On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a nonlinear inverse problem is reduced to a family of the linear integral equation of the first kind via the Lavrentiev approach and this reduction is used within the quasi-reversibility method. The analysis of its stability is carried out via Carleman estimates. The computational effectiveness of these methods is tested in the numerical experiments with the smooth and discontinuous coefficients to be recovered from the tomographic data.

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Completeness of asymmetric products of harmonic functions and uniqueness of the solution to the Lavrent'ev equation in inverse wave sounding problems

Кокурин¹,

Kokurin

2022

Известия Российской академии наук. Серия математическая

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Устанавливается, что семейство всех попарных произведений регулярных гармонических функций в области $D \subset \mathbb{R}^3$ и ньютоновых потенциалов точек, расположенных на луче вне $D$, полно в пространстве $L_2(D)$. Результат используется при обосновании единственности решения линейного интегрального уравнения, к которому сводятся обратные задачи волнового зондирования в $\mathbb{R}^3$. Единственность решений соответствующих обратных задач устанавливается для пространственно непереопределенных постановок, когда размерность пространственного носителя данных совпадает с размерностью носителя искомой функции. Теоремы единственности используются для доказательства осевой симметрии решений рассматриваемых обратных задач при наличии осевой симметрии входных данных этих задач. Библиография: 31 наименование.

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On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

Cited by 4 publications

References 3 publications

Uniqueness Conditions and Numerical Approximation of the Solution to M. M. Lavrentiev’s Integral Equation

Uniqueness Conditions and Numerical Approximation of the Solution to M. M. Lavrentiev’s Integral Equation

A comparative study of two globally convergent numerical methods for acoustic tomography

Completeness of asymmetric products of harmonic functions and uniqueness of the solution to the Lavrent'ev equation in inverse wave sounding problems

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