Lecture 1 Model Reduction Methods Martin A. Grepl 1 , Gianluigi Rozza 2 1 IGPM, RWTH Aachen University, Germany 2 MATHICSE - CMCS, Ecole Polytechnique Fédérale de Lausanne, Switzerland Summer School "Optimal Control of PDEs" Cortona (Italy), July 12-17, 2010 Grepl, Rozza Model Reduction Methods
Lecture 1 Acknowledgements & Sponsors Acknowledgements A.T. Patera K. Veroy D. B. P. Huynh A. Manzoni C. N. Nguyen Sponsors ◮ AFOSR, DARPA ◮ Swiss National Science Foundation ◮ European Research Council ◮ Singapore-MIT Alliance ◮ Progetto Rocca Politecnico di Milano-MIT ◮ German Excellence Initiative Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Focus Model Order Reduction by Reduced Basis Method for the efficient resolution of parametrized PDEs Examples in heat and mass transfer, linear elasticity, potential and viscous flows Some Pre-requisites Numerical Analysis, FEM, PDEs, Physical Mathematics Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) References and materials ( http://www.mat.uniroma1.it/cortona10/courses.html ): ◮ Rozza G., Huynh D.B.P., Patera A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch Comput Methods Eng (2008) 15: 229-275 ◮ Patera A.T., Rozza G., Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006–2010 ◮ Rozza G., Nguyen N.C., Huynh D.B.P., Patera A.T., Real-Time Reliable Simulation of Heat Transfer Phenomena, Proceedings of HT2009 ASME Summer Heat Transfer Conference, paper HT2009-88212 Links : ◮ http://augustine.mit.edu/methodology/methodology/.. _rbMIT_System.htm , Matlab Software, rbMIT (C)MIT Library ◮ http://augustine.mit.edu/methodology/methodology_book.htm First part of RB book, A.T. Patera, G. Rozza (C)MIT ◮ http://augustine.mit.edu/workedProblems.htm Worked problems, examples, Webserver (C)MIT Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Outline ◮ Lecture 1: Motivation, Coercive Elliptic Problems 1. Introduction/Motivation (a) Notation and Examples (b) Goal/Relevance 2. Elliptic Problems I (coercive, affine, compliant) (a) Problem Statement, Truth Approximation, Affine Representation (b) Reduced Basis Approximation (c) Offline-Online Computational Procedures (d) Sampling/Spaces Strategies: POD, Greedy, ... Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Outline ◮ Lecture 2: Elliptic Problems II, Parabolic Problems 1. Elliptic Problems II (e) A Posteriori Error Estimation (elements) (f) General Outputs (non-compliant), Non-symmetric Forms (Dual Problem, A Posteriori Error Estimation) 2. Parabolic Problems (a) Problem Statement, Truth Approximation (b) Reduced Basis Approximation (c) Offline-Online Computational Procedures (d) A Posteriori Error Estimation Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Outline ◮ Lecture 3: Parabolic problems, Non-affine Problems, Software 1. Parabolic Problems (d) A Posteriori Error Estimation (e) Offline-Online Decomposition (f) POD/Greedy Sampling (g) NOn-symmetric problems 2. Non-Affine Problems (a) Empirical Interpolation Method (b) EIM + RB 3. Summary on Software: RB@MIT ◮ Lecture 4: Applied Talk 1. Optimization & Optimal Control ◮ Parameter Optimization, GMA Welding Process, Advection-Diffusion (Environmental and thermal) 2. Shape Optimization ◮ Potential, thermal, (Navier)-Stokes flows Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Lecture 1 ◮ Lecture 1: Motivation, Coercive Elliptic Problems 1. Introduction/Motivation (a) Notation and Examples (b) Goal/Relevance 2. Elliptic Problems I (coercive, affine, compliant) (a) Problem Statement, Truth Approximation, Affine Representation (b) Reduced Basis Approximation (c) Offline-Online Computational Procedures (d) Sampling/Spaces Strategies: POD, Greedy, ... Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Statement: simple elliptic µ PDEs R P , evaluate s e ( µ ) = ℓ ( u e ( µ )) † Given µ ∈ D ⊂ I where u e ( µ ) ∈ X e satisfies ∀ v ∈ X e . a ( u e ( µ ) , v ; µ ) = f ( v ) , µ : input parameter; P -tuple D : input domain; s e : output; ℓ : linear bounded output functional; u e : field variable; 0 (Ω)) ν ⊂ X e ⊂ ( H 1 (Ω)) ν ; X e : function space ( H 1 † Here e refers to “exact.” Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Statement: hypotheses and definitions a ( · , · ; µ ): bilinear, continuous, symmetric, µ PDE coercive form, ∀ µ ∈ D ; f : linear bounded functional. Compliant case: l = f , a ( · , · ; µ ) symmetric Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Statement: hypotheses and definitions ◮ a symmetric: a ( u, v ; µ ) = a ( v, u ; µ ) , ◮ a bilinear: a ( λu + γv, w ; µ ) = λa ( u, w ; µ ) + γa ( v, w ; µ ) , ∀ λ, γ ∈ R , ∀ u, v, w ∈ X e , or a ( u, λv + γw ; µ ) = λa ( u, v ; µ ) + γa ( u, w ; µ ) , ∀ λ, γ ∈ R , ∀ u, v, w ∈ X e , ◮ a continuous: | a ( u, v ; µ ) | ≤ M || u || X e || v || X e , ∀ u, v ∈ X e , ◮ a coercive: ∃ α > 0 : a ( u, u ; µ ) ≥ α e || u || 2 X e , ∀ u ∈ X e , ◮ f (and l ) bounded/continuous: | f ( v ) | ≤ C || v || X e , ∀ v ∈ X e , ◮ f linear: f ( γv + ηw ) = γf ( v ) + ηf ( w ) , ∀ γ, η ∈ R , v ∈ X e . Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Statement: affine parameter dependence † Definition : Q � Θ q ( µ ) a q ( w, v ) a ( w, v ; µ ) = q =1 for q = 1 , . . . , Q Θ q : D → I µ -dependent functions R , X e × X e → I a q : µ - independent forms R . Stiffness matrix : Q � Θ q ( µ ) a q ( w a ( w , v ; µ ) = , v ) q =1 ζ j ζ i ζ j ζ i for q = 1 , . . . , Q † In fact, broadly applicable to many instances of geometry and property parametric variation. Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Little example: heat conduction � 1 u e Given k I ∈ [0 . 1 , 10] , evaluate u e I ( k I ) = | Ω I | Ω I where u e ( k I ) ∈ H 1 0 (Ω) satisfies ( Q = 2 ) � � � ∇ u e · ∇ v + ∇ u e · ∇ v = v, ∀ v ∈ H 1 k I 0 (Ω) . Ω I Ω II Ω X N ≡ { v | T h ∈ I P I ( T h ) , ∀ T h ∈ T h } ∩ X e ; dim( X N ) ≡ N . Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Classical Approximation: FEM (Galerkin projection) Given µ ∈ D , evaluate s N = ℓ ( u N ( µ )) , where u N ( µ ) ∈ X N satisfies ∀ v ∈ X N . a ( u N ( µ ) , v ; µ ) = f ( v ) , Typically: | s e ( µ ) − s N ( µ ) | small ⇒ N large . Surrogate for s e ( µ ) , u e ( µ ) : “truth” ◮ upon which we build reduced-basis approximation ; † ◮ relative to which we measure reduced-basis error . † † Require stability and efficiency as N → ∞ . Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Reduced-Basis Approximation: basic idea † M e = parameter-induced manifold R P ) , very smooth) (low-Dimensional ( D ⊂ I Classical Approach X N ≡ { v | T h ∈ I P I ( T h ) , ∀ T h ∈ T h } ∩ X e ; dim( X N ) ≡ N . Reduced Basis Approach W N ≡ span { ζ n ≡ u N ( µ n ) , 1 ≤ n ≤ N } † Pioneering works by Almroth, Stern& Brogan (1978), Noor & Peters (1980). Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Reduced-Basis Approximation: formulation Samples : S N = { µ 1 ∈ D , . . . , µ N ∈ D} Spaces : W N = span { ζ n ≡ u N ( µ n ) , 1 ≤ n ≤ N } µ ∈ D , Given evaluate s N ( µ ) = ℓ ( u N ( µ )) , u N ( µ ) ∈ W N satisfies where a ( u N ( µ ) , v ; µ ) = f ( v ) , ∀ v ∈ W N . Grepl, Rozza Model Reduction Methods
Introduction/Motivation Lecture 1 Elliptic Problems I (coercive, affine, compliant) Reduced-Basis Approximation: convergence Classical arguments yield a ( u N ( µ ) − u N ( µ ) , u N ( µ ) − u N ( µ ); µ ) = w N ∈ W N a ( u N ( µ ) − w N , u N ( µ ) − w N ; µ ) inf Properties of M e suggest w N ∈ W N a ( u N ( µ ) − w N , u N ( µ ) − w N ; µ ) → 0 inf rapidly (exponentially): N small . Grepl, Rozza Model Reduction Methods
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