Project 10: MRE Inverse Problem for the Wave Equation
Goal ¶ Formulate and implement a classical inverse problem motivated by Magnetic
Resonance Elastography (MRE): infer tissue elasticity parameters from
measured displacement data.
Let Ω \Omega Ω y d y_d y d
A scalar model problem is
{ y t t − ∇ ⋅ ( u ∇ y ) = f in Ω × ( 0 , T ) , y = 0 on ∂ Ω × ( 0 , T ) , y ( 0 ) = y 0 , y t ( 0 ) = y 1 in Ω . \begin{cases}
y_{tt}-\nabla\cdot(u\nabla y)=f & \text{in }\Omega\times(0,T),\\
y=0 & \text{on }\partial\Omega\times(0,T),\\
y(0)=y_0,\qquad y_t(0)=y_1 & \text{in }\Omega.
\end{cases} ⎩ ⎨ ⎧ y tt − ∇ ⋅ ( u ∇ y ) = f y = 0 y ( 0 ) = y 0 , y t ( 0 ) = y 1 in Ω × ( 0 , T ) , on ∂ Ω × ( 0 , T ) , in Ω. The inverse problem is
min ( y , u ) 1 2 ∫ 0 T ∥ y ( t ) − y d ( t ) ∥ L 2 ( Ω ) 2 d t + α 2 ∥ u − u r e f ∥ L 2 ( Ω ) 2 + γ 2 ∥ ∇ u ∥ L 2 ( Ω ) 2 \min_{(y,u)}
\frac12\int_0^T\|y(t)-y_d(t)\|_{L^2(\Omega)}^2\,dt
+
\frac\alpha2\|u-u_{\mathrm{ref}}\|_{L^2(\Omega)}^2
+
\frac\gamma2\|\nabla u\|_{L^2(\Omega)}^2 ( y , u ) min 2 1 ∫ 0 T ∥ y ( t ) − y d ( t ) ∥ L 2 ( Ω ) 2 d t + 2 α ∥ u − u ref ∥ L 2 ( Ω ) 2 + 2 γ ∥∇ u ∥ L 2 ( Ω ) 2 subject to the wave equation and bounds
0 < u a ≤ u ≤ u b . 0<u_a\le u\le u_b. 0 < u a ≤ u ≤ u b . The adjoint equation is backward in time:
{ p t t − ∇ ⋅ ( u ∇ p ) = y − y d in Ω × ( 0 , T ) , p = 0 on ∂ Ω × ( 0 , T ) , p ( T ) = 0 , p t ( T ) = 0 in Ω . \begin{cases}
p_{tt}-\nabla\cdot(u\nabla p)=y-y_d
& \text{in }\Omega\times(0,T),\\
p=0 & \text{on }\partial\Omega\times(0,T),\\
p(T)=0,\qquad p_t(T)=0 & \text{in }\Omega.
\end{cases} ⎩ ⎨ ⎧ p tt − ∇ ⋅ ( u ∇ p ) = y − y d p = 0 p ( T ) = 0 , p t ( T ) = 0 in Ω × ( 0 , T ) , on ∂ Ω × ( 0 , T ) , in Ω. The coefficient gradient is formally
g ( u ) = α ( u − u r e f ) − γ Δ u + ∫ 0 T ∇ y ( t ) ⋅ ∇ p ( t ) d t . g(u)
=
\alpha(u-u_{\mathrm{ref}})
-
\gamma\Delta u
+
\int_0^T \nabla y(t)\cdot\nabla p(t)\,dt. g ( u ) = α ( u − u ref ) − γ Δ u + ∫ 0 T ∇ y ( t ) ⋅ ∇ p ( t ) d t . The constrained optimality condition is
( g ( u ) , v − u ) L 2 ( Ω ) ≥ 0 ∀ v ∈ U a d . (g(u),v-u)_{L^2(\Omega)}\ge 0
\qquad \forall v\in U_{ad}. ( g ( u ) , v − u ) L 2 ( Ω ) ≥ 0 ∀ v ∈ U a d . Implementation Hints ¶ Start in 1D or 2D.
Use synthetic data generated from a known coefficient.
Implement the forward wave solve, then the backward adjoint solve.
Use projected gradient before attempting Newton or PDAS.
Relevant Examples ¶ Course code: codes/dealii/execs/inverse_poisson_kkt.cc for coefficient
identification structure.
Course lectures: lecture14.md, lecture15.md.
deal.II: step-23 for wave equation time stepping; elasticity examples for
vector-valued extensions.
Deliverables ¶ Forward and adjoint wave solvers.
Gradient verification by finite differences.
Reconstruction of a piecewise smooth coefficient.
Discussion of regularization and ill-posedness.