Inverse Problems and Applications - MAI

Inverse Problems and Applications

Linköping University, Sweden, April 2nd, 2013

Analysis of a forward problem in optical tomography H. Egger, M. Schlottbom

Dept. of Mathematics

Numerical Analysis and Scientific Computing TU Darmstadt

Optical tomography: Model & basic equations (Forward problem)

(σ,k) g

M ₁

Radiative Transfer Equation (RTE) v· ∇u+σu=

Z

V

k(·,·,v⁰)u(·,v⁰) dv⁰+f u(x,v) =g(x,v) ifn(x)·v<0

I σattenuation coefficient

I kscattering kernel

I uphoton density

I ObservationB:u7→M:=M(u) (linear)

I Forward operatorF: (σ,k)7→M(nonlinear)

1Scheel et al, First clinical evaluation of sagittal laser OT for detection of synovitis in arthritic finger joints,Ann Rheum Dis, 2005;64:239-245

Solvability of the radiative transfer equation inL^pspaces Setting

Solvability inL^∞ Solvability inL¹ Solvability inL^p Inverse Problem

Radiative Transfer Equation and basic assumptions

v· ∇u(x,v) +σ(x,v)u(x,v) = Z

V

k(x,v,v⁰)u(x,v⁰) dv⁰+f(x,v) inR ×V u= 0 onΓ₋:={(x,v)∈∂R ×V :n(x)·v<0}

I R ⊂R^d bounded and convex,∂R ∈C^0,1,V ⊂R^d.

I 0≤στ∈L^∞(R ×V),k(x,v,v⁰)≥0 σ_p(x,v⁰) :=

Z

V

k(x,v,v⁰) dv≤σ(x,v⁰) σ⁰_p(x,v) :=

Z

V

k(x,v,v⁰) dv⁰ ≤σ(x,v)

I weightω:= max{σ,τ⁻¹}>0 a.e.

L^p(ω) ={u:ω|u|^p∈L¹(R ×V)},L^∞(ω) =L^∞.

I W^p(ω) :={u∈L^p(ω) : _ω¹v· ∇u∈L^p(ω)}

R

x v

x+τ(x,v)v

Solvability without scattering

v∇u+σu=f, u|Γ₋= 0. (*)

Lemma:Forf/ω∈L^∞(R ×V) there exists a uniqueu∈W^∞(ω) solution of (*) and kukL^∞(R×V)≤ kf/ω|kL^∞(R×V)

Proof:Useω= max{σ,τ⁻¹}and define u(x,v) =u(x₀+tv,v) :=

Z t

0

e^{−Rstσ(x0+rv,v) dr}f(x₀+sv,v) ds. (1) Observe

I v∇u(x,v) = _dt^du(x₀+tv,v) =f(x,v)−σ(x,v)u(x,v).

I Rt

0e^{−Rstσ(x0+rv,v) dr}ω(x₀+sv,v) ds≤1.

Remark:k_ω¹v∇ukL^∞(R×V)≤ kf/ωkL^∞(R×V)+kukL^∞(R×V).

Contraction property

(v∇+σ)u=Ku, u|Γ₋= 0.

Lemma:Foru∈L^∞(R ×V) there holds

k(v∇+σ)⁻¹KukL^∞(R×V)≤ 1−e^{−kσp0τkL∞(R×V)}

kukL^∞(R×V), whereKu(x,v) :=R

Vk(x,v,v⁰)u(x,v⁰) dv⁰. Proof:By usingf =Kuin (1) there holds

|((v∇+σ)⁻¹Ku)(x,v)| ≤ Z t

0

e^{−Rstσ0p(x0+rv,v) dr}σ⁰_p(x0+sv,v) dskukL^∞(R×V)

≤ 1−e^{−kσp0τ|kL∞(R×V)}

kuk_L^∞_(R×V).

Solvability of RTE inL^∞

v∇u+σu=Ku+f, u|Γ₋= 0.

Theorem:Forf/ω∈L^∞(R ×V) the RTE has a unique solutionu∈W^∞(ω) with a-priori estimate

kuk_L^∞_(R×V)≤e^{kσ0pτ|kL∞(R×V)}kf/ωk_L^∞_(R×V) k1

ω∇ukL^∞(R×V)≤3e^{kσ0pτ|kL∞(R×V)}kf/ωkL^∞(R×V)

Proof:Φ:L^∞(R ×V)→L^∞(R ×V)

Φ(u) := (v∇+σ)⁻¹(Ku+f) is a contraction. Banach fixed point theorem.

Solvability inL¹: A duality argument(L¹(R ×V))⁰=L^∞(R ×V)

K⁰:L^∞(R ×V)→L^∞(R ×V), K⁰u(x,v⁰) :=

Z

V

k(x,v,v⁰)u(x,v) dv Lemma:Forw ∈L^∞(R ×V) there holds

k(−v· ∇+σ)⁻¹K⁰wk_L^∞_(R×V)≤ 1−e^{−kσpτ|kL∞(R×V)}

kwk_L^∞_(R×V) Corollary:[cf Bal, Jollivet 2008] Foru∈L¹(R ×V) there holds

kK(v∇+σ)⁻¹uk_L¹_(R×V)≤ 1−e^{−kσpτkL∞(R×V)}

kuk_L¹_(R×V)

Corollary:Φ:L¹(R ×V)→L¹(R ×V),Φ(w) :=K(v∇+σ)⁻¹w+fis a contraction.

Theorem:u:= (v· ∇+σ)⁻¹w ∈W¹(ω), withw =Φ(w), solves RTE v· ∇u+σu=Ku+f, u_|Γ− = 0.

Solvability of RTE inL^p,1≤p≤ ∞

v∇u+σu=Ku+f, u|Γ₋= 0.

Theorem:For any _ω^f ∈L^p(ω) the RTE has a unique solutionu∈W^p(ω) and kukL^p(ω) ≤Ckf/ωkL^p(ω)

k1

ωv· ∇ukL^p(ω) ≤3^1−1pCkf/ωkL^p(ω)

withC:= exp(¹_pkσ_pτk_L^∞_(R×V)+ (1−¹_p)kσ_p⁰τk_L^∞_(R×V)) andω={σ,τ⁻¹}.

Proof:Interpolation.

Remark:

I In general, constants are sharp.

I Ifσ >0, constants can be improved.

Conclusions

I GeneralL^p-solvability theory for RTE

I Parameters are allowed to vanish

I explicit constants

I Fixed point iteration≡source iteration converges even ifσ_p =σin allL^p

I Positive data implies positive solutions

I L^p-theory can be used to investigate continuity and differentiability of the parameter-solution map (σ,k)7→F(σ,k) (Hölder inequality)

Optical tomography: Reconstruction (Inverse Problem)

1. Given:Measurements (g,M) = (g,F(σ,k)).

2. Aim:Determineσ,k :F(σ,k) =M.

I Uniqueness/ Identifiability: [Choulli & Stefanov]

I Stability estimates: [Bal & Jollivet]

I noisy data→ill-posedness→regularization

Example: Lipschitz-continuityu=u(σ,k), ˜u=u( ˜σ,k):U:=u−u˜satisfies:

v· ∇U+σU=KU+ ( ˜σ−σ) ˜u.

A-priori estimates

kF(σ,k)−F( ˜σ,k)k₂ ≤CkUk₂≤Ck( ˜σ−σ) ˜uk₂

I L²regularization:k( ˜σ−σ) ˜uk₂ ≤ kσ˜−σk₂kuk˜ _∞

I H¹regularization:k( ˜σ−σ) ˜uk₂≤Ckσ˜ −σk_H¹kuk˜ _6/5

References

I V. Agoshkov.Boundary Value Problems for Transport Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, 1998.

I S. R. Arridge. Optical tomography in medical imaging.Inverse Problems, 15(2):R41–R93, 1999.

I G. Bal and A. Jollivet. Stability estimates in stationary inverse transport.Inverse Probl. Imaging, 2(4):427–454, 2008.

I K. M. Case and P. F. Zweifel. Existence and uniqueness theorems for the neutron transport equation.Journal of Mathematical Physics, 4(11):1376–1385, 1963.

I M. Choulli and P. Stefanov. An inverse boundary value problem for the stationary transport equation.Osaka J. Math., 36(1):87–104, 1998.

I R. Dautray and J. L. Lions.Mathematical Analysis and Numerical Methods for Science and Technology, Evolution Problems II, volume 6. Springer, Berlin, 1993.

I H. Egger and M. Schlottbom. A mixed variational framework for the radiative transfer equation.M3AS, 03(22), 2012.

I H. Egger and M. Schlottbom. On unique solvability for stationary radiative transfer with vanishing absorption. submitted, 2012.

I H. Egger and M. Schlottbom. Solvability of the stationary radiative transfer equation inLpspaces. in preparation, 2013.

I P. Stefanov and G. Uhlmann. An inverse source problem in optical molecular imaging.Anal. PDE, 1:115–126, 2008.

I V. S. Vladimirov. Mathematical Problems in the one-velocity Theory of Particle Transport.Atomic Energy of Canada Ltd. AECL-1661, translated from Transactions of the V.A. Steklov Mathematical Institute (61), 1961