IntroductionWave equation

Model Order Reduction for Wave Equations

Rob F. Remis and Jorn T. Zimmerling

DCSE Fall School, Delft, November 4 – 8, 2019

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IntroductionWave equation

Introduction

Rob F. Remis

Fac. of Electrical Engineering, Mathematics and Computer ScienceDelft University of Technology

r.f.remis@tudelft.nl

Jorn T. Zimmerling

Department of MathematicsUniversity of Michigan

j.zimmerl@umich.edu

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IntroductionWave equation

Introduction

Lecture 1: basic wave equations

Lecture 2: Discretization and symmetry

Lecture 3: Symmetry and Krylov model-order reduction

Lecture 4: Capita selecta

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IntroductionWave equation

Introduction

Reduced-order modeling (ROM) is a vast research area

We focus on some ROM techniques for wave field problems

Reduced-order model(ing) – ROM:Replace a large-scale system by a much smaller one

Before you start, you should have good reasons to believe thata significant reduction can be achieved

ROM for wave fields (hyperbolic problems) is hard, in general

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IntroductionWave equation

Introduction

Possible ROM scenarios:

Solve a forward problem efficiently

Replace a large-scale model in a design/imaging/inversionprocess

Replace a large-scale system, while preserving

input-output characteristics andessential system properties (e.g. stability, passivity)

of the unreduced system – structure preserving ROM

Apply ROM to directly solve an imaging/inversion problem –data driven imaging

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IntroductionWave equation

Introduction - Example 1

500 1000 1500y-direction [m]

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x-di

rect

ion

[m]

ReceiverSourcePML

1500

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ed [m

/s]

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Normalized Time

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pons

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.u.]

×10-5

ComparisonROM

2000 2500 3000 3500 4000 4500 5000

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IntroductionWave equation

Introduction - Example 2

21

22

T1 GRE

T2 TSE

Without pad With pad

b+1 = b+1;rom(p), p = pad design parameter vector

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IntroductionWave equation

The Wave Equation

The wave equationinstantaneous reacting material

∇2u − 1

c2∂ttu = −q

u = u(x, t): wave field quantity of interest∇2: Laplacian∂tt : double derivative with respect to timeq = q(x, t): sourcec(x): wave speed profilec(x) = c0 for a homogeneous medium

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IntroductionWave equation

The Wave Equation

Source has a bounded support in space

Solve wave equation for given initial and boundary conditions

The wave equationmaterial exhibiting relaxation

∇2u − 1

c20∂tt(u + χ ∗ u) = −q

χ(x, t) is called the relaxation function of the material

This function must be causal

χ(x, t) = 0 for t < 0 and x ∈ R3

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IntroductionWave equation

The Wave Equation

Applying a one-sided Laplace transform gives Helmholtz’sequation

∇2u − γ2u = −q

with a propagation coefficient

γ = γ0 = s/c0 instanteneous reacting and homogeneous

γ = s/c(x) instanteneous reacting and inhomogeneous

γ2 = γ20 [1 + χ(x, s)] with relaxation and inhomogeneous

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IntroductionWave equation

Maxwell’s equations

Maxwell’s equations

−∇×H + Jc + ∂tD = −Jext

and

∇× E + ∂tB = 0

Jext [A/m2]: external electric-current source (antenna)

E: electric field strength [V/m]

H: magnetic field strength [A/m]

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IntroductionWave equation

Maxwell’s equations

Jc [A/m2]: electric conduction current

D [C/m2]: electric flux density

B [T = Vs/m2]: magnetic flux density

How matter reacts to the presence of an EM field is describedby the constitutive relations

Jc = Jc(E), D = D(E), and B = B(H)

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IntroductionWave equation

Maxwell’s equations

Maxwell’s equations in one dimension

∂yHx + Jc;z + ∂tDz = −Jextz

and

∂yEz + ∂tBx = 0

Vacuum

Instantaneously reacting material

Matter exhibiting relaxation

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IntroductionWave equation

Maxwell’s equations

Vacuum

Jc;z = 0, Dz = ε0Ez , and Bz = µ0Hz

Parameters of vacuum

permittivity of vacuum ε0permeability of vacuum µ0

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IntroductionWave equation

Maxwell’s equations

Instantaneously reacting material

Jc;z = σ(x)Ez , Dz = ε(x)Ez , and Bz = µ(x)Hz

Medium parameters

conductivity σpermittivity εpermeability µ

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IntroductionWave equation

Maxwell’s equations

Material exhibiting relaxation

Jc;z = 0, Dz = ε0Ez+ε0

∫ t

τ=0χe(x, t−τ)Ez(x, τ) dτ, Bz = µ0Hz

χe relaxation function of the material

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IntroductionWave equation

Maxwell’s equations

Model that is often used (for gold at optical frequencies, forexample)

χe(t) = (ε∞ − 1)δ(t) + χe(t)

with

χe(t) =∆ε ω2

p√ω2p − δ2p

exp(−δpt) sin(√

ω2p − δ2p

)U(t)

U(t): Heaviside unit step function

ωp: plasma frequency

δp: damping coefficient

∆ε = εs − ε∞

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