CME 345: MODEL REDUCTION

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1 CME 345: MODEL REDUCTION Parameterized Partial Differential Equations Charbel Farhat Stanford University 1 / 19

2 Outline 1 Initial Boundary Value Problems 2 Typical Parameters of Interest 3 Semi-discretization Processes and Dynamical Systems 4 5 Subspace Approximation 2 / 19

3 Initial Boundary Value Problems Linear Partial Differential Equation (PDE) L(W, x, t) = 0 W = W(x, t) R q : State variable x Ω R d, d 3: Space variable t 0: Time variable Examples linearized Navier-Stokes equations elastodynamic equations of motion wave equation Boundary Conditions (BCs) B(W, x BC, t) = 0 Dirichlet BCs Neumann BCs Initial Condition (IC) W(x, 0) = W 0 (x) = W IC (x) 3 / 19

4 Initial Boundary Value Problems Parameterized PDE Parameter domain: D R p parameter vector (also referred to as parameter point ): µ = [µ 1,, µ p] T D R p where the superscript T designates the transpose operation Parameterized PDE Boundary conditions L(W, x, t; µ) = 0 B(W, x BC, t; µ) = 0 Initial condition W 0 (x) = W IC (x; µ) 4 / 19

5 Typical Parameters of Interest Shape parameters Material (properties) parameters Operation parameters (for example, flight conditions, cruise conditions, ) Boundary conditions Initial condition 5 / 19

6 Semi-discretization Processes and Dynamical Systems Semi-discretized problem The PDE is discretized in space using, for example a finite difference method a finite volume method a finite element method a discontinuous Galerkin method a spectral method... This leads to a system of N = q N space Ordinary Differential Equations (ODEs) that can be written as dw dt = f(w, t; µ) where w = w(t; µ) R N with the initial condition w(0; µ) = w 0 (µ) This is the High-Dimensional Model (HDM) 6 / 19

7 Multi-query context routine analysis uncertainty quantification design optimization inverse problems optimal control model predictive control 7 / 19

8 Multi-query Context Routine analysis 8 / 19

9 Multi-query Context Routine analysis 9 / 19

10 Multi-query Context Uncertainty quantification Monte-Carlo simulations / 19

11 Multi-query Context Design optimization 11 / 19

12 Multi-query Context Model predictive control 12 / 19

13 Model Parameterized PDE Advection-diffusion-reaction equation: W = W(x, t; µ) solution of W t + U W κ W = f R (W, t, µ R ) for x Ω with appropriate boundary and initial conditions Parameters of interest W(x, t; µ) = W D (x, t; µ D ) for x Γ D W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x; µ IC ) for x Ω µ = [U 1,, U d, κ, µ R, µ D, µ IC ] 13 / 19

14 Parameterized Solutions Two-dimensional advection-diffusion equation W + U W κ W = 0 for x Ω t with boundary and initial conditions W(x, t; µ) = W D (x, t; µ D ) for x Γ D W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x) for x Ω 14 / 19

15 Parameterized Solutions Two-dimensional advection-diffusion equation W t with boundary and initial conditions + U W κ W = 0 for x Ω W(x, t; µ) = W D (x, t; µ D ) for x Γ D 4 parameters of interest p = 4 w R N with N = 2, 701 W(x, t; µ) n(x) = 0 for x Γ N W(x, 0; µ) = W 0 (x) for x Ω µ = [U 1, U 2, κ, µ D ] R 4 15 / 19

16 Parameterized Solutions Solution snapshots at some time t i, for six sampled parameter points µ (j), j = 1,, 6 16 / 19

17 Subspace Approximation Question: Can we reuse the pre-computed snapshots to reconstruct a solution for an unsampled parameter point µ? Idea: Use a linear combination of these snapshots such as, for example w(t; µ) N s s (k) q (1) i (t; µ)w(t i ; µ (1) ) + + q (k) i (t; µ)w(t i ; µ (k) ) N (1) i=1 where N s (j), j = 1,, k denotes the number of snapshots pre-computed using the sampled parameter point µ (j), and k denotes the total number of parameter points sampled in the parameter space D w(t i ; µ (j) ) R N denotes the snapshot pre-computed at time t i using the sampled parameter point µ (j) q (j) i (t; µ) R denotes the expansion coefficient associated with w(t i ; µ (j) ) i=1 17 / 19

18 Subspace Approximation The linear expansion w(t; µ) N s s (k) q (1) i (t; µ)w(t i ; µ (1) ) + + q (k) i (t; µ)w(t i ; µ (k) ) N (1) i=1 can be written as where [ W = and [ q(t; µ) = w(t 1 ; µ (1) ),, w(t N (1) s q (1) 1 (t; µ),, q(1) w(t; µ) Wq(t; µ) N s (1) i=1 ] ; µ (1) ),, w(t 1 ; µ (k) ),, w(t (k) N ; µ (k) ) s (t; µ),, q (k) 1 (t; µ),, q(k) N s (k) ] T (t; µ) 18 / 19

19 Subspace Approximation The parameterized approximation w(t; µ) Wq(t; µ) is a subspace approximation of w(t; µ), where the subspace is { } S = span w(t 1 ; µ (1) ),,, w(t (k) N ; µ (k) ) s and its dimension is [ dim (S) = rank w(t 1 ; µ (1) ),,, w(t N (k) s ] ; µ (k) ) k j=1 N (j) s This approximation constitutes one of the pillars of projection-based model reduction. It raises however the following questions How to sample the parameter space D? How to reduce the dimensionality of W and therefore the approximation subspace S below k N s (j)? These are some of the questions that this course addresses j=1 19 / 19

Partial Differential Equations

Simulation in Computer Graphics Partial Differential Equations Matthias Teschner Computer Science Department University of Freiburg Motivation various dynamic effects and physical processes are described