Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Abstract

Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography (PAT), we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first-and secondorder linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purpose, we also establish the stability of the reconstruction of the absorption coefficients with respect to the change in the scattering coefficient.

Recovery of coefficients in semilinear transport equations

“…For small parameter ε, the well-posedness result in Theorem 2.6 yields that there is a unique small solution f (t, x, v) ≡ f (t, x, v; ε) to (3.1) with initial data f | t=0 = εh and boundary data f | ∂ − SΩ T = εg. We can obtain the differentiability of the solution f = f (t, x, v; ε) with respect to ε by adapting the proof of [ [29], Proposition A.4], where the differentiability is discussed for a nonlinear transport equation, to our setting. Hence, we have the k-th derivative of f with respect to ε at ε = 0, which is defined by…”

“…For instance, when ℓ = 2, N 0 (f ) can represent the quadratic nonlinearity such as N 0 (f ) = f 2 or f S d−1 f dω(v ′ ). The latter example finds applications in photoacoustic tomography with nonlinear absorption effect and we refer the interested readers to the references [29,44]. Theorem 1.2.…”

“…The main objectives are to determine the nonlinearity N , absorption σ and the scattering coefficient µ by the boundary data. The problem is motivated by applications in the photoacoustic tomography, in which the nonlinear excitation is observed due to two-photon absorption effect of the underlying medium, see [29,45,44,48] and the references therein.…”

“…Then differentiating it multiple times with respect to these parameters to earn simpler and new linearized equations. For more detailed discussions and related studies, see for instance [11,25,35] for hyperbolic equations, [17,20,23,24,28,31,32,27,34,33,37] for elliptic equations, and [29,30,36] for kinetic equations.…”

An inverse problem for the minimal surface equation

“…Different cases of nonlinear conductivity equations have had a treatment in [CFKKU21], [KKU20]. This method has also been used in the case of a nonlinear magnetic Schrödinger equation ( [LZ20]) and in inverse transport and diffusion problems [LRZ21]. See also [KU20] for a semilinear elliptic equation with gradient nonlinearities and [LL20] for the case of fractional semilinear elliptic equations.…”