Geometric inverse problems for linear and non-linear wave equations - - PowerPoint PPT Presentation

Geometric inverse problems for linear and non-linear wave equations

Matti Lassas

University of Helsinki

Finnish Centre of Excellence in Inverse Modelling and Imaging

2018-2025 2018-2025

Outline:

◮ Inverse problem for linear wave equation ◮ Generalised time reversal methods and point sources ◮ Inverse problem for non-linear wave equation

Boundary measurements for anisotropic wave operator

Let n ≥ 2, M ⊂ Rn have a smooth boundary and g(x) be a positive definite matrix depending on x ∈ M. Let ν be the unit normal of the boundary ∂M and (x, t) ∈ M × R+. Let u(x, t) = uf (x, t) solve the wave equation (∂2

t − ∇ · g(x)∇)u(x, t) = 0

• n (x, t) ∈ M × R+,

ν · g∇u(x, t)|∂M×R+ = f (x, t), u|t=0 = 0, ∂tu|t=0 = 0, with the Neumann boundary value f . Then the Neumann-to-Dirichlet map is Λgf = uf (x, t)

• (x,t)∈∂M×R+

Observation: There are different matrix valued functions g(1)(x) and g(2)(x) such that Λg(1) = Λg(2).

Next we consider M ⊂ Rn as Riemannian manifold with a boundary. Then the travel time metric is given by ds2 = n

j,k=1 gjk(x)dxjdxk,

where (gjk) = (gjk)−1. The distance of points x, y ∈ M is distM(x, y) = travel time between x and y. Manifold is given by a collection of local coordinate charts.

Inverse problem on manifold

Let M be a Riemannian manifold with boundary, (gjk(x))n

j,k=1 be

the Riemannian metric and ∆g be the Laplace operator ∆gu =

n

• j,k=1

|g|−1/2 ∂ ∂xj (|g|1/2gjk ∂ ∂xk u), |g| = det (gjk), (gjk) = (gjk)−1. Let uf = uf (x, t) be the solution of ∂2

t u − ∆gu = 0

• n M × R+,

∂νu|∂M×R+ = f , u|t=0 = 0, ∂tu|t=0 = 0, where ν is unit interior normal of ∂M. Define Λgf = uf |∂M×R+. Assume that we are given the boundary data (∂M, Λg).

Inverse problem on a manifold

Inverse problem: Let ∂M and the map Λg : ∂νu|∂M×R+ → u|∂M×R+ be given. Can we determine the manifold (M, g) by representing it as a collection of images of g on local coordinate charts? Near the boundary ∂M, we can represent the metric tensor in the time migration coordinates, i.e., the boundary normal coordinates. (Figure by Cameron, et al, Berkeley.)

Results of Belishev-Kurylev 1992 and Tataru 1994 yield

Theorem

Let M ⊂ Rn and g(1)(x) and g(2)(x) be smooth matrix-valued functions such that Λg(1) = Λg(2) for the equation ∂2

t u − ∆gu = 0.

Then (M, g(1)) and (M, g(2)) isometric, that is, there is a diffeomorphism F : M → M, F|∂M = Id such that g(2)(x) = DF(y)t · g(1)(y) · DF(y)

• y=F −1(x)

(1) If Λg(1) = Λg(2) for the equation (∂2

t − ∇ · g(x)∇)u = 0 then the

map F in (1) satisfies also det (DF(x)) = 1, that is, F preserves the Euclidean volume [Katchalov-Kurylev-L. 2001].

Results on the hyperbolic inverse problem:

◮ Uniqueness for inverse problem (∂2 t − c(x)2∆)u = 0 in

Ω ⊂ Rn by combining the Boundary Control method by Belishev ’87, Belishev-Kurylev ’87 and the controllability results by Tataru ’95.

◮ Spectral problem for ∆g on manifold, Belishev-Kurylev 1992. ◮ Bingham-Kurylev-L.-Siltanen: Solution by modified time

reversal and focusing of waves 2008.

◮ de Hoop-Kepley-Oksanen: Numerical methods for focusing of

waves 2016.

◮ Partial data: L.-Oksanen 2014, Mansouri-Milne 2016.

Outline:

◮ Inverse problem for linear wave equation ◮ Generalised time reversal methods and point sources ◮ Inverse problem for non-linear wave equation

Figure on the left: Inverse problems group of Kuopio, J. Kaipio et al. Figure on the right: University of Karlsruhe, R. Riedlinger et al.

Time reversal methods:

• M. Fink;
• G. Bal, L. Borcea, G. Papanicolaou.

Basic steps of the iterated time reversal are

◮ Send a wave that hits to a point scatterer ◮ Record the scattered signals sj(t) ◮ Send the time reversed signals sj(T − t)

Next we apply a modified time reversal iteration to solve an inverse

• problem. In particular, we do not assume the existence of point

scatterers.

Time reversal methods:

• M. Fink;
• G. Bal, L. Borcea, G. Papanicolaou.

Basic steps of the iterated time reversal are

◮ Send a wave that hits to a point scatterer ◮ Record the scattered signals sj(t) ◮ Send the time reversed signals sj(T − t)

Next we apply a modified time reversal iteration to solve an inverse

• problem. In particular, we do not assume the existence of point

scatterers.

Time reversal methods:

• M. Fink;
• G. Bal, L. Borcea, G. Papanicolaou.

Basic steps of the iterated time reversal are

◮ Send a wave that hits to a point scatterer ◮ Record the scattered signals sj(t) ◮ Send the time reversed signals sj(T − t)

Next we apply a modified time reversal iteration to solve an inverse

• problem. In particular, we do not assume the existence of point

scatterers.

Recall that u(x, t) = uf (x, t) solves (∂2

t − ∆g)u(x, t) = 0

• n (x, t) ∈ M × R+,

∂νu(x, t)|∂M×R+ = f (x, t), u|t=0 = 0, ∂tu|t=0 = 0. Let T > 0 and denote Λ2Tf = uf (x, t)

• (x,t)∈∂M×[0,2T]

. Moreover, let uf (T), uh(T)L2(M) =

• M

uf (x, T)uh(x, T)dVg(x) and uf (T)L2(M) = uf (T), uf (T)

1 2 .

Inner products of waves

By Blagovestchenskii identity, uf (T), uh(T)L2(M) =

• ∂M×[0,2T]

(Kf )(x, t)h(x, t) dS(x)dt, where K = JΛ2T − RΛ2TRJ, Rf (x, t) = f (x, 2T − t) “time reversal operator”, Jf (x, t) = 1 2 2T−t

t

f (x, s)ds “low-pass filter”. Moreover, uf (T), 1L2(M) = f , φTL2(∂M×[0,2T]), φT(t) = (T − t)+

Domains of influence

Let distM(x, y) denote the travel time from x to y.

Definition

Let s > 0 and Γ ⊂ ∂M. The set M(Γ, s) = {x ∈ M : distM(x, Γ) ≤ s} is the domain of influence of Γ at time s.

s Γ

❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄

Γ M(Γ, s) t = 0 t = s

Lemma

Let f ∈ L2(∂M × [0, s]) be such that f (x, t) = 0 for x ∈ Γ. Then uf (x, s) = 0, for x ∈ M(Γ, s).

• Proof. The result follows the finite velocity of the wave

propagation.

• ❈

❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ✄

Γ M(Γ, s) t = 0 t = s

We use following Tataru’s approximate controllability result.

Theorem

Let s > 0 and Γ ⊂ ∂M be an open subset. Let v(x) ∈ L2(M) be zero outside M(Γ, s). Then for any ε > 0 there is f ∈ L2(∂M × [0, s]), that is zero outside Γ × [0, s], such that uf (s) − vL2(M) =

• M

|uf (x, s) − v(x)|2dV (x) 1/2 < ε. Stability estimates of the equivalent unique continuation result by Bosi-Kurylev-L. 2016, Laurent-Léautaud 2017

❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄

Γ M(Γ, s) t = 0 t = s

Blind control problem: Find a boundary source f such that uf (x, s) ≈ χM(Γ,s)(x) =

• 1, when x ∈ M(Γ, s),

0, otherwise. The above control problem is equivalent to the minimisation min

f

uf (T) − 12

L2(M) = Kf , f − 2f , φT + const.

where f ∈ L2(Γ × [T − s, T]), that is, f is zero outside Γ × [T − s, T].

❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄

M(Γ, s) t = T − s t = T

The minimization problem has usually no solution and is ill-posed. We consider the regularized minimization problem min

f ∈L2(Γ×[T−s,T]) Gα(f )

where α ∈ (0, 1), Gα(f ) = Kf , f L2 − 2f , φTL2 + αf 2

L2.

This leads to a linear equation (K + α)f = φT. As K = JΛ2T − RΛ2TRJ, this equation can be solved using an iteration of operators R, Λ2T and J.

The modified time reversal iteration is the following algorithm: Let f0 = 0 and define fn+1 := (1 − α2)fn + α(RΛ2TRJfn − JΛ2Tfn) + αφT, where fn ∈ L2(Γ × [T − s, T]) and α > 0.

Theorem

Let Γ ⊂ ∂M, 0 ≤ s ≤ T. For α > 0, the modified time reversal iteration gives sources fn = fn(α). Then f (α) = lim

n→∞ fn(α),

in L2(∂M × [0, 2T]), and lim

α→0 uf (α)(x, T) = χM(Γ,s)(x),

in L2(M).

s Γ

Bingham-Kurylev-L.-Siltanen 2008, de Hoop-Kepley-Oksanen 2016

The iteration to focus waves and numerical methods have been further developed by M. de Hoop, P. Kepley, and L. Oksanen

(a) (c) (d)

Above, the wave u(x, s) is non-zero near γz,ν(s), where z ∈ ∂M, s > 0. Figures by M. de Hoop, P. Kepley, and L. Oksanen, SIAP 2016.

Theorem

Let z ∈ ∂M and s > 0. Using an iteration of Λ2T, R and J, we find boundary sources f (α), α > 0, such that lim

α→0 u

• f (α)(x, T) = Cδy(x)

in D′(M), y = γz,ν(s).

s

z y s Bingham-Kurylev-L.-Siltanen 2008, de Hoop-Kepley-Oksanen 2016

Construction of artificial point source

Dahl-Kirpichnikova-L. 2009 and Kirpichnikova-Korpela-L.-Oksanen:

Theorem

Let T > 2diam(M), z ∈ ∂M, s > 0, and y = γz,ν(s) such that dist(y, ∂M) = s. Minimizing a quadratic form F energy

α

(f ) involving the Neumann-to-Dirichlet map Λ, we find f(α) such that lim

α→0

• uf(α)(x, T)

∂tuf(α)(x, T)

• =
• Cδy(x)
• in D′(M).

s

z y s

When t > T, uf(α)(t) is close to Green’s function C · G(x, t; y, T), (∂2

t − ∆g)G(x, t; y, T) = δ(t − T)δy(x)

• n (x, t) ∈ M × R+,

G(x, t; y, T)|t=0 = 0, ∂tG(x, t; y, T)|t=0 = 0. This Green’s function is the wave produced by a point source at the time T at the point y = γz,ν(s), where z ∈ ∂M and s > 0. z Next we consider the reconstruction of a manifold using

• bservations from point sources.

Corollary (Bingham-Kurylev-L.-Siltanen 2008)

Assume we are given ∂M and Λ. The modified time reversal iteration gives the set of the boundary distance functions {rx : x ∈ M} ⊂ C(∂M), where rx(z) = distM(x, z), for z ∈ ∂M. Above, rx(z) is the travel time of the wave produced by a point source at x to the boundary point z.

Corollary (Bingham-Kurylev-L.-Siltanen 2008x)

Assume we are given ∂M and Λ. The modified time reversal iteration gives the set of the boundary distance functions {rx : x ∈ M} ⊂ C(∂M), where rx(z) = distM(x, z), for z ∈ ∂M. Above, rx(z) is the travel time of the wave produced by a point source at x to the boundary point z.

Reconstruction of (M, g) from R(M).

Assume that (M, g) is a compact Riemannian manifold with boundary, rx(z) = distM(x, z), z ∈ ∂M.

Theorem (Kurylev 1997, Katcalov-Kurylev-L. 2001)

When ∂M and R(M) = {rx : x ∈ M} are given, we can uniquely determine the topological and differentiable structure of M and the metric g, up to an isometry. The same holds for complete manifolds (L.-Salo-Uhlmann 2011). Next we prove this for a simple manifold on which all points can be connected by a unique distance minimizing curve.

For a continuous function f : ∂M → R, denote f C(∂M) = max

z∈∂M |f (z)|.

Recall that for x ∈ M we define a function rx : ∂M → R, rx(z) = distM(x, z), z ∈ ∂M. In a simple case when all geodesics (i.e., rays) are distance minimising curves, we have rx − ryC(∂M) = distM(x, y), x, y ∈ M.

r r

z x

For a continuous function f : ∂M → R, denote f C(∂M) = max

z∈∂M |f (z)|.

Recall that for x ∈ M we define a function rx : ∂M → R, rx(z) = distM(x, z), z ∈ ∂M. In a simple case when all geodesics (i.e., rays) are distance minimising curves, we have rx − ryC(∂M) = distM(x, y), x, y ∈ M.

r r r

z x y

For a continuous function f : ∂M → R, denote f C(∂M) = max

z∈∂M |f (z)|.

Recall that for x ∈ M we define a function rx : ∂M → R, rx(z) = distM(x, z), z ∈ ∂M. In a simple case when all geodesics (i.e., rays) are distance minimising curves, we have rx − ryC(∂M) ≤ distM(x, y), x, y ∈ M.

r r r

z x y

For a continuous function f : ∂M → R, denote f C(∂M) = max

z∈∂M |f (z)|.

Recall that for x ∈ M we define a function rx : ∂M → R, rx(z) = distM(x, z), z ∈ ∂M. In a simple case when all geodesics (i.e., rays) are distance minimising curves, we have rx − ryC(∂M) ≥ distM(x, y), x, y ∈ M.

r r r

z x y

For a continuous function f : ∂M → R, denote f C(∂M) = max

z∈∂M |f (z)|.

Recall that for x ∈ M we define a function rx : ∂M → R, rx(z) = distM(x, z), z ∈ ∂M. In a simple case when all geodesics (i.e., rays) are distance minimising curves, we have rx − ryC(∂M) = distM(x, y), x, y ∈ M.

r r r

z x y

Summary: When all points x, y ∈ M can be connected by a unique distance minimising curve, we have rx − ryC(∂M) = distM(x, y), x, y ∈ M, where rx(z) = distM(x, z). Let R(M) = {rx : x ∈ M} ⊂ C(∂M). Then, the metric spaces (M, distM) and (R(M), · C(∂M)) are isometric. This means that we can construct the set R(M) ⊂ C(∂M) and consider it as an isometric copy of the original manifold (M, distM).

Outline:

◮ Inverse problem for linear wave equation ◮ Generalised time reversal methods and point sources ◮ Inverse problem for non-linear wave equation

Non-linear wave equation in space-time

Let (x, t) ∈ R3 × (−∞, T) and gu(x, t) + a(x, t) u(x, t)2 = f (x, t), u(x, t) = 0 for t < 0, where g = (gjk(x))3

j,k=1 is an anisotropic wave speed and

gu = ∂2 ∂t2 u(x, t) − ∆gu(x, t). We have generalisations for a more general wave equations (Kurylev-L.-Uhlmann 2018, L.-Uhlmann-Wang 2018).

Theorem (Kurylev-L.-Uhlmann 2018)

Assume that the coefficient of the non-linear term, a(x, t), is nowhere zero. Consider the non-linear wave equation gu + a u2 = f in R3 × (−∞, T), u(x, t) = 0 in t < 0, where supp(f ) ⊂ V = (0, T) × Bg(x0, r). Then the measurement

• perator LV : f → u|V , defined for small sources f , determine the

wave speed gjk(x) in the ball Bg(x0, r + T/2), in the travel time coordinates.

The idea of the proof for non-linear wave equation gu + a u2 = f .

(Load video)

Video can be loaded from

https://www.mv.helsinki.fi/home/lassas/inverse_problems_for_GR_and_non-linear-hyperbolic_equations.html

• Non-linearity helps to solve

the inverse problem,

• The non-linear interaction
• f distorted plane waves

create artificial point sources.

• These point sources determine

functions rx.

The Einstein-Maxwell equations

Similar techniques can be used for the inverse problem for the Einstein-Maxwell equations for space-time metric g, electromagnetic 4-field F = dφ, and 4-current J, Einαβ(g) = Tαβ, in (−∞, T) × N, Tαβ = F λ

αFβλ − 1

4gαβF λµFλµ − 1 2(Jµφµ)gαβ, F = dφ, δgdφ = J♭, Div gJ = 0. (Kurylev-L.-Oksanen-Uhlmann 2018, L.-Uhlmann-Wang 2018) Hopefully similar techniques can be used also in non-linear elasticity.

Thank you for your attention!