Inverse Problems and Imaging最新文献

We prove that a continuous potential begin{document}$ q $end{document} can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator begin{document}$ Delta_g^2+q $end{document} on a conformally transversally anisotropic Riemannian manifold of dimension begin{document}$ ge 3 $end{document} with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [56]. In particular, our result is applicable and new in the case of smooth bounded domains in the begin{document}$ 3 $end{document}–dimensional Euclidean space as well as in the case of begin{document}$ 3 $end{document}–dimensional admissible manifolds.

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:

begin{document}$ begin{align*} boldsymbol{u}_t-mu Deltaboldsymbol{u}+(boldsymbol{u}cdotnabla)boldsymbol{u}+alphaboldsymbol{u}+beta|boldsymbol{u}|^{r-1}boldsymbol{u}+nabla p = boldsymbol{F}: = boldsymbol{f} g, nablacdotboldsymbol{u} = 0, end{align*} $end{document}

in bounded domains begin{document}$ Omegasubsetmathbb{R}^d $end{document} (begin{document}$ d = 2, 3 $end{document}) with smooth boundary, where begin{document}$ alpha, beta, mu>0 $end{document} and begin{document}$ rin[1, infty) $end{document}. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function begin{document}$ boldsymbol{u} $end{document}, the pressure gradient begin{document}$ nabla p $end{document} and the vector-valued function begin{document}$ boldsymbol{f} $end{document}. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for begin{document}$ rgeq 1 $end{document} in two dimensions and for begin{document}$ r geq 3 $end{document} in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.

For an integer begin{document}$ rge0 $end{document}, we prove the begin{document}$ r^{mathrm{th}} $end{document} order Reshetnyak formula for the ray transform of rank begin{document}$ m $end{document} symmetric tensor fields on begin{document}$ {{mathbb R}}^n $end{document}. Roughly speaking, for a tensor field begin{document}$ f $end{document}, the order begin{document}$ r $end{document} refers to begin{document}$ L^2 $end{document}-integrability of higher order derivatives of the Fourier transform begin{document}$ widehat f $end{document} over spheres centered at the origin. Certain differential operators begin{document}$ A^{(m,r,l)} (0le lle r) $end{document} on the sphere begin{document}$ {{mathbb S}}^{n-1} $end{document} are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any begin{document}$ r $end{document} although the volume of calculations grows fast with begin{document}$ r $end{document}. The algorithm is realized for small values of begin{document}$ r $end{document} and Reshetnyak formulas of orders begin{document}$ 0,1,2 $end{document} are presented in an explicit form.

Consider the transmission eigenvalue problem

begin{document}$ (Delta+k^2mathbf{n}^2) w = 0, (Delta+k^2)v = 0 mbox{in} Omega;quad w = v, partial_nu w = partial_nu v mbox{on} partialOmega. $end{document}

It is shown in [16] that there exists a sequence of eigenfunctions begin{document}$ (w_m, v_m)_{minmathbb{N}} $end{document} associated with begin{document}$ k_mrightarrow infty $end{document} such that either begin{document}$ {w_m}_{minmathbb{N}} $end{document} or begin{document}$ {v_m}_{minmathbb{N}} $end{document} are surface-localized, depending on begin{document}$ mathbf{n}>1 $end{document} or begin{document}$ 0. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions begin{document}$ (w_m, v_m)_{minmathbb{N}} $end{document} associated with begin{document}$ k_mrightarrow infty $end{document} such that both begin{document}$ {w_m}_{minmathbb{N}} $end{document} and begin{document}$ {v_m}_{minmathbb{N}} $end{document} are surface-localized, no matter begin{document}$ mathbf{n}>1 $end{document} or begin{document}$ 0. Though our study is confined within the radial geometry, the construction is subtle and technical.

We consider two kinds of inverse problems on determining multiple parameters simultaneously for one-dimensional time-fractional diffusion-wave equations with derivative order begin{document}$ alpha in (0, 2) $end{document}. Based on the analysis of the poles of Laplace transformed data and a transformation formula, we first prove the uniqueness in identifying multiple parameters, including the order of the derivative in time, a spatially varying potential, initial values, and Robin coefficients simultaneously from boundary measurement data, provided that no eigenmodes are zero. Our main results show that the uniqueness of four kinds of parameters holds simultaneously by such observation for the time-fractional diffusion-wave model where unknown orders begin{document}$ alpha $end{document} vary order (0, 2) including 1, restricted to neither begin{document}$ alpha in (0, 1] $end{document} nor begin{document}$ alpha in (1, 2) $end{document}. Furthermore, for another formulation of the fractional diffusion-wave equation with input source term in place of the initial value, we can also prove the simultaneous uniqueness of multiple parameters, including a spatially varying potential and Robin coefficients by means of the uniqueness result in the case of non-zero initial value and Duhamel's principle.

We construct counterexamples to inverse problems for the wave operator on domains in begin{document}$ mathbb{R}^{n+1} $end{document}, begin{document}$ n ge 2 $end{document}, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On begin{document}$ mathbb{R}^{n+1} $end{document} the metrics are conformal to the Minkowski metric.