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Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012
Hybrid Inverse Problems and Internal Functionals Guillaume Bal
Department of Applied Physics & Applied Mathematics Columbia University Joint w.: C´ edric Bellis, Eric Bonnetier, Matias Courdurier, Chenxi Guo, S´ ebastien Imperiale, Alexandre Jollivet, Vincent Jugnon, Fran¸ cois Monard, Shari Moskow, Kui Ren, Gunther Uhlmann, Faouzi Triki, Ting Zhou.
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High Contrast and High Resolution in Imaging
Some imaging techniques are high-contrast low-resolution. Typically based on elliptic models that do not propagate singularities. WF(data) not affected by (most of) WF(parameters). Such methods include Op- tical, Electrical Impedance Tomography, and Elastography. Other techniques are (sometimes) high-resolution low-contrast. They are based on the Fourier transform, wave propagation, or integral geom- etry. WF(data) determines WF(parameters) and injectivity sometimes
- holds. Such methods include M.R.I, Ultrasound, X-ray CT.
Hybrid (coupled-physics) Inverse Problems result from the physical coupling of one modality in each category. They combine high-contrast with high-resolution. Mathematically, they take the form of inverse prob- lems for the high-contrast parameters from knowledge of internal func- tionals obtained from the high-resolution modality.
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Hybrid Inverse Problems (HIP) are typically Low Signal.
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Examples of physical couplings
In Photo-acoustic Tomography, light propagates through a domain. Ab- sorbed photons generate a thermal expansion and the emission of ul-
- trasound. This is the photo-acoustic effect. Boundary ultrasound mea-
surements are first inverted to provide high resolution photon absorption maps, which are internal functionals of high-contrast optical parameters. In Transient Elastography, elastic waves are generated. The resulting displacement is imaged by high-resolution ultrasound. The elastic dis- placement is a functional of high-contrast elastic parameters. Other hybrid inverse problems include Acousto-Optics, Thermo-acoustics, Magnetic Resonance Elastography, Magnetic Resonance Electrical Impedance Tomography, Ultrasound Modulated Tomography, etc.
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Examples of Hybrid Inverse Problems
- We consider PDE models of the form:
−∇ · γ(x)∇u + σ(x)u = 0 in X, u = f on ∂X −∇ × ∇ × E + n(x)k2E + iσ(x)E = 0 in X, ν × E = g on ∂X
- We consider internal functionals of the form:
H(x) = Γ(x)σ(x) u(x) Photo-acoustics H(x) = u(x) Transient (MR) Elastography H(x) = σ(x) |u|2(x) or σ(x)|E|2(x) Thermo-acoustics H(x) = γ(x) ∇u(x) · ∇u(x) Ultrasound Modulation
- We have one or several illuminations f = fj (and thus H = Hj).
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Inverse Problems with Internal Functionals
- Applications in hydrology: ∇ · γ∇u = 0, 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; Kuchment&Steinhauer 12
- TE/MRE (H = u); J. McLaughlin et al. IP 04; IP 09; IP 10; G. Naka-
mura 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
- Books: Ammari, Springer 08; O. Scherzer (Handbook) Springer 11.
- Exponentially increasing Bio-Engineering literature.
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B.-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 of coefficients in scalar second-order elliptic equations from knowledge of their solutions, CPAM, 2012
B.-Ren-Uhlmann-Zhou, Quantitative Thermo-acoustics and related problems, I.P. 27(5), 055007, 2011 B.-Zhou, 2013 B.-Ren, Non-uniqueness results for a hybrid inverse problem, Cont. Math, 559, 2011
B., Cauchy problem and Ultrasound Modulated EIT, 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, 2013 B.-Guo-Monard 2013 B.-Imperiale 2013 B.-Moskow 2013
B., Hybrid inverse problems and internal measurements, Inside Out 2012
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Ultrasound modulation problem
Consider the 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.
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Ultrasound modulation problem
Consider the problem −∇ · 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 The left-hand side is a polynomial of γ, uj and their derivatives. This forms a 4 × 3 redundant system of nonlinear PDEs.
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Ultrasound modulation problem
Consider the problem −∇ · 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 The left-hand side is a polynomial of γ, uj and their derivatives. This forms a 4 × 3 redundant system of nonlinear PDEs in X.
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Systems of coupled nonlinear equations
The above hybrid inverse problems may be recast as F(γ, {uj}1≤j≤J) = H, (1) where γ is the collection of unknown parameters and uj are PDE solu-
- tions. For instance, for the ultrasound modulation problem, we have
F(γ, {uj}1≤j≤J) =
γ|∇uj|2
H =
2J − rows . so that (1) is a redundant 2J × (J + m) system of nonlinear equations with m number of unknowns in γ so that m = 1 if γ is scalar. HIP theory therefore concerns uniqueness, stability estimates, recon- struction procedures for typically redundant (over-determined) systems
- f the form (1) with appropriate boundary conditions.
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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 is possible and yields the 0-Laplacian −∇ · H(x) |∇u|2∇u = 0 in X, u = f
The above equation with Cauchy data may be transformed as (I − 2 ∇u ⊗ ∇u) : ∇2u+∇ ln H·∇u = 0 in X, u = f and ∂u ∂ν = j
Here ∇u = ∇u
|∇u|. Thus gij = (I − 2
∇u ⊗ ∇u)ij is a definite matrix of sig- nature (1, n − 1). We thus have a quasilinear strictly hyperbolic equation with ∇u(x) the “time” direction. Cauchy data generate stable solutions
- n “space-like” part of ∂X for the metric g.
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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. 12]. Let Σ1 ⊂ Σg the space-like component of ∂X and O the domain of influence of Σ1. For θ the distance of O to the boundary
- f the 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 +
where γ =
H |∇u|2 and ˜
γ =
˜ H |∇˜ u|2 are the reconstructed conductivities. We
- bserve the loss of one derivative from δH to δγ (sub-elliptic estimate).
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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).
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Linearization and Ellipticization
Consider as an example for R ∋ α > 0 a redundant system of the form −∇·γ(x)∇uj = 0, γα(x)|∇uj|2(x)−Hj(x) = 0, uj|∂X = fj, 1 ≤ j ≤ J. With 1 ≤ j ≤ J = 1 and α < 2, the system is hyperbolic. How about the redundant system 2J × (J + 1) for J ≥ 2? It may be elliptic. Linearizing gives the system: ∇ · δγ∇uj + ∇ · γ∇δuj = (2) αγα−1δγ|∇uj|2 + 2γα∇uj · ∇δuj = δHj. (3) Rewrite this system as Av = S 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.
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SLIDE 16 Let us define Fj = ∇uj. The symbol of this 2J × (J + 1) system is: pJ(x, ξ) =
αγα−1|F1|2 2γαF1 · iξ . . . F1 · iξ −γ|ξ|2 . . . . . . . . . ... . . . αγα−1|FJ|2 . . . 2γαFJ · iξ FJ · iξ . . . −γ|ξ|2
. The system is elliptic when pJ(x, ξ) is maximal rank (J+1) for all ξ ∈ Sn−1. Take the row |F1|2 and all rows with γ|ξ|2. The determinant is propor- tional to α − 2( ˆ F1 · ˆ ξ)2. When α > 2, the system is elliptic with J = 1. Repeat for all rows with |Fj|2. When α < 2, the system becomes ellip- tic for a sufficiently rich family of ˆ Fj =
Fj · ˆ ξ)2 for all 1 ≤ j ≤ J cannot be satisfied by any ˆ ξ ∈ Sn−1. (i) Redundant concatenation of hyperbolic systems becomes elliptic. (ii) Possible IF we choose fj s.t. the following qualitative statement on quadratic forms holds: α|ξ|2 − 2( ˆ Fj · ξ)2 = 0, 1 ≤ j ≤ J implies ξ = 0.
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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 ho- mogeneous differential operator of order si + tj. For the above system, we choose s2k+1 = 0, s2k = 1, t1 = 0, tk≥2 = 1. The system needs to be augmented with boundary conditions that sat- isfy the Lopatinskii condition. Dirichlet conditions on δuj and no con- dition 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∂2
z 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 α|Fj|2 − 2(Fj · N)2 = 0 for some j. This is the condition for joint ellipticity.
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We can then use the theory of Agmon-Douglis-Nirenberg extended to redundant systems by Solonnikov (J. Sov. Math. 73) to obtain that the system Av = S (including b.c.) admits a left-parametrix R so that RA = I − T, T compact. Moreover, we have elliptic stability (regularity) estimates.
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Elliptic stability estimates
Stability (regularity) estimates for redundant DN systems take the form
J+1
vjHl+tj(X) ≤ C
2J
SiHl−si(X) + C2
vjL2(X). For the example (2)-(3), we have for v1 = δγ that si = 0 when Si = 0 and t1 = 0. Thus, δγHl(X) +
δujHl+1(X) ≤ C
δHjHl(X) + C2
δujL2(X). This implies no loss of derivatives from δH to δγ. However, we do not have injectivity of the system since C2 = 0 a priori. This shows that the system A can be inverted up to a finite dimensional kernel with RA Fredholm of index 0.
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Invertibility and Local Uniqueness for Nonlinear I.P.
Let us recast the original nonlinear I.P. as F(v0 + v) = H, H0 := F(v0), with A = F′(v0) the linearization about v0. When A admits a bounded left inverse (F′)−1(v0) := (I − T)−1R, then we can recast the I.P. as v = G(v) := (F′)−1(v0)(H−H0)−(F′)−1(v0)
Since F is polynomial and hence Lipschitz, we obtain that G(v) is a contraction when H − H0 is sufficiently small. This provides a local uniqueness result for the nonlinear I.P. To obtain such a result, we need a framework in which the linearized problem is injective. This is often a difficult problem.
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Holmgren, Carleman, and Calder´
Assume that the system A is elliptic in the regular sense, i.e., tj = t and si = 0. (Else, AtA is not necessarily DN elliptic.) 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 Calder´
- n’s theorem, for a restricted class of operators A.
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Holmgren and local results
Holmgren’s theorem can be used for A with analytic coefficients and A
- n a sufficiently small domain X.
When A = AA has analytic coefficients and AAv = 0, then an application
- f H¨
- rmander’s theorem shows that WFA(v) ⊂ WFA(det(At
AAA)v) so
that v is analytic. With vanishing Cauchy data, v = 0 and injectivity follows. This provides genericity for hybrid inverse problems (invertibility of linear and nonlinear IP on open, dense, set). When spatial domain X is small, write A = A0 + (A − A0) with A0 the
- perator with coefficients frozen at x = 0.
We then apply Holmgren’s theorem to A0 and then to A by perturbation on a small domain.
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Carleman estimates and Calder´
When A is not analytic and X is not small, proving injectivity is more difficult. Recalling that A = P + R with P leading term, we seek injectivity results depending on leading term P and not R. This essentially forces p(ξ+τN) for ξ ∈ Sn−1 and N ∈ Sn−1 to be a diagonal (diagonalized) symbol with diagonal terms that are polynomials in τ with at most simple real roots and at most double complex roots. When these assumptions do not hold, then UCP depends on the structure of lower-order terms. Applies to modified form of ultrasound modulation problem and systems
C P2
- u = 0 with P1 satisfying UCP, P2 elliptic with simple
complex roots (saving one to control C; all operators of order m here).
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Summary of elliptization
- Non-elliptic determined systems can be augmented to redundant elliptic
systems by acquiring more internal functionals.
- A-D-N theory (Solonnikov) provides left parametrix for system satisfying
Lopatinskii conditions and optimal stability estimates but no injectivity.
- Injectivity of linear problem crucial to obtain local injectivity of nonlinear
problem as dimension of kernel of linear problem is not stable (index is).
- Injectivity can be proved by augmenting the boundary conditions, for
instance by solving AtAv = AtS with Dirichlet B.C. (e.g., v = ∂νv = 0).
- On small domain, Holmgren’s theory applies.
On larger domains, a UCP needs to be proved using Carleman/Calder´
more problem-dependent.
- Applied to reconstruction of optical parameters (γ, σ) (S. Moskow) and
Maxwell systems (T. Zhou).
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Constraints for ellipticity and beyond
- In some settings, ellipticity is not true, independent of J, although A is
injective: the next-order term P1 after the principal part P is necessary to invert P + P1 (with sub-elliptic estimates and the loss of one derivative; see Chenxi Guo’s talk).
- In some settings, redundant functionals give more than ellipticity.
- The ellipticity of the linearized system requires certain constraints
- n the values of the functionals Hj (linear independence of gradients,
quadratic forms, etc.). Such constraints depend on the choice of the boundary conditions {fj} that are obtained by explicit solutions such as, for instance, complex geometric optics (CGO) solutions.
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Redundant Internal Functionals in UMEIT
−∇·γ(x)∇uj = 0 X, uj = fj ∂X, Hij(x) = γ(x)2α∇ui·∇uj(x), 1 ≤ i, j ≤ J. Reconstructions in UMEIT (α = 1
2) are obtained by acquiring redundant
internal functionals Hij = Si · Sj(x) with Si(x) = γα ∇ui(x). Then ∇ · Sj = (α − 1)F · Sj, dS♭
j = αF ♭ ∧ S♭ j,
1 ≤ j ≤ J, F = ∇(log γ). Strategy: (i) Eliminate F and find closed-form equation for S = (S1| . . . |Sn) . (ii) Solve for the redundant system of ODEs for S. Works IF H is invertible in M(n; R), i.e., det(∇u1, . . . , ∇un) = 0. This qualitative property on elliptic solutions holds for well-chosen {fj}.
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Elimination and system of ODEs in UMEIT
Lemma [B.-Bonnetier-Monard-Triki 12; Monard-B. 12]. Let Ω ⊂ X. IF infx∈Ω det(S1(x), . . . , Sn(x)) ≥ c0 > 0, then with D(x) =
F(x) = cF D
n
- i,j=1
- ∇(DHij) · Si(x)
- Sj(x), cF =
1 1 + (n − 2)α, H−1 = (Hij). Moreover, we find ∇ ⊗ Sj =
i,k,l,m Hik(Sk · ∇Sj) · SlHlmSi ⊗ Sm with
2(Si · ∇Sj) · Sk = Si · ∇Hjk − Sj · ∇Hik + Sk · ∇Hij − 2F · SkHij + 2F · SjHik. Theorem [idem; Capdeboscq et al. SIIS 09 in n = 2]. There exists an
- pen set of illuminations fj for J = n in even dimension and J = n + 1
in odd dimension such that for γ and γ′ the conductivities corresponding to H and H′, we have the following global stability result: γ − γ′W 1,∞(X) ≤ C
- ε0 + H − H′W 1,∞(X)
- ε0 = |γ(x0) − γ′(x0)| + J
i=1 Si(x0) − S′ i(x0).
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Anisotropic conductivities and internal data in n = 2
∇ · γ∇ui = 0 X, ui = fi ∂X, Hij = γ∇ui · ∇uj, 1 ≤ i, j ≤ I. Define γ = A2 and A = |A| ˜ A with det( ˜ A) = 1. Then for appropriate boundary conditions fi on ∂X and for I = 4, we have: Theorem [Monard B. 12] The internal functionals H = {Hij}4
i,j=1 uniquely
determine the tensor ˜ A via explicit algebraic equations. Moreover, we have the (sub-elliptic) stability estimate ˜ A − ˜ A′L∞(X) ≤ CH − H′W 1,∞. Theorem [Monard B. 12] Let ˜ A be known. Then |A| is uniquely deter- mined by {Hij}1≤i,j≤2 ∈ W 1,∞. Moreover, we have the (elliptic) estimate |A| − |A′|W 1,∞(X) ≤ CH − H′W 1,∞.
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Reconstructions from solution measurements
Consider a general scalar elliptic equation ∇ · a∇u + b · ∇u + cu = 0 in X, u = f
with a, b, c, ∇·a of class C0,α( ¯ X) for α > 0, complex-valued, and α0|ξ|2 ≤ ξ · (ℜa)ξ ≤ α−1
0 |ξ|2. For τ a non-vanishing function on X, define
aτ = τa, bτ = τb − a∇τ, cτ = τc and the equivalence class c := (a, b, c) ∼ (aτ, bτ, cτ). Let I ∈ N∗ and (fi)1≤i≤I be I boundary conditions. Define f = (f1, . . . , fI). The measurement operator Mf is Mf : c → Mf(c) = (u1, . . . , uI), with Hj(x) = uj(x) solution of the above elliptic problem with f = fj.
Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals
SLIDE 30 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012
Unique reconstruction up to gauge transformation
∇ · a∇uj + b · ∇uj + cuj = 0 in X, uj = fj
1 ≤ j ≤ I. We assume the above elliptic equation well posed for c = (a, b, c). Theorem [B. Uhlmann 2012]. Let c and ˜ c be two classes of coefficients with (a, b, c) and ∇ · a of class Cm,α( ¯ X) for α > 0 and m = 0 or m = 1. For I sufficiently large and an open set of boundary conditions f = (fj)1≤j≤I, then Mf(c) uniquely and stably determines c: (a, b + ∇ · a, c) − (˜ a,˜ b + ∇ · ˜ a, ˜ c)W m,∞(X) ≤ CMf(c) − Mf(˜ c)W m+2,∞(X), b − ˜ bL∞(X) ≤ CMf(c) − Mf(˜ c)W 3,∞(X), for m = 0, 1 and for an appropriate (˜ a,˜ b, ˜ c) of ˜ c. The explicit reconstructions and proofs are based on linear indepen- dence of gradients ∇uj and Hessians ∇⊗2uj of solutions to elliptic model.
Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals
SLIDE 31 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012
The IFs and the CGOs
Several HIPs require to verify qualitative properties of elliptic solutions:
- the absence of critical points in Photo-acoustics and Elastography
- the hyperbolicity of a given Lorentzian metric in UMOT
- the linear independence of gradients of elliptic solutions in UMOT
- the joint ellipticity of quadratic forms in UMEIT
(i) One way to solve this is to use CGO solutions whenever available: we verify the property on unperturbed CGOs (for constant-coefficient equa- tion), then by continuity on perturbed CGOs, and then by continuity for illuminations fj on ∂X close to CGO traces. (ii) When CGO solutions are not available (anisotropic or complex val- ued leading coefficients), we construct local solutions (by freezing coef- ficients) that satisfy such conditions. We then use UCP and the Runge approximation to control such solutions from ∂X.
Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals
SLIDE 32 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012
Vector fields and complex geometrical optics
- Take ρ = (ρr + iρi) ∈ Cn with ρ · ρ = 0.
Then ∆eρ·x = 0. Let u1 = ℜeρ·x and u2 = ℑeρ·x so that ∇u1 = eρr·x cos(ρi · xρr) − sin(ρi · xρi)
sin(ρi · xρr) + cos(ρi · xρi)
|∇u1| > 0, |∇u2| > 0, ∇u1 · ∇u2 = 0.
2eρ·x
1 + ψρ(x)
- solution of −∇ · γ∇uρ + σuρ = 0.
Theorem[B.-Uhlmann 10]. For q sufficiently smooth and k ≥ 0, we have |ρ|ψρH
n 2+k+ε(X) + ψρH n 2+k+1+ε(X) ≤ CqH n 2+k+ε(X).
Thus the perturbed gradient directions θ1 =
satisfy |θ1| > 0, |θ2| > 0, and |θ1 · θ2| ≪ 1 locally so that (θ1, θ2) are linearly independent on the bounded domain X of interest.
Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals
SLIDE 33 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012 Gunther’s 60th, Irvine June 20, 2012
First Conclusions
- Mathematically hybrid imaging modalities are stable inverse problems
combining high resolution with high contrast (though they are Low Signal).
- They typically involve the analysis of redundant (overdetermined) sys-
tems of nonlinear partial differential equations, with optimal stability es- timate are obtained for elliptic systems.
- Additional redundancy may provide algebraic/explicit reconstructions.
- Tensors and Complex-valued coefficients can be reconstructed to
account for anisotropy and dispersion effects. The reconstruction of anisotropic coefficients may lead to non-elliptic, yet invertible, systems.
- Reconstructions require qualitative properties of PDE solutions (linear
independence of gradients, Hessians). Well-chosen boundary conditions chosen by CGO constructions or local constructions with boundary con- trol (by Runge approximation and UCP).
Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals Guillaume Bal Hybrid Inverse Problems & Internal Functionals
SLIDE 34
HAPPY BIRTHDAY GUNTHER !
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HAPPY BIRTHDAY GUNTHER !
