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Mathematical Optimization and INSTITUTE OF MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Inverse problems with L 1 data fitting Christian Clason, Bangti JIN, Karl Kunisch Institute for Mathematics and Scientific

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Inverse problems with L1 data fitting

Christian Clason, Bangti JIN, Karl Kunisch

Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Graz

Applied Inverse Problems 2009 Wien, July 24, 2009

1 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Inverse problem

Find x such that Kx = yδ K : L2(Ω) → L2(Ω) bounded linear operator

Ω ⊂ Rn bounded domain

yδ ∈ L2(Ω) noisy measurement Impulsive noise (e.g., salt & pepper, random valued)

2 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Inverse problem

L1 data fitting

(P) min

x∈L2Kx − yδL1 + α

2 x2

L2

More robust in the presence of outliers Applicable in image processing, signal processing (single pixel failure) Challenge: Nondifferentiable functional, noise level unknown Regularization assumes smooth solution; alternative: TV

3 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Fenchel duality

V, Y Banach spaces, topological duals V ∗, Y ∗

Λ ∈ L(V, Y), F : V → R ∪ {∞}, G : Y → R ∪ {∞}

Fenchel conjugate of F:

F∗ : V ∗ → R ∪ {∞} F∗(v∗) = sup

v∈V

v∗, vV ∗,V − F(v)

4 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Fenchel duality

Fenchel duality theorem F, G convex and lower semicontinuous ∃v0 ∈ V: F(v0) < ∞, G(Λv0) < ∞, G continuous at Λv0:

(FD) inf

v∈V F(v) + G(Λv) = sup q∈Y ∗ −F∗(Λ∗q) − G∗(−q)

Extremality relations: v, q solutions of (FD) iff (ER)

  • Λ∗q ∈ ∂F(v),

−q ∈ ∂G(Λv),

4 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Dual problem

Define F : L2 → R, F(v) = α

2 v2

L2 ,

G : L2 → R, G(v) = v − yδL1, Λ : L2 → L2, Λv = Kv. Fenchel conjugates F∗ : L2 → R, F∗(q) = 1

2α q2

L2 ,

G∗ : L2 → R ∪ {∞}, G∗(q) =

  • q, yδL2

if qL∞ ≤ 1,

∞

if qL∞ > 1.

5 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Dual problem

Dual problem

(P∗)

  

min

p∈L2

1 2α K ∗p2

L2 − p, yδL2

s.t.

pL∞ ≤ 1,

Fenchel duality theorem: (P∗) has solution pα Solution not unique if ker K ∗ = {0}!

6 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Dual problem

Solutions xα, pα related by

  • K ∗pα = αxα,

0 ≤ Kxα − yδ, p − pαL2, for all p ∈ L2 with pL∞ ≤ 1. Given a solution pα, unique solution of (P): xα = 1

αK ∗pα

7 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Characterization of minimizer

For all p ∈ L2, p ≥ 0:

Kxα − yδ, pL2 =

if supp p ⊂ {x : |pα(x)| < 1} ,

Kxα − yδ, pL2 ≥

if supp p ⊂ {x : pα(x) = 1} ,

Kxα − yδ, pL2 ≤

if supp p ⊂ {x : pα(x) = −1} .

Interpretation:

Box constraint on pα active where data is not attained by xα Sign of pα gives sign of noise

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Regularization of dual problem

(P∗)

  

min

p∈L2

1 2α K ∗p2

L2 − p, yδL2

s.t.

pL∞ ≤ 1,

Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0

9 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Regularization of dual problem

(P∗

c )

      

min

p∈H1

1 2α K ∗p2

L2 − p, yδL2

+ 1

2c max(0, c(p − 1))2

L2 + 1

2c min(0, c(p + 1))2

L2

Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0

9 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Regularization of dual problem

(P∗

β,c)

      

min

p∈H1

1 2α K ∗p2

L2 + β

2 ∇p2

L2 − p, yδL2

+ 1

2c max(0, c(p − 1))2

L2 + 1

2c min(0, c(p + 1))2

L2

Non-differentiable problem replaced by smooth box-constrained problem Moreau-Yosida regularization for c > 0 ⇒ efficient solution by semismooth Newton method Superlinear convergence needs norm gap: Add smoothing term with β > 0

9 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Existence and convergence of minimizers

Regularized problem

min

p∈H1

1 2α K ∗p2

L2 + β

2 ∇p2

L2 − p, yδL2

+ 1

2c max(0, c(p − 1))2

L2 + 1

2c min(0, c(p + 1))2

L2

Strictly convex if ker K ∗ ∩ ker ∇ = {0}

⇒ Existence of unique minimizer pβ,c Theorem (Convergence)

pβ,c

H1

− − − →

c→∞ pβ L2

− − − ⇀

β→0 pα,

pβ,c

L2

− − − ⇀

β→0 pc

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Optimality system (regularized)

  

1

αKK ∗pc − β∆pc − yδ + λc = 0, λc = max(0, c(pc − 1)) + min(0, c(pc + 1))

Nonlinear equation for pc Pointwise max, min semismooth

⇒ solution by generalized Newton method

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Semismoothness in function spaces

X, Y Banach spaces, D ⊂ X open

Definition

F : D ⊂ X → Y Newton differentiable at x ∈ D, if there is neighborhood N(x), G : N(x) → L(X, Z)

F(x + h) − F(x) − G(x+h)h = o(h)

Set {G(s) : s ∈ N(x)} Newton derivative of F at x.

Definition

F semismooth if N-differentiable and G(s)−1 uniformly bounded. F semismooth ⇒ generalized Newton method G(sk)δx = −F(xk), sk ∈ N(xk), converges locally superlinearly.

12 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Semismoothness of projection operator

Projection operator

P(p) := max(0, (p − 1)) + min(0, (p + 1)) is semismooth from Lq to Lp, if and only if q > p,

Newton derivative

DNP(p)h = hχ{|p|>1} :=

  • h(x)

if |p(x)| > 1, if |p(x)| ≤ 1.

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Application to optimality system

Can be written as F(p) = 0 with F : H1 → (H1)∗, F(p) := 1

αKK ∗p − β∆p + max(0, c(p − 1)) + min(0, c(p + 1)) − yδ

Sobolev embedding, sum and chain rule for Newton derivatives

→ F is semismooth, Newton derivative Newton derivative

DNF(p)h = 1

αKK ∗h − β∆h + chχ{|p|>1}

14 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Computation of Newton step

Active sets A+

k :=

  • x : pk(x) > 1
  • A−

k :=

  • x : pk(x) < −1
  • Ak :=A+

k ∪ A− k

Newton step

Given pk, solve for pk+1 1

αKK ∗pk+1 − β∆pk+1 + cχAk pk+1 = yδ + c(χA+

k − χA− k ) 15 / 35 Problem formulation Solution of optimality system Parameter choice method Numerical results

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Convergence of semismooth Newton method

Theorem

If pc − p0H1 is sufficiently small, semismooth Newton method converges superlinearly in H1 to pβ,c.

Theorem (Termination criterion) A+

k+1 = A+ k and A− k+1 = A− k ⇒ pk+1 satisfies F(pk+1) = 0.

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Choice of β

Sufficiently large β > 0 necessary for numerical stability of Newton step

1 αKK ∗ − β∆ + cχAk

  • pk+1 = yδ + c(χA+

k − χA− k )

(KK ∗ ill-conditioned, χAk rank deficient) but introduces unwanted smoothing

⇒ optimal β: as small as possible!

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Choice of β

Observation:

System matrix well-conditioned ⇒ solution pβ,c feasible System matrix (numerically) singular ⇒

  • pk+1
  • ∞ ≈ c ≫ 1

Continuation strategy

Choose β0 > 0 Compute pβn,c While pβn,c∞ ≤ 1 + ε, set βn+1 = 1

τ βn, continue

minimization Else take last feasible iterate pβN−1,c

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Choice of α

xα minimizer for fixed α:

Value function

F(α) = Kxα − yδL1 + α 2 xα2

L2

F(α) continuous, increasing, differentiable

Derivative

F ′(α) = 1 2 xα2

L2

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Balancing principle

Idea:

Balance data residual (increasing in α)

ϕ(α) = Kxα − yδL1

with penalty term (decreasing in α)

αF ′(α) = α

2 x2

L2

Choose α∗ such that (σ − 1)ϕ(α∗) = α∗F ′(α∗) σ > 1 controls relative weight

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Model function

Padé approximation of value function

m(α) = b + c t + α. c, t from interpolation conditions m(α) = F(α), m′(α) = F ′(α), b from asymptotic α → ∞: F(α) → yδL1, m(α) → b b = yδL1

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Fixed point iteration

1 Compute xαk (semismooth Newton method) 2 Evaluate F(αk) and F ′(αk) 3 Construct model function mk(α) = b + ck tk+α 4 Calculate intercept ˆ

m of tangent of mk(α) at (αk, F(αk)):

ˆ

m = F(αk) − αkF ′(αk),

5 Solve for αk+1 in mk(αk+1) = σ ˆ

m

6 repeat

Fixed point α∗ satisfies

F(α∗) = σ(F(α∗) − α∗F ′(α∗))

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Fixed point iteration

Theorem (Existence)

For σ sufficiently close to 1 and yδ = 0, there exists at least one positive solution α∗ to the balancing equation

(σ − 1)ϕ(α) − αF ′(α) = 0. Theorem (Convergence)

If initial guess α0 satisfies

(σ − 1)ϕ(α0) − α0F ′(α0) < 0,

fixed point iteration monotonically is decreasing, converges to α∗. (Other inequality needs assumption on sign change close to α∗)

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Noise estimate

Noise level satisfies

F(0) = min

x Kx − yδL1 ≤ Kx∗ − yδL1

Model function at final iteration k good approximation of F:

Noise level estimate

mk(0) ≈ δ := y† − yδL1 y† exact solution

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Numerical examples

Examples from Regularization Tools

http://www2.imm.dtu.dk/~pch/Regutools/

Comparison with iteratively reweighted least squares (IRLS) Rodriguez, Wohlberg, IEEE Trans. Imag. Process., 18 (2009) Implementation in Matlab Code available:

http://www.uni-graz.at/~clason/codes/l1fitting.zip

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Noise

Impulsive random noise

yδ =

  • y† + εξ,

with probability r, y,

  • therwise,

y† exact data

ξ normally distributed with mean 0 and standard deviation 1

r = 0.3, ε = 1 not salt & pepper!

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

Inverse heat conduction: heat

K is Volterra integral operator of the first kind,

(Kx)(t) = t

k(s, t)x(s) ds, k(s, t) = (s − t)− 3

2

2√π e−

1 4(s−t)

Exponentially ill-posed, condition number ≈ 1037

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heat: Results

10 20 30 40 50 60 70 80 90 100 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 exact noisy 10 20 30 40 50 60 70 80 90 100 −0.2 0.2 0.4 0.6 0.8 1 1.2 L1 (α=2.24e−02) exact

left: noisy data, right: adaptive α

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heat: Results

10 20 30 40 50 60 70 80 90 100 −0.2 0.2 0.4 0.6 0.8 1 1.2 L1 (α=2.24e−02) exact 10 20 30 40 50 60 70 80 90 100 −0.2 0.2 0.4 0.6 0.8 1 1.2 exact L1 (α=2.01e−02) L2 (α=3.13e−03)

left: adaptive α, right: optimal α and L2 minimizer

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heat: Results

10 20 30 40 50 60 70 80 90 100 −0.2 −0.15 −0.1 −0.05 0.05 0.1 10 20 30 40 50 60 70 80 90 100 −1 1 noise p

noise and dual solution p

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

heat: Semismooth Newton method

Iterates in the path-following method for β:

β

iterations e F(x)

∇pL2

1.000e+0 2 2.248e-1 3.274e-2 2.951e-2 4.000e-2 2 1.855e-1 1.963e-2 1.090e-1 1.600e-3 2 1.610e-1 1.713e-2 2.191e-1 6.400e-5 2 1.556e-1 1.737e-2 1.408e+0 2.560e-6 6 2.250e-1 1.644e-2 5.852e+0 1.024e-7 4 7.042e-2 1.435e-2 6.436e+0 4.096e-9 3 2.043e-2 1.415e-2 6.902e+0 1.638e-10 10 1.563e-2 1.414e-2 7.361e+0 3.277e-11 10 1.546e-2 1.414e-2 8.960e+0

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE OF MATHEMATICS AND SCIENTIFIC COMPUTING

heat: Comparison with IRLS

Reconstruction quality identical (hence not shown) Semismooth Newton method faster: Computing time (in seconds) for the SSN vs. IRLS method n 50 100 200 400 800 1600 tssn (sec) 0.011 0.034 0.188 1.163 8.052 39.07 tirls (sec) 0.165 0.430 2.009 14.09 110.8 723.0

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heat: Model function

Convergence in one iteration b: balancing principle, opt: optimal, e: error

(r, ǫ) δ δb αb αopt eb eopt (0.3,0.1) 1.390e-3 1.335e-3 1.402e-3 1.830e-2 1.860e-1 2.026e-2 (0.3,0.3) 4.170e-3 4.155e-3 6.638e-3 1.830e-2 4.515e-2 2.026e-2 (0.3,0.5) 6.950e-3 6.939e-3 1.135e-2 1.830e-2 2.706e-2 2.026e-2 (0.3,0.7) 9.731e-3 9.719e-3 1.604e-2 1.830e-2 2.103e-2 2.026e-2 (0.3,0.9) 1.251e-2 1.249e-2 2.083e-2 1.830e-2 2.037e-2 2.026e-2 (0.1,0.3) 7.438e-4 7.439e-4 1.227e-3 7.742e-4 1.727e-3 5.980e-4 (0.3,0.3) 4.170e-3 4.155e-3 6.638e-3 1.830e-2 4.515e-2 2.026e-2 (0.5,0.3) 7.871e-3 7.799e-3 1.225e-2 3.199e-2 5.635e-2 3.772e-2 (0.7,0.3) 1.110e-2 1.074e-2 2.254e-2 7.390e-2 1.118e-1 1.034e-1 (0.9,0.3) 1.570e-2 1.470e-2 2.247e-2 5.094e-2 1.662e-1 1.388e-1

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Inverse source problem (2D)

K is ∆−1 Discretization 64 × 64, n = 4096 Mildly ill-posed, condition number ≈ 103 Convergence of fixed point iteration in 3 iterations,

α = 8.797 × 10−3

Noise estimate 5.475 × 10−3 (exact: 5.490 × 10−3)

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Inverse source problem: Results

left: exact solution, right: noisy data

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Inverse source problem: Results

left: exact solution, right: reconstruction

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Conclusion

L1 data fitting term for impulsive noise Treatment of nondifferentiability via Fenchel duality and semismooth Newton method Automatic regularization parameter choice from balancing principle via model function and fixed point iteration Current work: Extension to L1 − TV

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