CA Cme334 Ch2

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CME 345: MODEL REDUCTION - Parameterized PDEs

CME 345: MODEL REDUCTION

Pameterized Partial Differential Equations

David Amsallem & Charbel FarhatStanford [email protected]

David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Parameterized PDEs

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Outline

1 Initial Boundary Value Problems

2 Parameters of Interest

3 Semi-discretization Processes and Dynamical Systems

4 The Case for Model Reduction

5 Subspace Approximation


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Initial Boundary Value Problems

Partial differential equation (PDE)

L(W , x, t ) = 0

W = W (x, t ) ∈ Rq : state variablex ∈ Ω ⊂ Rd , d ≤ 3: space variablet ≥ 0: time variable

Examples:Navier-Stokes equationsHogkin-Huxley equationsSchrodinger equations

Associated boundary conditions

B (W , xBC, t ) = 0

Dirichlet BCNeumann BC

Initial condition

W 0(x) = W IC(x)David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Parameterized PDEs

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Initial Boundary Value Problems

Parameterized PDE

Parametrized partial differential equation (PDE)

L(W , x, t ;µ) = 0

Associated boundary conditions

B (W , xBC, t ;µ) = 0

Initial conditionW 0(x) = W IC(x,µ)

µ ∈ D ⊂ Rp : parameter vector


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Parameters of Interest

Material parameters

Shape parameters


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Parameters of Interest

Initial condition

Boundary conditions


3 O O

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Semi-discretization Processes and Dynamical Systems

Semi-discretized problem

The PDE is discretized in space by one of the following methods

Finite Differences approximationFinite Element methodFinite Volumes methodDiscontinuous Galerkin methodSpectral methods....

This leads to a system of N = q × N space ordinary differentialequations (ODEs)

d w

dt = f (w, t ;µ)

in terms of the discretized state variable

w = w(t ;µ) ∈ RN

with initial condition w(0;µ) = w0(µ)

This is the high-dimensional model (HDM)


CME 345 MODEL REDUCTION P i d PDE

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The Case for Model Reduction

Multi-query context

Routine analysisUncertainty quantificationDesign optimizationInverse problemsOptimal controlModel predictive control


CME 345 MODEL REDUCTION P t i d PDE

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Multi-query Context

Routine analysis


CME 345: MODEL REDUCTION Parameterized PDEs

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Multi-query Context

Routine analysis



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Multi-query Context

Uncertainty quantification

Monte-Carlo simulations,...David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Parameterized PDEs


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Multi-query Context

Design optimization



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Multi-query Context

Model predictive control



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Multi-query Context

Model predictive control



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Model Parameterized PDE

Advection-diffusion-reaction equation: W = W (x, t ;µ) solution of

∂ W

∂ t + U · ∇W − κ∆W = f R(W , t ,µR) for x ∈ Ω

with appropriate boundary and initial conditions

W (x, t ;µ) = W D (x, t ;µD) for x ∈ ΓD

∇W (x, t ;µ) · n(x) = 0 for x ∈ ΓN

W (x, 0;µ) = W 0(x;µIC) for x ∈ Ω

Parameters of interest

µ = [ U 1, · · · , U d , κ,µR,µD,µIC]



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Parameterized Solutions

Two dimensional advection-diffusion equation

∂ W ∂ t

+ U · ∇W − κ∆W = 0 for x ∈ Ω

with boundary and initial conditions


∇W (x, t ;µ) · n(x) = 0 for x ∈ ΓNW (x, 0;µ) = W 0(x) for x ∈ Ω

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Two dimensional advection-diffusion equation∂ W

∂ t + U · ∇W − κ∆W = 0 for x ∈ Ω

with boundary and initial conditions


∇W (x, t ;µ) · n(x) = 0 for x ∈ ΓN

W (x, 0;µ) = W 0(x) for x ∈ Ω

p = 4 parameters of interest

µ = [ U 1, U 2, κ,µD] ∈ R4

w ∈ RN with N = 2, 701



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Subspace Approximation

Can we reuse solutions for different parameter values?

Idea: use linear combinations of pre-computed solutions (snapshots)

w(t ;µ) ≈ q 1(t ;µ)w(t 1;µ1) + · · · + q k (t ;µ)w(t k ;µk )

w(t i ;µ

i )∈ RN

: pre-computed solution at (t i ,µ

i )q i (t ;µ) ∈ R: coefficient in the expansion

The parameterized solution w(t ;µ) is then constrained to belongingto the subspace

S = span {w(t 1;µ1), · · · , w(t k ;µk )}

Subspace dimension

dim(S ) = rank [w(t 1;µ1), · · · , w(t k ;µk )] ≤ k


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CA Cme334 Ch2