hybrid inverse problems and systems of pde guillaume balcolton/archive/bal.pdf · elastography and...

University of Delaware August 2, 2013University of Delaware August 2, 2013University of Delaware August 2, 2013

Hybrid Inverse Problems and Systems of PDE

Guillaume Bal

Department of Applied Physics & Applied Mathematics

Columbia University

Parts joint with: Eric Bonnetier, Chenxi Guo, Francois Monard, Shari Moskow,

Kui Ren, John Schotland, Gunther Uhlmann, Faouzi Triki, Ting Zhou.


Hybrid Inverse Problems and Systems of PDE

Inverse NON scattering problems

Guillaume Bal

Department of Applied Physics & Applied Mathematics

Columbia University

Parts joint with: Eric Bonnetier, Chenxi Guo, Francois Monard, Shari Moskow,

Kui Ren, John Schotland, Gunther Uhlmann, Faouzi Triki, Ting Zhou.


High contrast

Electrical Impedance Tomography is an imaging modality reconstructing

high contrast conductivity from current-to-voltage measurements. It is

however low resolution. The same type of statements holds for (limited

frequency) inverse scattering problems [Colton-Kress 98; Cheney et al. 99;

Borcea 02].


Mathematical model: Calderon problem

EIT is modeled by:

−∇ · γ(x)∇u = 0 in X and u = f on ∂X.

Calderon problem: Reconstruction of γ(x) from knowledge of the Dirichlet-

to-Neumann map Λγ, where f 7→ Λγf = γν · ∇u|∂X on the boundary ∂X.

The Calderon problem is injective: Λγ = Λγ =⇒ γ = γ

[Sylvester-Uhlmann 87, Nachman 88, Novikov 88, Astala-Paivarinta 06, Haberman-Tataru 11].

The Calderon problem is “unstable”: The modulus of continuity is log-

arithmic [Alessandrini 88], which results in low resolution:

‖γ − γ‖X ≤ C∣∣∣ ln ‖Λγ − Λγ‖Y

∣∣∣−δ.


High resolution

Ultrasound Imaging (ultrasonography) is an imaging modality that pro-

vides high resolution. However, it displays low contrast in soft tissues.


High Contrast and High Resolution

High-contrast low-resolution modalities: OT, EIT, Elastography. Based

on elliptic models that do not propagate singularities (well).

High-resolution low-contrast (soft tissues): M.R.I, Ultrasound, X-ray

CT. Singularities propagate: WF(data) determines WF(parameters).

High-contrast & High-resolution:

Hybrid Inverse Problems (HIP): Physical

Coupling between one modality in each

category.

HIP are typically Low Signal.


The Photo-acoustics Effect

Coupling between (Near-Infra-Red) Radiation and Ultrasound.


Experimental results in Photoacoustics

Reconstruction of Ultrasound generated by Photo-Acoustic effect.

From Paul Beard’s Lab, University College London, UK.


Experimental results in Photoacoustics

Reconstruction of H(x) generated by Photo-Acoustic effect.

From Lihong Wang’s Lab (Washington University)


Elastography and Magnetic Resonance

Assessment of Hepatic Fibrosis by Liver Stiffness

Coupling between Elastic Waves and Magnetic Resonance Imaging

From Richard L. Ehman’s Lab (Mayo Clinic, Rochester, MN)


Elastography and Ultrasound

Electromechanical Wave Imaging (EWI) of the heart

Coupling between Transient Elastic Waves and Ultrasound

From Elisa Konofagou’s Lab (Columbia University)


Ultrasound Modulation

Optical tissue properties Modulated by Ultrasound.

Coupling between Optical or Electromagnetic Waves and Ultrasound

From Biomedical Photonic Imaging Lab (University of Twente).


Inverse Problems with Internal Functionals

• Applications in hydrology: ∇ · γ∇u = 0, H = u known. Richter, SIAP 81;

Alessandrini, ANS Pisa 87; Kohn&Lowe, M2AN 88

• MREIT and CDII (H = γ|∇u|; medical imaging); Nachman, Tamasan et al.

IP 07, IP 09, 11; Seo et al. SIAM Rev 11

• UMEIT (H = γ|∇u|2; medical); Ammari et al. SIAP 08; Gebauer&Scherzer

SIAP 08; Capdeboscq et al. SIIS 09; Kuchment&Kunyansky, IP 11;

• TE/MRE (H = u); J. McLaughlin et al. IP 04; IP 09; IP 10; G. Nakamura et al.

JAA 08; SIAP 11

• QPAT/QTAT and related (H = Γ|u|α; medical imaging); Cox et al. IP

07, JOSA 09; Ammari et al. 11; Gao et al. 11; Triki IP 11; Patrolia 12

• Elliptic Theory; Kuchment & Steinhauer IP 12

• Books: Ammari, Springer 08; O. Scherzer (Handbook) Springer 11;

Arridge and Scherzer, editors, special issue IP 12

• Exponentially increasing Bio-Engineering literature.


Papers available at www.columbia.edu/∼gb2030/pubs.html:• QPAT, QTAT, MRE, TEB.-Jollivet-Jugnon, Inverse transport theory of Photoacoustics, I.P. 26, 025011, 2010

B.-Uhlmann, Inverse Diffusion Theory of Photoacoustics, I.P. 26(8), 085010, 2010

B.-Ren, Multi-source Quantitative PAT in diffusive regime, I.P. 27(7), 075003, 2011

B.-Uhlmann, Reconstruction in second-order elliptic equations CPAM, 2013

B.-Ren-Uhlmann-Zhou, Quantitative Thermo-acoustics and related problems, I.P.

27(5), 055007, 2011 B.-Zhou, 2013

• Ultrasound Modulation

B., Cauchy problem and Ultrasound Modulated EIT, Anal. & PDE 2012

B.-Bonnetier-Monard-Triki, Inverse diffusion from knowledge of power densities 2012

Monard-B., Inverse diffusion problem with redundant internal information 2012

Monard-B., Inverse anisotropic diffusion in 2D, 2012; M.-B. Higher dimensions, 2012

B.-Guo-Monard 2013 B.-Naetar-Scherzer-Schotland 2013 B.-Moskow 2013

• Review paper

B., Hybrid inverse problems and internal measurements, Inside Out 2012

• Elliptic Theory and Uniqueness Theory

B., Hybrid inverse problems and redundant systems of P.D.E., arXiv:1210.0265.


Quantitative Thermo-Acoustic Tomography (QTAT)

• In the Maxwell model for radiation, quantitative TAT is −∇×∇× E + n(x)k2E + ikσ(x)E = 0 in X, ν × E = f on ∂X

H(x) = σ(x)|E|2(x) in X Internal Functional.

• Using a (scalar) Helmholtz model for radiation, quantitative TAT is ∆u+ n(x)k2u+ ikσ(x)u = 0 in X, u = f on ∂X Illumination,

H(x) = σ(x)|u|2(x) in X Internal Functional.

QTAT consists of uniquely and stably reconstructing σ(x) from knowl-

edge of H(x) for appropriate illuminations f .

This is an Inverse NON scattering problem: H(x) sits on top of singular-

ities of σ(x): we do not need (far field) scattering.


Stability result for scalar QTAT

∆u+ k2u+ iσ(x)u = 0 in X, u = f on ∂X, H(x) = σ(x)|u|2.

Theorem [B.,Ren,Uhlmann,Zhou’11] Let σ and σ be uniformly bounded

functions in Y = Hp(X) for p > n with X the bounded support of the

unknown conductivity.

Then there is an open set of illuminations f such that

H(x) = H(x) in Y implies that σ(x) = σ(x) in Y.

Moreover, there exists C such that ‖σ − σ‖Y ≤ C‖H − H‖Y .

The inverse scattering problem with internal data (inverse NON scat-

tering problem) is well posed by application of a Banach fixed point.


CGO solutions and Banach fixed point

• Let ρ ∈ Cn with |ρ| large and ρ · ρ = 0. Define q(x) = k2 + ikσ(x) on X

appropriately extended to x ∈ Rn.

• Then the reconstruction of σ is based on finding the unique fixed point

to the equation

σ(x) = e−(ρ+ρ)·xd(x)−Hg[σ](x) in Y.

• The functional Hg[σ] defined as

Hg[σ](x) = σ(x)(ψg(x) + ψg(x) + ψg(x)ψg(x)),

vanishes when q = 0 and is a contraction map for g properly chosen,

where ψg is defined as the solution to

(∆ + 2ρ · ∇)ψg = −q(1 + ψg), X, ψg = g ∂X.


Discontinuous conductivity in TAT

1

0 20 40 60 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Discontinuous conductivity in TAT

0 20 40 60 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Stability under smallness constraint

Consider more general equations of the form

P (x,D)u = σ(x)u in X with (Pu, u) ≥ α(u, u) and appropriate b.c.

This includes: the Maxwell case P = −ik−1(∇×∇×−k2) and the (scalar)

Helmholtz case P = ik−1(∆ + k2).

Then for small conductivities 0 < σ0 ≤ σ ≤ σM < α, H(x) = σ(x)|u|2

uniquely determines σ. Moreover, we have the stability result

‖σ1 − σ2‖L2(X) ≤C(g1, g2)

α− σM‖H1 −H2‖L2(X).

Extension to large σ for Maxwell? The same result should follow from

generalization of scalar CGOs to Maxwell CGOs. But this is wrong...


Back to QTAT

• Using a (scalar) Helmholtz model for radiation, QTAT is ∆u+ n(x)k2u+ ikσ(x)u = 0 in X, u = f on ∂X Illumination,

H(x) = σ(x)|u|2(x) in X Internal Functional.

• In the Maxwell model for radiation, quantitative TAT is −∇×∇× E + n(x)k2E + ikσ(x)E = 0 in X, ν × E = f on ∂X

H(x) = σ(x)|E|2(x) in X Internal Functional.

• With E ∼ ∇u, k small, and taking divergence, we get−∇ · σ(x)∇u = 0 in X, u = f on ∂X

H(x) = σ(x)|∇u|2(x) in X Internal Functional.

QTAT for Maxwell looks like a 0−Laplacian, which is a difficult problem.


The 0−Laplacian with J = 1

−∇ · γ(x)∇u = 0, γ(x)|∇u|2(x)−H(x) = 0 u = f on ∂X.

The elimination of γ yields the 0-Laplacian

−∇ ·H(x)

|∇u|2∇u = 0 in X, u = f on ∂X.

The above equation with Cauchy data may be transformed as

(I − 2∇u⊗ ∇u) : ∇2u+∇ lnH·∇u = 0 in X, u = f and∂u

∂ν= j on ∂X.

Here ∇u = ∇u|∇u|. This is a quasilinear strictly hyperbolic equation with

∇u(x) a “time-like” direction. Cauchy data generate stable solutions on

“space-like” part of ∂X for the Lorentzian metric (I − 2∇u⊗ ∇u).


Stability on domain of influence

Let u and u be two solutions of the hyperbolic equation and v = u− u.

IF (appropriate) Lorentzian metric is uniformly strictly hyperbolic, then:

Theorem [B. Anal&PDE 13]. Let Σ1 ⊂ Σg space-like component of ∂X

and O domain of influence of Σ1. For θ distance of O to boundary of

domain of influence of Σg, we have the local stability result:

∫O|v|2 + |∇v|2 + (γ − γ)2 dx ≤

C

θ2

( ∫Σ1

|δf |2 + |δj|2 dσ +∫O|∇δH|2 dx

),

where γ = H|∇u|2 and γ = H

|∇u|2. We observe the loss of one derivative

from δH to δγ (sub-elliptic estimate).


Domain of Influence

Domain of influence (blue) for metric g = I − 2ez ⊗ ez on sphere (red).

Null-like vectors (surface of cone) generate instabilities. Right: Sphere

(red), domains of uniqueness (blue) and with controlled stability (green).


Theoretical analyses of HIP

Can we find general theories for stability/uniqueness of (many) HIPs?

Can we understand role of number of measurements J, of B.C. fj?

Consider as an example the UMT (0−Laplacian with J = 2) problem

−∇ · γ(x)∇u1 = 0 in X, u1 = f1 on ∂X

−∇ · γ(x)∇u2 = 0 in X, u2 = f2 on ∂X

H1(x) = γ(x)∇u1(x) · ∇u1(x) in X

H2(x) = γ(x)∇u2(x) · ∇u2(x) in X

The left-hand side is a polynomial of γ, uj and their derivatives. This

forms a 4× 3 redundant system of nonlinear PDEs.






−∇ · γ(x)∇u1 = 0 in X, u1 = f1 on ∂X

−∇ · γ(x)∇u2 = 0 in X, u2 = f2 on ∂X

γ(x)∇u1(x) · ∇u1(x) = H1(x) in X

γ(x)∇u2(x) · ∇u2(x) = H2(x) in X


forms a 4× 3 redundant system of nonlinear PDEs.






−∇ · γ(x)∇u1 = 0 in X, u1 = f1 on ∂X

−∇ · γ(x)∇u2 = 0 in X, u2 = f2 on ∂X

γ(x)∇u1(x) · ∇u1(x) = H1(x) in X

γ(x)∇u2(x) · ∇u2(x) = H2(x) in X


forms a 4× 3 redundant system of nonlinear PDEs in X.


Systems of coupled nonlinear equations

Hybrid inverse problems may be recast as the system of PDE:

F(γ, uj1≤j≤J) = H, (1)

where γ are unknown parameters and uj are PDE solutions.

For UMEIT, we have

F(γ, uj1≤j≤J) =

(−∇ · γ∇ujγ|∇uj|2

), H =

(0Hj

), 2J − rows .

Then, (1) is a redundant 2J × (J + m) system of nonlinear equations with J + m

unknowns (m = 1 if γ is scalar).

HIP theory concerns uniqueness, stability, reconstruction procedures for

typically redundant (over-determined) systems of the form (1) with ap-

propriate boundary conditions.


Elliptic Theory for larger J

Consider a redundant system of the form

−∇ · γ(x)∇uj = 0, γ(x)|∇uj|2(x) = Hj(x), uj|∂X = fj, 1 ≤ j ≤ J.

• With J = 1, the system is hyperbolic.

• With J ≥ 2, the redundant system 2J × (J + 1) may be elliptic.

• After linearization, we obtain the system:

∇ · δγ∇uj +∇ · γ∇δuj = 0 (2)

δγ|∇uj|2 + 2γ∇uj · ∇δuj = δHj. (3)

With v = (δγ, δu1, . . . , δuJ), we recast the above system for v as

Av := (PJ +RJ)v = S

where PJ is the principal part and RJ is lower order.

Let us define Fj = ∇uj. The symbol of this 2J × (J + 1) system is:

pJ(x, ξ) =

|F1|2 2γF1 · iξ . . . 0F1 · iξ −γ|ξ|2 . . . 0

... ... . . . ...|FJ |2 0 . . . 2γFJ · iξFJ · iξ 0 . . . −γ|ξ|2

.

• System said elliptic when pJ(x, ξ) maximal rank (J+1) for all ξ ∈ Sn−1.

(i) Redundant concatenation of hyperbolic systems (J = 1) may be elliptic.

(ii) pJ elliptic IF we choose fj s.t. the following qualitative statement on

quadratic forms holds:|ξ|2 − 2(Fj · ξ)2 = 0, 1 ≤ j ≤ J

implies ξ = 0.

For ellipticity, we thus want the light cones generated by the directions

Fj to intersect to 0. (shown to hold for appropriate boundary conditions fj.)


Theory of Redundant elliptic systems

• The system is elliptic in the sense of Douglis and Nirenberg.

Each row and column is given an index si and tj and the principal term is the homogeneous differential

operator of order si + tj. For the above system, we choose s2k+1 = 0, s2k = 1, t1 = 0, tk≥2 = 1.

• We need boundary conditions that satisfy the Lopatinskii condition.

Dirichlet conditions on δuj and no condition on δγ satisfy the LC.

Indeed, we need to show that v(z) = (δγ(z), . . . , δuJ(z)) ≡ 0 is the only solution to

δuj(0) = 0, Fj ·N∂zδγ + γ∂2z δuj = 0, |Fj|2δγ + 2γFj ·N∂zδuj = 0, z > 0

vanishing as z →∞ for N = ν(x) at x ∈ ∂X and z coordinate along −N . We observe that this is the case

if |Fj|2 − 2(Fj ·N)2 6= 0 for some j. This is the condition for joint ellipticity.

• Theory of Agmon-Douglis-Nirenberg extended to over-determined sys-

tems by Solonnikov shows that Av = S (including b.c.) admits a left-

parametrix R so that RA = I − T with T compact.


Elliptic stability estimates

From the ADN-Sol. theory results the Stability estimates

J+1∑j=1

‖vj‖Hl+tj(X)≤ C

2J∑i=1

‖Si‖Hl−si(X)+ C2

∑tj>0

‖vj‖L2(X).

For the UMEIT example (Hj = γ|∇uj|2), this is:

‖δγ‖Hl(X) +∑j

‖δuj‖Hl+1(X) ≤ C∑j

‖δHj‖Hl(X) + C2∑j

‖δuj‖L2(X).

• No loss of derivatives from δH to δγ: Optimal Stability (unlike J = 1).

• We do not have injectivity of the system (C2 6= 0): A can be inverted

up to a finite dimensional kernel with RA Fredholm of index 0.


Injectivity: Holmgren, Carleman, and Calderon

• Assume A is elliptic in the regular sense, i.e., tj = t and si = 0. Consider, with

t = 2, the two problems

Av = S, v|∂X = 0, and AtAv = AtS, v|∂X = ∂νv|∂X = 0.

The second system is (J+1)×(J+1)- determined even if the first one is

2J × (J + 1) redundant. It provides an explicit reconstruction procedure.

Moreover, injectivity of the second one implies injectivity of the redundant

(both in X and on ∂X) system:

Av = 0, v|∂X = ∂νv|∂X = 0.

• Injectivity for such a system can be proved by Holmgren’s theorem

when A has analytic coefficients and by Carleman estimates, as obtained

for systems in Calderon’s theorem, for a restricted class of operators A.


Invertibility and Local Uniqueness for Nonlinear I.P.

Recast original nonlinear I.P. as

F(v0 + v) = H, H0 := F(v0), A = F ′(v0).

IF A admits a bounded left inverse (F ′)−1(v0), then:

v = G(v) := (F ′)−1(v0)(H−H0)−(F ′)−1(v0)(F(v0+v)−F(v0)−F ′(v0)v

).

G(v) contraction when H−H0 small: local uniqueness result for HIP.

Similarly, a Newton algorithm of the following form may converge

vk+1 − vk = (AtkAk)−1Atk

(H−F(v0 + vk)

), Ak = F ′(v0 + vk).


UMOT reconstructions, algo 1

Reconstruction (Newton iterations based on system AtAv = AtS) with:

(i) one H; (ii) two H without ellipticity; (iii) two H with ellipticity;

(iv) true conductivity.

Calculations by Kristoffer Hoffmann (DTU and CU).


Back to the QTAT problem

Consider the QTAT problem with several measurements J > 1 and two

unknown (real valued) coefficients

−∇×∇× Ej + (k2n(x) + ikσ(x))Ej = 0 in X, ν × Ej = fj on ∂X.

The internal information is Hj(x) = σ(x)|Ej|2(x) for 1 ≤ j ≤ J.

Theorem [B. Zhou 13]. For J = 4, there is an open set of b.c. fjs.t. the linearization of the above system is elliptic. Under additional

assumptions, it is injective, and for l > 72, we obtain the local stability

‖n− n‖Hl(X) + ‖σ − σ‖Hl(X) + ‖Ej − Ej‖Hl(X;CJ) . ‖Hj − Hj‖Hl(X;CJ)


Elements of proof of ellipticity

After linearization about (σ, n, Ej), the principal symbol of the linearized

operator is elliptic iif we have

|Ej|2|Ek|2(|Ej · ξ|2 − |Ek · ξ|2) = 0 for 1 ≤ j < k ≤ J ⇒ ξ = 0

We need enough solutions of −∇×∇×Ej+qEj = 0 s.t. above holds. We

use CGO solutions for Maxwell of Colton-Paivarinta for ρ ⊥ ρ⊥, s > 0,

a ∈ C3, b ∈ C3 of the form Eζ(x) = γ−12eix·ζ(ηζ +Rζ(x))

with γ =q

k2, ζ = −isρ+

√s2 + k2ρ⊥, ηζ =

−ζ · aζ − kζ × b+ k2a

|ζ|For J = 4 well-chosen members of such a family, we verify that Rζj is

negligible (in appropriate topology) and the above ellipticity condition is

satisfied.


Comparison Helmholtz vs Maxwell

Assume n known (we thus look to reconstruct σ).

Linearization of Helmholtz (∆ + k2n+ ikσ)u = 0 with H = σ|u|2:

|u|2δσ = δH + lower order terms.

Llinearization of Maxwell (−∇×∇×+k2n+ ikσ)E = 0 with H = σ|E|2:(|E|2∆−

2k2σ2

k2σ2 + k4n2E ⊗ E : ∇⊗∇

)δσ = ∆δH + lower order terms.

The Helmholtz operator is always elliptic since |u|2 > 0.

The Maxwell operator is elliptic provided that

2k2σ2

k2σ2 + k4n2< 1 or σ < kn.

We recover that QTAT-Maxwell is elliptic-stable when σ is small. When it

is not, then the Helmholtz and Maxwell models are qualitatively different.


Anisotropic reconstructions

Consider Maxwell’s system of equations

∇× Ej = ıωµ0Hj and ∇×Hj = γEj

where γ = σ − ıωε in X with boundary conditions ν × Ej|∂X = fj. We

assume µ0 constant known and γ an unknown symmetric tensor with real

and imaginary parts uniformly elliptic.

The first step of MREIT or CDII provides knowledge of Hj1≤j≤J. The

second step consists of reconstructing the complex-valued tensor γ.

Theorem. [Guo-B. 2013] There exists an open set of boundary condi-

tions fj1≤j≤J such that Hj uniquely determines γ. Moreover,

‖γ − γ‖Hs(X;C6) ≤ C‖H − H‖Hs+2(X;CJ×3), s sufficiently large.

In some cases, we may choose J = 6 or J = 7.


Elements of derivation

Algebraic manipulations show that

γ−1 : (ΩZkST )sym = tr(ΩMT

k ), A : B = tr(ATB)

for all anti-symmetric Ω and all 1 ≤ k ≤ J−3, where Zk, S, and M depend

on the information Hj. For instance the columns of Zk are given by

Zk,i = −∇det(F1,...,

i︷︸︸︷Fn+k,...,Fn)

det(F1,...,Fn) , Fj = ∇×Hj.

We thus need to construct enough (ΩZkST )sym to span all symmetric

complex-valued matrices.

• For γ close to isotropic, this is done by CGO solutions, as in Colton-

Paivarinta 92, Chen-Yang 2013, B.-Zhou 2013.

• For more general γ, this is done locally and controlled from the boundary

by a Runge approximation, as in B. Uhlmann 2013 in the scalar case.


UMEIT Data Hij(x)

Three maps H11, H12, and H22 among (Hij)1≤i,j≤4.


Explicit reconstructions from UMEIT data

Anisotropy No noise 1% noise Determinant


Conclusions

• Hybrid imaging modalities provide stable inverse problems combining

high resolution with high contrast (though they are Low Signal).

• Tensors and Complex-valued coefficients can be reconstructed to

account for anisotropy and dispersion effects.

• Inverse NON scattering and many other Hybrid inverse problems

well posed problems.

• Thank you David for NOT having worked on Inverse NON scattering.


Happy Birthday David !

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