Parametrische Modellreduktion mit dünnen Gittern

Transcription

1 Parametrische Modellreduktion mit dünnen Gittern (Parametric model reduction with sparse grids) Ulrike Baur Peter Benner Mathematik in Industrie und Technik, Fakultät für Mathematik Technische Universität Chemnitz GMA FA.3 Workshop Anif/Salzburg, 26. September 28 Projekt mit J. Korvink, IMTEK (Universität Freiburg) Automatic, Parameter-Preserving Model Reduction for Applications in Microsystems Technology /7 U. Baur P. Benner Parametric model reduction with sparse grids

2 Outline Parametric model reduction with sparse grids Parametric linear systems Balanced Truncation/Interpolatory MOR Use of sparse grids Numerical results Outlook 2/7 U. Baur P. Benner Parametric model reduction with sparse grids

3 Motivation - parametric model reduction Parametric systems appear in many applications, e.g. MEMS design - optimization of geometry and topology Example from Oberwolfach Benchmark collection thermal conduction model: film coefficients {p i } 3 i= describe heat exchange on interfaces 3 Eẋ(t) = (A + p i A i )x(t) + Bu(t), i= y(t) = C T x(t) preserve the parameters in reduced-order system! recent approaches: multivariate moment matching, e.g. [Benner/Feng 7, Bond/Daniel 5, Daniel et al. 4, Eid et al. 7, Farle et al. 6, Feng 5, Moosmann/Korvink 6, Weile et al. 99] 3/7 U. Baur P. Benner Parametric model reduction with sparse grids

4 Parametric linear systems Consider parametric linear systems with p R d : ẋ(t) = A(p) x(t) + B(p) u(t) y(t) = C(p) T x(t) (A(p),B(p),C(p)) R n n R n m R n q and m,q n First: single parameter case p [a, b] ẋ(t) = A(p)x(t) + B u(t) y(t) = C T x(t) with and A(p) = A + p A G(s,p) = C T (si n (A + p A )) B 4/7 U. Baur P. Benner Parametric model reduction with sparse grids

5 Balanced Truncation/Interpolatory MOR Choose interpolation points p,...,p k [a, b] 2 Compute reduced-order systems by balanced truncation (BT): Ĝ j (s) = ĈT j (si rj Âj) ˆBj for j =,...,k 3 Parametric reduced-order system by interpolation: with k Ĝ(s,p) = l j (p)ĝj(s) j= Ĝ(s,p j ) = Ĝj(s) for j =,...,k 5/7 U. Baur P. Benner Parametric model reduction with sparse grids

6 2 Balanced Truncation For G j (s) := G(s,p j ) = C T (si n (A + p j A )) B compute reduced-order systems by BT: Ĝ j (s) = Ĉ T j (si rj  j ) ˆB j for j =,...,k BT preserves stability, computable H -error bound: n G j Ĝj 2 < tol i=r j + For numerical solution of Lyapunov equations: LR-ADI [Penzl, Li/White 2], Smith [Penzl, Antoulas/Gugercin/Sor. 3], Krylov [Jaimoukha/Kasenally 94, Saad 9, Simoncini 7], sign function method [Benner/Quintana-Ortí 99, Baur 8] 6/7 U. Baur P. Benner Parametric model reduction with sparse grids σ i

7 3 Interpolatory MOR Parametric reduced-order system by interpolation: k Ĝ(s,p) = l j (p)ĝj(s) = Ĉ (p). Ĉ k (p) = T j= k j= k i=,i j p p i p i p j (si r  )... Ĉ T j (si rj Âj) ˆBj (si rk Âk) But: for high-dimensional parameter space many interpolation points many times BT, i.e. very high complexity! ˆB. ˆB k 7/7 U. Baur P. Benner Parametric model reduction with sparse grids

8 Sparse grids [Zenger 9, Griebel 9, Bungartz 92] On [, ] construct equidistant grid with mesh size h l = 2 l and associated (2 l )-dim. space S l of piecewise linear functions. Hierarchical basis decomposition [Yse86]: S l = T T l Use subspaces of S l : For f C 2 ([, ]) and interpolant f I S l l f I = f i, f i T i, i= the interpolation error is bounded T T 2 T 3 f f I O(h 2 l ) and f i 2 4 i 2 f x 2. 8/7 U. Baur P. Benner Parametric model reduction with sparse grids

9 Sparse grids [Zenger 9, Griebel 9, Bungartz 92] On [, ] d construct grid with mesh size h l (i := (i,...,i d ) N d ). For f : [, ] d R, 2d f x 2... x2 d C ([, ] d ) search interpolant f I in space of piecewise d-linear functions: dimension full grid space sparse grid space S l = l l T i Sl = i = i d = T i i l+d O(h d l ) O(h l (log(h l )) d ) f f I O(h 2 l ) O(h2 l (log(h l )) d ) We employ sparse grids for high-dimensional parameter space p [, ] d. 9/7 U. Baur P. Benner Parametric model reduction with sparse grids

10 Sparse grids [Klimke/Wohlmuth 5, Klimke 7] MATLAB Sparse Grid Interpolation Toolbox: Clenshaw-Curtis grid Points: 5, Level: Points full grid: 9 Points: 3, Level: 2 Points full grid: 25 Points: 29, Level: 3 Points full grid: 8 Points: 65, Level: 4 Points full grid: Points: 7, Level: Points full grid: 27 Points: 25, Level: 2 Points full grid: 25 Points: 69, Level: 3 Points full grid: 729 Points: 77, Level: 4 Points full grid: 493 /7 U. Baur P. Benner Parametric model reduction with sparse grids

11 Numerical results - convection-diffusion equation x (t,ξ) = x(t,ξ) + p x(t,ξ) + b(ξ)u(t) ξ (,)2 t using FDM with n = ẋ(t) = (A + p A + p 2 A 2 )x(t) + B u(t) B,C R n random parameter space: p,p 2 [, ] MATLAB Sparse Grid Interpolation Toolbox [Klimke/Wohlmuth 5, Klimke 7] absolute tolerance for grid refinement: 4 level l = 2 number of sparse grid points: k = 3 tolerance for BT: 4 systems of reduced order r j {3,4} for j =,...,k /7 U. Baur P. Benner Parametric model reduction with sparse grids

12 Numerical results - convection-diffusion equation G Gr x 5 5 p2 (parameter) p (parameter) 2/7 U. Baur P. Benner Parametric model reduction with sparse grids

13 Numerical results - convection-diffusion equation freq.e 2 freq 2.4e freq.8e+ x 5 x 5 x p2 p p2 p p2 p x 5 freq 4.89e+2 x 5 freq 2.2e+4 x 5 freq.e p2 p p2 p p2 p 3/7 U. Baur P. Benner Parametric model reduction with sparse grids

14 Numerical results ẋ(t) = (p 3 A+p A +p 2 A 2 )x(t)+b u(t), p,p 2 [, ], p 3 [., ] absolute tolerance for grid refinement: 4 level l = 2 number of sparse grid points: k = 25 tolerance for BT: 4 systems of reduced order r j {3,...,8} G Gr for p 3 =. G Gr for p 3 = x 5 x p2 (parameter) p (parameter) p2 (parameter) p (parameter) 4/7 U. Baur P. Benner Parametric model reduction with sparse grids

15 Conclusions Summary: We have developed a Balanced Truncation/Interpolatory method for parametric model reduction. The method can be applied to higher dimensional parameter spaces. Next steps: only function for evaluation of reduced-order system, search for explicit description of TFM, state-space model; derive global error bound by combination of error estimates for BT, interpolation; use sparse grids also for other interpolatory methods as proposed in [Beattie/Benner/Gugercin 8]; combine sparse grid interpolation with H 2 -optimal model reduction. 5/7 U. Baur P. Benner Parametric model reduction with sparse grids

16 References Beattie, C.; Benner, P.; Gugercin, S.: Interpolatory Projection Methods for Parameterized Model Reduction, in preparation. B. Bond and L. Daniel, Parameterized model order reduction of nonlinear dynamical systems, in Proceedings of the IEEE Conference on Computer-Aided Design, 25. L. Daniel, O. Siong, L. Chay, K. Lee, and J. White, Multiparameter Moment Matching Model Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models, Trans. on Computer Aided Design of Integrated Circuits, 24. R. Eid, B. Salimbahrami, B. Lohmann, E. B. Rudnyi, and J. G. Korvink, Parametric order reduction of proportionally damped second-order systems, Sensors and Materials, 27. O. Farle, V. Hill, P. Ingelström, and R. Dyczij-Edlinger, Ordnungsreduktion linearer zeitinvarianter Finite-Elemente-Modelle mit multivariater polynomieller Parametrierung, at-automatisierungstechnik, 26. 6/7 U. Baur P. Benner Parametric model reduction with sparse grids

17 References L. Feng, Parameter independent model order reduction, Math. Comput. Simulation, 25. D. S. Weile, E. Michielssen, E. Grimme, and K. Gallivan, A method for generating rational interpolant reduced order models of two-parameter linear systems, Appl. Math. Lett., 999. Bungartz, H.: Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, Dissertation, TU München, 992. J. Garcke, Sparse grid tutorial, TU Berlin, 27. A. Klimke and B. Wohlmuth, Algorithm 847: spinterp: Piecewise multilinear hierarchical sparse grid interpolation in MATLAB, ACM Transactions on Mathematical Software, 25. Klimke, A.: Sparse Grid Interpolation Toolbox user s guide, Technical Report, University of Stuttgart, 27. Zenger, C.:Sparse grids. In Parallel algorithms for partial differential equations, Notes Numer. Fluid Mech., 99. 7/7 U. Baur P. Benner Parametric model reduction with sparse grids