2021/04/22 by Michael V. Klibanov, Klibanov, Michael V., Thuy T. Le +7
Engineering · Mathematics · Medicine · #FOS: Mathematics #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Photoacoustic and Ultrasonic Imaging
paper · pdf · doi:10.48550/arxiv.2104.11392
openalex publication_date 2021/04/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
To compute the spatially distributed dielectric constant from the\nbackscattering data, we study a coefficient inverse problem for a 1D hyperbolic\nequation. To solve the inverse problem, we establish a new version of Carleman\nestimate and then employ this estimate to construct a cost functional which is\nstrictly convex on a convex bounded set with an arbitrary diameter in a Hilbert\nspace. The strict convexity property is rigorously proved. This result is\ncalled the convexification theorem and is considered as the central analytical\nresult of this paper. Minimizing this convex functional by the gradient descent\nmethod, we obtain the desired numerical solution to the coefficient inverse\nproblems. We prove that the gradient descent method generates a sequence\nconverging to the minimizer and we also establish a theorem confirming that the\nminimizer converges to the true solution as the noise in the measured data and\nthe regularization parameter tend to zero. Unlike the methods that are based on\noptimization, our convexification method converges globally in the sense that\nit delivers a good approximation of the exact solution without requiring any\ninitial guess. Results of numerical studies of both computationally simulated\nand experimental data are presented.\n