Dynamic Inverse Problems: Schmitt Efficient Algorithms and - - PowerPoint PPT Presentation

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Dynamic Inverse Problems: Schmitt Efficient Algorithms and - - PowerPoint PPT Presentation

Mar 11, 2024 240 likes •546 views

Inverse Problems A.K. Louis, U. Dynamic Inverse Problems: Schmitt Efficient Algorithms and Approximate Inverse Problems Definitions Inverse Two Constraints Semi - discrete Case Examples A.K. Louis 1 U. Schmitt 2 Approximate Inverse 1

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Dynamic Inverse Problems: Efficient Algorithms and Approximate Inverse

A.K. Louis1

  • U. Schmitt2

1Institut für Angewandte Mathematik

Universität des Saarlandes http://www.num.uni-sb.de

2Minewave GmbH

Saarbrücken

Vancouver, 28.06.2007

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Content

1

Static and Dynamic Inverse Problems Definitions Two Constraints

2

Semi - discrete Case

3

Examples

4

Approximate Inverse

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Definitions

Static Inverse Problems Continuous and semi-discrete versions Af(t, x) =

  • k(t, x, y)f(y)dy = g(t, x)

Aℓf = gℓ

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Definitions

Static Inverse Problems Continuous and semi-discrete versions Af(t, x) =

  • k(t, x, y)f(y)dy = g(t, x)

Aℓf = gℓ Dynamic Inverse Problems Af(t, x) = k(t, τ, x, y)f(τ, y)dydτ Aℓfℓ = gℓ NEED OF REGULARIZATION BOTH IN TIME AND SPACE

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Temporal Regularization

Activation curves without and with temporal regularization

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Undertermined Problems

Consider A is matrix which has more colums than rows A maps from an infintite-dimensional Hilbert space to finitely many data A maps from function spaces of different dimensions

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Undertermined Problems

Consider A is matrix which has more colums than rows A maps from an infintite-dimensional Hilbert space to finitely many data A maps from function spaces of different dimensions ⇒ Prefer to solve Af = g as AA∗u = g and put f = A∗u

  • r its Tikhonov - Phillips variants.
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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Content

1

Static and Dynamic Inverse Problems Definitions Two Constraints

2

Semi - discrete Case

3

Examples

4

Approximate Inverse

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Two Constraints

Minimize Af − g2 + γ2f2 + µ2Bf2 where in our application one of the terms is used as spatial the other as temporal smoothness condition.

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Two Constraints

Minimize Af − g2 + γ2f2 + µ2Bf2 where in our application one of the terms is used as spatial the other as temporal smoothness condition. Only when A and B commute then this can be written in the above mentioned form.

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Formulation with slack variable

Minimize with respect to f and d Af − g2 + γ2f2 + µ2d2 + α2Bf − d2

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Formulation with slack variable

Minimize with respect to f and d Af − g2 + γ2f2 + µ2d2 + α2Bf − d2 Change of variables d := γ

µy Then this is

Af − g2 + γ2f2 + γ2y2 + αBf − αγ µy2

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Formulation with slack variable

Minimize with respect to f and d Af − g2 + γ2f2 + µ2d2 + α2Bf − d2 Change of variables d := γ

µy Then this is

Af − g2 + γ2f2 + γ2y2 + αBf − αγ µy2 With the new variables ξ = f y

  • and h =

g

  • this is equivalent to
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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Compact Formulation

Minimze Tαξ − h2 + γ2ξ2 with Tα =

  • A

αB −α γ

µI

  • The solution of

TαT ∗

α

u v

  • = h

with ξ = T ∗

α

u v

  • leads to the desired result.
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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Efficient Minimization with two constraints

Theorem (L., Schmitt, 2002, 2007) The minimzation of Af − g2 + γ2f2 + µ2Bf2 is computed in two steps. First we solve for u with ω = γ2

µ2

A

  • I − B∗(BB∗ + ωI)−1B
  • A∗u + γ2u = g

and put fγ,µ =

  • I − B∗(BB∗ + ωI)−1B
  • A∗u
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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Semi - discrete Case

Aℓ : H → Gℓ, ℓ = 1, . . . , L AℓFℓ = Gℓ Minimize J(f) =

L

  • ℓ=1

Aℓfℓ − gℓ2 + γ2

L

  • ℓ=1

fℓ2 + µ2

L−1

  • ℓ=1

fℓ+1 − fℓ2 (tℓ+1 − tℓ)2 Define A = diag(Aℓ) ∈ L(HL, G1 ⊕ · · · ⊕ GL)

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

f = (f1, · · · fL) B = D ⊕ IH ∈ L(HL, HL−1 D = τ1 −τ1 τ2 −τ2 · · τL−1 −τL−1

  • ∈ RL×(L−1)

τi = (ti+1 − ti)−1 Generalized Sylvester Equations

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Dynamic Computerized Tomography

Rℓf(s) = Rf(ωℓ, s) =

  • f(sω + tω⊥)dt
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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Dynamic Computerized Tomography

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Current Density Reconstruction

Application: study of neurological activity in the brain Forward Model div(σ∇Φ) = divj in Ω σ∇Φ, n = 0 at δΩ σ conductivity tensor Φ electrical potential Data Φ at the boundary

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Current Density Reconstruction

Reconstruction without temporal smoothness constrains

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Current Density Reconstruction

Reconstruction with temporal smoothness constrains

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Approximate Inverse

L., Inverse Problems,1996, 1999 Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Given: A : L2(Ω1, µ1) → L2(Ω2, µ2) linear, continuous Mollifier δx ≈ eγ(x, ·) or δ′

x ≈ eγ(x, ·)

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Approximate Inverse

L., Inverse Problems,1996, 1999 Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Given: A : L2(Ω1, µ1) → L2(Ω2, µ2) linear, continuous Mollifier δx ≈ eγ(x, ·) or δ′

x ≈ eγ(x, ·)

Compute fγ(x) = f, eγ(x, ·)L2(Ω1,µ1) = Af, ψγ(x)L2(Ω2,µ2)

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Approximate Inverse

L., Inverse Problems,1996, 1999 Compare: Backus-Gilbert, 76, Grünbaum, 80, Smith 80 Given: A : L2(Ω1, µ1) → L2(Ω2, µ2) linear, continuous Mollifier δx ≈ eγ(x, ·) or δ′

x ≈ eγ(x, ·)

Compute fγ(x) = f, eγ(x, ·)L2(Ω1,µ1) = Af, ψγ(x)L2(Ω2,µ2) Idea: Solve A∗ψγ(x) = eγ(x, ·)

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Approximate Inverse in L2-Spaces

Data g given

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Approximate Inverse in L2-Spaces

Data g given Approximate Inverse Sγg(x) := g, ψγ(x)L2(Ω2,µ2) Theorem (L, 1997) Let the operators T1, T2 intertwine with A∗; i.e., T1A∗ = A∗T2 and solve for a reference mollifier Eγ the equation A∗Ψγ = Eγ Then the general reconstruction kernel for the general mollifier eγ = T1Eγ is

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Linear Regularization

Y1

A+

ւ ↑ ˜ Mγ A : X − → Y Mγ ↑

A+

ւ X−1 Sγ = MγA+ = A+ ˜ Mγ

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Inverse Problems A.K. Louis, U. Schmitt Inverse Problems

Definitions Two Constraints

Semi - discrete Case Examples Approximate Inverse

Tikhonov - Phillips as Approximate Inverse

Solve Eγ = ΨγA via Tikhonov - Phillips, then Sγ = A∗ AA∗ + λI −1 Use Invariances to further speed up the method