Regularization of nonlinear ill-posed problems by the exponential - - PowerPoint PPT Presentation

Problem setting Asymptotic regularization Numerical examples

Regularization of nonlinear ill-posed problems by the exponential Euler method

Michael H¨

• nig

Lehrstuhl f¨ ur Angewandte Mathematik Heinrich Heine Universit¨ at D¨ usseldorf

September 2008

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Outline

1

Problem setting

2

Asymptotic regularization

3

Numerical examples

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Nonlinear ill-posed problems

F : D(F) ⊂ X → Y continous and Fr´ echet differentiable X, Y real Hilbert spaces, nonlinear problem: F(x) = y

1

x+ the x0-minimum-norm solution

2

• nly perturbed data yδ ∈ Y, y − yδY ≤ δ available

3

problem ill-posed e.g. F compact and D(F) weakly closed regularization required

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Assumptions

1

tangential cone condition

• F(

x) − F(x) − F′(x)( x − x)

• ≤ η F(

x) − F(x) ,

• x, x ∈ Br(x0)

2

source condition x0 − x+ = J(x+)γw, w ≤ ρ, J(x) := F′(x)∗F′(x)

3

local restriction of the derivative F′(x) = RxF′(x+) Rx − I ≤ C+ x − x+ , ∀x ∈ Br(x+)

4

w.l.o.g. F′(x) ≤ 1, x ∈ Br(x0) e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Assumptions

1

tangential cone condition

• F(

x) − F(x) − F′(x)( x − x)

• ≤ η F(

x) − F(x) ,

• x, x ∈ Br(x0)

2

source condition x0 − x+ = J(x+)γw, w ≤ ρ, J(x) := F′(x)∗F′(x)

3

local restriction of the derivative F′(x) = RxF′(x+) Rx − I ≤ C+ x − x+ , ∀x ∈ Br(x+)

4

w.l.o.g. F′(x) ≤ 1, x ∈ Br(x0) e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Assumptions

1

tangential cone condition

• F(

x) − F(x) − F′(x)( x − x)

• ≤ η F(

x) − F(x) ,

• x, x ∈ Br(x0)

2

source condition x0 − x+ = J(x+)γw, w ≤ ρ, J(x) := F′(x)∗F′(x)

3

local restriction of the derivative F′(x) = RxF′(x+) Rx − I ≤ C+ x − x+ , ∀x ∈ Br(x+)

4

w.l.o.g. F′(x) ≤ 1, x ∈ Br(x0) e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Assumptions

1

tangential cone condition

• F(

x) − F(x) − F′(x)( x − x)

• ≤ η F(

x) − F(x) ,

• x, x ∈ Br(x0)

2

source condition x0 − x+ = J(x+)γw, w ≤ ρ, J(x) := F′(x)∗F′(x)

3

local restriction of the derivative F′(x) = RxF′(x+) Rx − I ≤ C+ x − x+ , ∀x ∈ Br(x+)

4

w.l.o.g. F′(x) ≤ 1, x ∈ Br(x0) e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Asymptotic regularization

Showalter ode ˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

u(0) = x0 stopping time t∗ chosen by discrepancy principle (τ > 1) F(u(t∗)) − yδ ≤ τδ < F(u(t)) − yδ, 0 ≤ t < t∗ analyzed by Tautenhahn (1994) u(t∗) → x+, u(t∗) − x+ = O(δ

2γ 2γ+1 ),

as δ → 0

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Regularization with Runge-Kutta integrators

Ansatz: Application of time integration schemes for solving ˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

B¨

• ckmann & Pornsawad (2008):

1

use simplified Runge-Kutta methods

2

convergence under (severe) step size restrictions

3

numerical experiments show restrictions are due to analysis

4

no proof of optimal order

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Explicit Euler

˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

application of explicit Euler leads to un+1 = un + hnF′(un)∗(yδ − F(un)) nonlinear Landweber iteration for hn = 1

1

step size restriction

2

many (explicit) iterations

3

convergence and optimal order known Hanke, Neubauer & Scherzer (1995)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Explicit Euler

˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

application of explicit Euler leads to un+1 = un + hnF′(un)∗(yδ − F(un)) nonlinear Landweber iteration for hn = 1

1

step size restriction

2

many (explicit) iterations

3

convergence and optimal order known Hanke, Neubauer & Scherzer (1995)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Linearly implicit Euler

˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

application of implicit Euler leads to un+1 = un + hnF′(un+1)∗(yδ − F(un+1))

• ne Newton step with simplified Jacobian −J(un) = −F′(un)∗F′(un)

gives un+1 = un + hn(I + hnJ(un))−1F′(un)∗(yδ − F(un)) Newton type method with Tikhonov Philipps as inner regularization

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Linearly implicit Euler

un+1 = un + hn(I + hnJ(un))−1F′(un)∗(yδ − F(un)) J(u) = F′(u)∗F′(u) Newton type method with Tikhonov Philipps regularization

1

large step sizes

2

• ne linear system in each timestep

3

convergence (rates) known Rieder (2001)

4

related to iteratively regularized Gauss–Newton Bakushinskii (1992)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Exponential Euler

˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

exponential Euler un+1 = un + hnϕ(−hnJ(un))F′(un)∗(yδ − F(un)) ϕ(z) = ez − 1 z properties

1

solves linear problems exactly

2

“explicit” scheme

3

again: approximate Jacobian by J(u) = F′(u)∗F′(u)

4

equivalent to Newton method with regularized linear system

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Exponential Euler

˙ u(t) = F′(u(t))∗ yδ − F(u(t))

• t > 0

exponential Euler un+1 = un + hnϕ(−hnJ(un))F′(un)∗(yδ − F(un)) ϕ(z) = ez − 1 z properties

1

solves linear problems exactly

2

“explicit” scheme

3

again: approximate Jacobian by J(u) = F′(u)∗F′(u)

4

equivalent to Newton method with regularized linear system

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Exponential Euler

regularization properties: Theorem Under suitable assumptions:

1

exponential Euler iterates un converge to x+

2

convergence rates are of optimal order Proofs: see next talk by Marlis Hochbruck

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Recapitulation

nonlinear problem F(u) = yδ linearization✲ linear problems F′(un)∆un = yδ − F(un) nonlinear ode Showalter

❄

time integration

✲

nonlinear regularization method linear regularization method

❄

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Implementation

notation: h = hn, J = J(un) = F′(un)∗F′(un) h.psd. computation of ϕ(−hJ)v by Krylov subspace methods JVm = VmTm + Tm+1,mvm+1eT

m

Tm tridiagonal, Vm orthogonal ϕ(−hJ)v ≈ Vmϕ(−hTm)e1 Druskin, Khnizhnerman (1995) Hochbruck, Lubich (1997)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Implementation

a posteriori error estimates: (van den Eshof & Hochbruck 2006) approximation of the relative error in step m θm = wm − wm−1 wm , wm := ϕ(−hTm)e1 approximation of the error ǫm ǫm θm 1 − θm wm accuracy ǫm = O(δ) sufficient

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Implementation

properties

1

no linear systems

2

matrix free implementation

• nly functions providing F′(un)v, F′(un)∗v required

3

low accuracy low dimensional Krylov subspaces

4

fast computation of ϕ(−hTm) by Pad´ e approximation

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example I: parameter identification in pde’s (Rieder 1999)

reconstruction of x in −∆x + yx = v in Ω x = w on ∂Ω Ω = (0, 1)2, v, w and yδ known exact solution x+(ξ, η) = 1.5 sin(2πξ) sin(3πη) + 3

• (ξ − 0.5)2 + (η − 0.5)2

+ 2 x0 chosen such that source condition is satisfied with γ = 1

2

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example I: parameter identification in pde’s

0.5 1 0.5 1 1 2 3 4 0.5 1 0.5 1 1 2 3 4

coefficient x+(ξ, η) reconstruction uδ

n∗(ξ, η)

(rel. error 5%, noise δ = 10−2.5 ≈ 0.0032)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example I: parameter identification in pde’s

10

−4

10

−3

10

−2

10

−1

10

−3

10

−2

10

−1

10 √ δ cg

• exp. Euler

10

−4

10

−3

10

−2

10

−1

3 4 5 6 7 8 cg

• exp. Euler

reconstruction error number of outer iterations as functions of the perturbation parameter δ (exponential Euler method vs. cg-REGINN)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example II: groundwater hydrology (Hanke 1997)

−div(x grad y) = f in Ω y = g

• n ∂Ω

in Ω = [0, 6]2, mixed Dirichlet–Neumann boundary data y(ξ, 0) = 100, yξ(6, η) = 0, (xyξ)(0, η) = −500, yη(ξ, 6) = 0 right-hand side f(ξ, η) =      0 < η < 4, 137 4 < η < 5, 274 5 < η < 6.

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example II: groundwater hydrology

20 40 60 80 100 120 140 160

diffusivity x+ reconstruction uδ

n∗

(rel. error 18%, noise δ = 10−3)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example II: groundwater hydrology

10

−3

10

−2

10

−1

0.2 0.25 0.3 0.35 0.4 cg

• exp. Euler

reconstruction error as function of the perturbation parameter δ (exponential Euler method vs. cg-REGINN)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Example II: groundwater hydrology

10

−3

10

−2

10

−1

20 40 60 80 cg

• exp. Euler

10

−3

10

−2

10

−1

20 40 60 80 cg

• exp. Euler

number of outer iterations cpu-time as functions of the perturbation parameter δ (exponential Euler method vs. cg-REGINN)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

Conclusion

so far:

1

new regularization method introduced

2

method fits in known class of regularizations

3

regularizing properties stated

4

numerical experiments reflect theory

5

regularization method is competitive to do:

1

show convergence and optimal order ( following talk)

Michael H¨

• nig

Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples

References

• M. Hochbruck, M. H¨
• nig and A. Ostermann.

Regularization of nonlinear ill-posed problems by exponential integrators. M2NA, submitted.

• M. Hochbruck, M. H¨
• nig and A. Ostermann.

A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems. technical report. Preprints available at www.am.uni-duesseldorf.de

Michael H¨

• nig

Regularization by the exponential Euler method