Model reduction for multiscale problems Mario Ohlberger wissen - PowerPoint PPT Presentation

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Model reduction for multiscale problems Mario Ohlberger wissen leben Dec. 12-16, 2011 RICAM, Linz WWU Münster

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Outline Motivation: Multi-Scale and Multi-Physics Problems Model Reduction: The Reduced Basis Approach WWU Münster wissen leben A new Reduced Basis DG Multiscale Method , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Outline Motivation: Multi-Scale and Multi-Physics Problems Model Reduction: The Reduced Basis Approach WWU Münster wissen leben A new Reduced Basis DG Multiscale Method , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Example: PEM fuel cells Pore Cell Stack System WWU Münster wissen leben [BMBF-Project PEMDesign: Fraunhofer ITWM and Fraunhofer ISE] , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Security behavior of nuclear waste disposals WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Example: Hydrological Modeling WWU Münster wissen leben [BMBF-Project AdaptHydroMod: Wald & Corbe, Hügelsheim ] , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Real World Problem Continuous Mathematical Model ◮ Here: system of partial differential equations ◮ Problem: infinite dimensional solution space ◮ no solutions in closed form WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Continuous Mathematical Model Discretization!! WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Continuous Mathematical Model Discrete model on uniform grid (FEM, FV, DG, ...) ◮ Typical error estimates: || u − u h || ≤ c inf || u − v h || WWU Münster v h ∈ X h wissen leben ◮ Error related to approximation property of X h ◮ = ⇒ Very general approach, but in particular cases not very efficient!! , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Continuous Mathematical Model WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Continuous Mathematical Model Problem specific: Adaptive Mesh Refinement ◮ Typical error estimates: || u − u h || ≤ c η ( u h ) WWU Münster wissen leben ◮ Error related to approximate solution! ◮ = ⇒ Construct optimal mesh! ◮ Problem: Grid construction for every solve! Resulting system is still high-dimensional! , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Error Control and Adaptivity for HMM HMM for linear elliptic homogenization problems [Ohlberger: Multiscale Model. Simul., 2005] [Henning, Ohlberger: Numer. Math., 2009] HMM for multi-scale transport with large expected drift [Henning, Ohlberger: Netw. Heterog. Media. 2010] [Henning, Ohlberger: J. Anal. Appl. 2011] WWU Münster wissen leben HMM for nonlinear monotone elliptic problems [Henning, Ohlberger 2011] = ⇒ see poster (8) at this workshop , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Mathematical Modelling and Model Reduction Continuous Mathematical Model Problem class specific: Reduced Basis Method ◮ Typical error estimates: || ( u − u N )( µ ) || ≤ c η ( u N ( µ )) WWU Münster wissen leben ◮ Error related to reduced solution! ◮ = ⇒ Construct optimal reduced space for problem class!! Resulting system is low dimensional! , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Outline Motivation: Multi-Scale and Multi-Physics Problems Model Reduction: The Reduced Basis Approach WWU Münster wissen leben A new Reduced Basis DG Multiscale Method , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Reduced Basis Method for Evolution Equations Goal: Fast “Online”-Simulation of Complex Evolution Systems for • Parameter/Design Optimization • Optimal Control • Integration into System Simulation • Uncertainty Quantification Ansatz: • Reduced Basis Method (RB) WWU Münster dim ( W N ) < < dim ( W H ) ! wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Reduced Basis Method for Evolution Equations Goal: Fast “Online”-Simulation of Complex Evolution Systems for • Parameter/Design Optimization • Optimal Control • Integration into System Simulation • Uncertainty Quantification Ansatz: • Reduced Basis Method (RB) WWU Münster dim ( W N ) < < dim ( W H ) ! wissen leben Classical references: notation RB [Noor, Peters ’80], initial value problems [Porsching, Lee ’87], method [Nguyen et al. ’05], book [Patera, Rozza ’07], http://augustine.mit.edu, http://morepas.org , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Model Reduction: Reduced Basis Method Goal: Find c ( · , t ; µ ) ∈ L 2 (Ω) for t ∈ [ 0 , T ] , µ ∈ P ⊂ ❘ p with ∂ t c ( µ ) + L µ ( c ( µ )) = 0 in Ω × [ 0 , T ] , plus suitable Initial and Boundary Conditions. FV/DG Approximation c H ( µ ) ∈ W H for given Parameter µ Assumption: WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Model Reduction: Reduced Basis Method Goal: Find c ( · , t ; µ ) ∈ L 2 (Ω) for t ∈ [ 0 , T ] , µ ∈ P ⊂ ❘ p with ∂ t c ( µ ) + L µ ( c ( µ )) = 0 in Ω × [ 0 , T ] , plus suitable Initial and Boundary Conditions. FV/DG Approximation c H ( µ ) ∈ W H for given Parameter µ Assumption: Ansatz (RB): Define low dimensional Subspace W N ⊂ W H WWU Münster wissen leben and project FV/DG Scheme onto the Subspace RB Approximation c N ( µ ) ∈ W N . = ⇒ , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Model Reduction: Reduced Basis Method Goal: Find c ( · , t ; µ ) ∈ L 2 (Ω) for t ∈ [ 0 , T ] , µ ∈ P ⊂ ❘ p with ∂ t c ( µ ) + L µ ( c ( µ )) = 0 in Ω × [ 0 , T ] , plus suitable Initial and Boundary Conditions. FV/DG Approximation c H ( µ ) ∈ W H for given Parameter µ Assumption: Ansatz (RB): Define low dimensional Subspace W N ⊂ W H WWU Münster wissen leben and project FV/DG Scheme onto the Subspace RB Approximation c N ( µ ) ∈ W N . = ⇒ Requirement: • Efficient Choice of W N (Exponential Convergence in N) • Offline–Online Decomposition for all Calculations • Error Control for || c H ( µ ) − c N ( µ ) || , , M. Ohlberger Model reduction for multiscale problems

Institute for W ESTFÄLISCHE Computational and W ILHELMS -U NIVERSITÄT Applied Mathematics M ÜNSTER > Model Reduction: Reduced Basis Method Assumption: FV/DG Scheme for Evolution Equations c 0 L k I ( µ )[ c k + 1 ( µ )] = L k E ( µ )[ c k H ( µ )] + b k ( µ ) . H = P [ c 0 ( µ )] , H with time step counter k and c k H ( µ ) ∈ W H . WWU Münster wissen leben , , M. Ohlberger Model reduction for multiscale problems