! 19 / 32
Outline Introduction 1 Problem Definition and Applications Theoretical and Computational Challenges Early Numerical Results 2 Source Inversion as MIP with PDE Constraints Eliminating the PDE and States Control of Heat Equation 3 Design and Operation of Actuators Sum-Up Rounding Heuristic for Time-Dependent Controls Conclusions 4 20 / 32
Actuator Placement and Operation [Falk Hante] Goal: Control temperature with actuators Select sequence of control inputs (actuators) Choose continuous control (heat/cool) at locations Match prescribed temperature profile ... “de-mist bathroom mirror with hair-drier” Potential Actuator Locations l = 1 , . . . , L 21 / 32
Recommend
More recommend
