Model Order Reduction with pymor and NGSolve Stephan Rave

Transkript

1 Münster Model Order Reduction with pymor and NGSolve Stephan Rave living knowledge WWU Münster NGSolve User Meeting Vienna June

2 Outline Münster Model Order Reduction with pymor and NGSolve 2 /19 Model Order Reduction. Model Order Reduction with pymor. Model Order Reduction with pymor and NGSolve. pymor main developers Rene Milk Petar Mlinarić Stephan Rave Felix Schindler

3 Münster Model Order Reduction with pymor and NGSolve 3 /19 Model Order Reduction

4 Münster Model Order Reduction with pymor and NGSolve 4 /19 Parametric Model Order Reduction Consider parametric problems Φ : P V s : V R S where P R P compact set (parameter domain). V Hilbert space (solution state space dim V 0 possibly dim V = ). Φ maps parameters to solutions of parametric PDE (hard to compute). s maps state vectors to quantities of interest. Objective Compute s Φ : R P V R S for many µ P or quickly for unknown single µ P.

5 Münster Model Order Reduction with pymor and NGSolve 5 /19 The Reduced Basis Method in a Nutshell Objective Quickly compute s Φ : R P V R S. When Φ s are sufficiently smooth a quickly computable low-dimensional approximation of s Φ should exist.

6 Münster Model Order Reduction with pymor and NGSolve 5 /19 The Reduced Basis Method in a Nutshell Objective Quickly compute s Φ N : R P V N R S. When Φ s are sufficiently smooth a quickly computable low-dimensional approximation of s Φ should exist. Idea 1: State space reduction: Define approximation Φ N : P V N via Galerkin projection dim V N =: N dim V. Approximate s Φ s Φ N.

7 Münster Model Order Reduction with pymor and NGSolve 5 /19 The Reduced Basis Method in a Nutshell Objective Quickly compute s Φ N : R P V N R S. When Φ s are sufficiently smooth a quickly computable low-dimensional approximation of s Φ should exist. Idea 1: State space reduction: Define approximation Φ N : P V N via Galerkin projection dim V N =: N dim V. Approximate s Φ s Φ N. Idea 2: V N span{φ(µ 1 )... Φ(µ k )}.

8 Münster Model Order Reduction with pymor and NGSolve 5 /19 The Reduced Basis Method in a Nutshell Objective Quickly compute s Φ N : R P V N R S. When Φ s are sufficiently smooth a quickly computable low-dimensional approximation of s Φ should exist. Idea 1: State space reduction: Define approximation Φ N : P V N via Galerkin projection dim V N =: N dim V. Approximate s Φ s Φ N. Idea 2: V N span{φ(µ 1 )... Φ(µ k )}. Idea 3: Construct V N iteratively via greedy search of P using quickly computable surrogate η N (Φ N (µ) µ) Φ(µ) Φ N (µ).

9 An Example Münster Model Order Reduction with pymor and NGSolve 6 /19 Experimental Data Mathematical Modeling Multiscale Numerics Model Reduction MULTIBAT: Gain understanding of degradation processes in rechargeable Li-Ion Batteries through mathematical modeling and simulation. Integration Validation Focus: Li-Plating. Li-plating initiated at interface between active particles and electrolyte. Need microscale models which resolve active particle geometry. Huge nonlinear discrete models.

10 Münster Model Order Reduction with pymor and NGSolve 7 /19 An Example MOR Results DOFs Snapshots: 3 dim V N = cell voltage / V M = transferred charge / nah Rel. err.: < Full model: 13h Reduction: 9h Red. model: 5m avg. Li concentr. / kmol/m Speedup: na na na na na full model red. model

11 Münster Model Order Reduction with pymor and NGSolve 8 /19 Model Order Reduction with pymor

12 Münster Model Order Reduction with pymor and NGSolve 9 /19 pymor Model Order Reduction with Python Goal One tool for algorithm development and large-scale applications. Started late k lines of Python code 3k single commits. BSD-licensed fork us on Github! Quick prototyping with Python 2/3. Comes with small NumPy/SciPy-based discretization toolkit for getting started quickly. Seamless integration with high-performance PDE solvers.

13 Münster Model Order Reduction with pymor and NGSolve 10 /19 Generic Algorithms and Interfaces for MOR dealii model() fenics model() dune model() PDE solver deal.ii FEniCS DUNE... pymor-deal.ii dolfin dune-pymor d prod =... operator rhs LincombOperator Discretization pymor example.py U = d.solve(mu) d.visualize(u) apply2 Operator apply2 Operator... product Operator VectorArray axpy scal U = d.solve(mu) apply inverse apply2 greedy(d...) Greedy reductor(d basis) Reductor Gram-Schmidt User Code gram schmidt basis extension(basis U prod) Generic algorithms... VectorArray Operator Discretization classes represent objects in solver s memory. No communication of high-dimensional data. Tight low-level integration with external solver. No MOR-specific code in solver.

14 Münster Model Order Reduction with pymor and NGSolve 11 /19 Implemented Algorithms Gram-Schmidt POD. Greedy basis generation with different extension algorithms. Automatic (Petrov-)Galerkin projection of arbitrarily nested affine combinations of operators. Interpolation of arbitrary (nonlinear) operators EI-Greedy DEIM. A posteriori error estimation. Iterative linear solvers Newton algorithm time-stepping algorithms. New! System theory Methods: balanced truncation IRKA... (sys-mor branch)

15 Münster Model Order Reduction with pymor and NGSolve 12 /19 FEniCS Support Included Directly interfaces FEniCS LA backend no copies needed. Use same MOR code with both backends! Only 150 SLOC for bindings. Thermal block demo: 30 SLOC FEniCS + 15 SLOC wrapping for pymor. Easily increase FEM order etc. Figure: 3x3 thermal block problem top: red. solution bottom: red. error left: pymor solver right: FEniCS solver

16 Münster Model Order Reduction with pymor and NGSolve deal.ii Support I pymor-deall.ii support module deal.ii I Python bindings for I dealii::vector I dealii::sparsematrix. estimator effectivity maximum red. error I pymor wrapper classes. 0.8 I 10 7 MOR demo for linear elasticity example from tutorial Figure: top: Solutions for (µ λ) = (1 1) and (µ λ) = (1 10) bottom: red. errs. and max./min. estimator effectivities vs. dim VN. 13 /19

17 Münster Model Order Reduction with pymor and NGSolve 14 /19 Tools for interfacing MPI parallel solvers I Automatically make sequential bindings MPI aware. I Reduce HPC-Cluster models without thinking about MPI at all. I Interactively debug MPI parallel solvers. Figure: FV solution of 3D Burgers-type equation ( DOFs 600 time steps) using Dune. Table: Time (s) needed for solution using DUNE / DUNE with pymor timestepping. MPI ranks DUNE pymor overhead % 4.5% 5.0% 5.7% 5.3% 5.3% 6.0% 5.4% 11.8%

18 Münster Model Order Reduction with pymor and NGSolve 15 /19 Model Order Reduction with pymor and NGSolve

19 NGSolve Support Münster Model Order Reduction with pymor and NGSolve 16 /19 Based on NGSolve - Python. See ngs-user-meeting17 branch of pymor repo. pymor wrappers for GridFunction and BaseMatrix classes. 3d thermal block demo included. Joint work with Christoph Lehrenfeld. Figure: 3d thermal block problem top: full/red. sol. bottom: err. for worst approx. µ and max. red. error vs. dim V N.

20 Münster Model Order Reduction with pymor and NGSolve 17 /19 import live_demo

21 Münster Model Order Reduction with pymor and NGSolve 18 /19 Matrix DEIM with NGSolve Osmotic cell swelling problem tu α u = 0 V nu + α nu = 0 βκ + γ(u u 0 ) = V n in Ω(t) on Ω(t) on Ω(t) osmosis Transformed concentration equation on reference domain J n+1 û n+1 ˆv dx + t J n+1 V (F n+1 T ˆx ˆv)û n+1 dx ˆΩ ˆΩ + t αj n+1 (F n+1 T ˆx u) (F n+1 T ˆx ˆv) dx = ˆΩ J n û n ˆv dx ˆΩ Nonlinear in transformation Ψ F := x Ψ J := det(f ) V := tψ. Use Matrix-DEIM w.r.t. Ψ.

22 Münster Model Order Reduction with pymor and NGSolve 19 /19 Thank you for your attention! My homepage Jupyter notebook pymor Generic Algorithms and Interfaces for Model Order Reduction SIAM J. Sci. Comput. 38(5) pp. S194 S216 MULTIBAT: Unified Workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures electrochemical degradation models and model order reduction (submitted)