Time reversal for radiative transport with applications to inverse and control problems

Abstract

In this paper we develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small in the L ∞ norm. Then, using duality arguments, we show that the solvability of the inverse problem leads to exact controllability of the transport field with minimum-norm control obtained constructively. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery.

Inverse transport problems in quantitative PAT for molecular imaging

“…Following the control theory for transport equations developed in [1,3,38], we can show, under reasonable assumptions, the existence of sources g x such that u x = K I (u x ) holds for each pair (η, σ a,xf ) ∈ A. Such sources, however, might be complicated, for instance we might need to solve a control problem, to construct in practical applications.…”

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

“…Note that other control theory results for the time-dependent RTE have had applications to inverse problems as well. See for example [1,2].…”

Recovery of the absorption coefficient in radiative transport from a single measurement

“…Given arbitrary φ ∈ V 0 , the goal of the control problem is to find an outflow control condition η ∈ L 2 ([0, τ ]; T + ) to drive the solution ψ of ( 18)-( 20) from ψ(τ ) = 0 to ψ(0) = φ. The well-posedness of the control problem is described in the following theorem which is a direct consequence of [1].…”

“…Now we state the inverse problem for transient transport along with our main results. Our proof, presented in Section 3, is based on tools from control theory developed in [1,25]. We break the problem into two steps.…”

“…The precise statements of our results are presented in Section 2. Our proof, found in Section 3, is primarily based on boundary controllability for the RTE recently obtained in [1,25].…”