Model order reduction for PDE constrained optimization in vibrations - PowerPoint PPT Presentation

Model order reduction for PDE constrained optimization in vibrations Karl Meerbergen (Joint work with Yao Yue) KU Leuven SCEE12 – Z¨ urich

Examples of vibrating systems Car tyres Windscreens ◮ Structural damping ◮ Choice of connection (glue) to the car K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 2 / 35

Examples of vibrating systems Planes Maxwell-equation – electrical circuits micro-gyroscope for navigation Bridge vibrating under footsteps systems and Thames wind K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 3 / 35

Model order reduction (MOR) in optimization context Dynamical system: A ( ω, γ ) x = f d T x y = � ω max | y | 2 d ω g = (energy) ω min Objective: minimize g ( γ ) Reduce the cost of computing y and ∇ y : ◮ Model reduction using moment matching = Pad´ e approximation via Krylov methods ◮ Allows for cheap computation of g and ∇ γ g Assume uni-modal objective function K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 4 / 35

Outline Motivation 1 MOR: Krylov-Pad´ e methods 2 Parametric MOR 3 Trust region approach 4 Block Krylov method for low rank parameters 5 Conclusions 6 K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 5 / 35

Left and right Krylov spaces For the linear case: A ( ω ) = I − ω B : Right Krylov space: f , Bf , B 2 f , . . . , B k − 1 f Basis: V k = [ v 1 , . . . , v k ] Left Krylov space: d , B ∗ d , B 2 ∗ d , . . . , B ( k − 1) ∗ d Basis: W k = [ w 1 , . . . , w k ] Reduced model: � � A ( ω ) � x = f � d T � � y = x with � A = W k ∗ AV k , � f = W k ∗ f , � d = V k ∗ d . K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 6 / 35

Moment matching Let A ( ω ) = I − ω B System output: y = d T ( I − ω B ) − 1 f . Theorem Define the expansions: y 0 + ω y 1 + ω 2 y 2 + · · · y = y 1 + ω 2 � � y = y 0 + ω � � y 2 + · · · Let V k and W k be right and left Krylov basis resp., then y j = � y j with j = 0 , 1 , . . . , 2 k − 1 . Rational (Pad´ e) approximation: n k � � � ρ j ρ j y ( ω ) = � y ( ω ) = � λ j − ω λ j − ω j =1 j =1 poles are Ritz values (approximate eigenvalues) K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 7 / 35

Gradient Build two-sided reduced model for fixed value of γ for y = d T A ( ω, γ 0 ) − 1 f Gradient: � dA ( ω, γ 0 ) � � � � ∗ � dy A ( ω, γ 0 ) − 1 f A ( ω, γ 0 ) −∗ d d γ = d γ Blue part can be computed from right-Krylov space: A ( ω, γ 0 ) − 1 f ≈ V k ( � A − 1 ( ω, γ 0 ) � f ) ( k moments matched for A ( ω, γ 0 ) − 1 f .) Red part can be computed from left-Krylov space: A ( ω, γ 0 ) −∗ d ≈ W k ( � A −∗ ( ω, γ 0 ) � d ) ( k moments matched for A ( ω, γ 0 ) −∗ d .) K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 8 / 35

Gradient Full gradient � dA ( ω, γ 0 ) � � � � ∗ � dy A ( ω, γ 0 ) − 1 f d γ = A ( ω, γ 0 ) −∗ d d γ Reduced gradient � ∗ � dA ( ω, γ 0 ) � � � � d � y W k � A ( ω, γ 0 ) −∗ � V k � A ( ω, γ 0 ) − 1 � = d f d γ d γ � � � � � ∗ � d � A ( ω, γ 0 ) A ( ω, γ 0 ) −∗ � � A ( ω, γ 0 ) − 1 � � = d f d γ d γ and dy d � y d γ match k moments. [Antoulas, Beattie, Gugercin 2010] [Yue, M. 2011] Almost free computatation of the gradient, regardless the number of parameters! K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 9 / 35

Implementation issues Linear case: A ( ω ) = A 0 + ω A 1 . ( A 0 + ω A 1 ) x = f Replace by: B = A − 1 0 A 1 , b = A − 1 ( I + ω B ) x = b with 0 f d T x y = Krylov space: sequence of matrix vector multiplies with B Computational cost: ◮ Sparse matrix factorization of A 0 ◮ k sparse matrix vector products with A 1 ◮ k backward solves with A 0 . K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 10 / 35

Nonlinear frequency dependence ( K + i ω C − ω 2 M ) x = f ‘Linearization’: ◮ Define matrices A and B � � � � K iC − M A = B = I I so that � � � � x f ( A − ω B ) = ω x 0 This is called a linearization, a similar trick as the solution of second order ODE’s. K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 11 / 35

Linearizations Higher order polynomials ( A 0 + ω A 1 + · · · + A p ω p ) x = f Methods based on Companion ‘linearization’ [Amiraslani, Corless, Lancaster, 2009] Transform to ( A λ B ) x = 0 −   A 0 A 1 A p − 1   0 − A p    x  · · · · · · · · · I I 0 ω x         ω = 0     .    .  ... ... − . .         . .         ω p − 1 x I I 0 Can also be used for truly nonlinear problems (using Taylor expansion) [Van Beeumen, M., Michiels 2012] K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 12 / 35

Nonlinear output Quadratic output: ( A 0 − ω A 1 ) x = f y = x ∗ Sx with S a low rank matrix. reduced model: ( � A 0 − ω � � A 1 ) � x = f x ∗ � y = � S � x with � A 0 = W T A 0 V , � A 1 = W T A 1 V , � S = V T SV and � f = W T f . V : Krylov space with matrix A − 1 0 A 1 and starting vector A − 1 0 f W : Block-Krylov space with matrix A − 1 0 A 1 and SV . √ Matching between k + k and k + k moments [Van Beeumen, Van Nimmen, Lombaert, M., 2012] K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 13 / 35

Design optimization Determination of optimal parameters of a vibrating system Example: optimal parameters for a damper of a floor in a building near a noisy road k 1 c 1 m 1 K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 14 / 35

Design optimization Parametrized linear system: � ( K ( γ ) + i ω C ( γ ) − ω 2 M ( γ )) x = f d T x y = Find parameters γ so that � ω max ◮ � y � 2 = | y | 2 d ω is minimal 0 | y | 2 is minimal ◮ � y � ∞ = sup ω max 0 Assume: uni-modal, non necessarily smooth Expensive evaluation of y and the gradient K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 15 / 35

Overview γ ( i ) Use MOR for function and gradient evaluation K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 16 / 35

Algorithm We use the Damped BFGS optimization method On iteration i : ◮ Build reduced model for γ ( i ) g ◮ Compute g and ∇ g from the reduced model Objective may not be smooth: use sufficient decrease condition and possibly backtracking after the BFGS step K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 17 / 35

Numerical examples Floor with damper ( n = 29800) Reduced model k = 7 Determine optimal parameters c 1 and k 1 Direct method MOR Matrix size 29800 7 Optimizer computed (12231609 , 106031 . 18) (12231614 , 106031 . 22) 1 . 316093349 10 10 1 . 316093349 10 10 Function value CPU time 7626s 179s K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 18 / 35

Overview γ ( i ) γ ( i ) Use PMOR for Use MOR for line search function and optimization + gradient backtracking evaluation K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 19 / 35

Parametric MOR: PIMTAP Reduced model for equations of the following form: ( G 0 + λ G 1 + s ( C 0 + λ C 1 )) x = b d ∗ x y = Perform moment matching, i.e. terms associated with s i λ j . Algorithms ◮ Many, many papers, e.g. Lihong Feng. ◮ PIMTAP [Li, Bai, Su, Zeng 2007], [Li, Bai, Su, Zeng 2008], [Li, Bai, Su 2009] ◮ Subspace that contains the vectors: r j = 0 i < 0 , or j < 0 i r 0 G − 1 = 0 b 0 r j − G − 1 0 ( C 0 r j i − 1 + G 1 r j − 1 + C 1 r j − 1 = i − 1 ) i i These are the moments in the expansion x ∼ � r j i s i λ j Two-sided PIMTAP ◮ One PIMTAP for b and one for d . K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 20 / 35

Moment matching of the gradient Moment matching patterns for left and right Krylov spaces λ λ ✻ ✻ 4 4 3 3 2 2 1 1 ✲ ✲ s s 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Moment matching pattern for y λ ✻ 4 3 2 1 ✲ s 0 1 2 3 4 5 6 7 8 9 10 K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 21 / 35

Moment matching of the gradient Moment matching pattern for y λ ✻ 4 3 2 1 ✲ s 0 1 2 3 4 5 6 7 8 9 10 Moment matching pattern for ∂ y ∂ s and ∂ y ∂λ λ λ ✻ ✻ 4 4 3 3 2 2 1 1 ✲ ✲ s s 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 22 / 35

The MOR/PMOR Framework We can make a reduced model for ω and all parameters γ This is usually expensive and overkill: only reduce on the important directions So, we use PIMTAP for efficient line search: γ ( i +1) γ ( i ) ◮ The MOR Framework generates a reduced model for each γ accessed. ◮ The PMOR Framework generates a reduced model for each line search iteration. K. Meerbergen (KU Leuven) MOR - Optimization SCEE12 – Z¨ urich 23 / 35