研究目的
To recover the law of a random potential from empirical correlations in the exterior of the potential, specifically by studying scattering of waves from a time-independent random potential and deriving conditions for uniqueness in the inverse problem.
研究成果
The paper establishes that the kth moment map of a random potential can be uniquely determined from empirical correlations of scattered waves, and for Gaussian fields, the probability distribution is uniquely recoverable. It also shows that if two random fields yield the same correlations under sufficient regularity, they have identical laws. This provides a foundation for correlation-based imaging in inverse scattering problems.
研究不足
The approach assumes specific regularity conditions (C1, C2, C3) for the random field, such as bounded H2-norm and compact support. The method may not be directly applicable to non-smooth potentials without additional regularization. Numerical implementation and practical experimental setups are not addressed, limiting immediate applicability to real-world scenarios.
1：Experimental Design and Method Selection:
The study uses theoretical models involving the wave equation with a random potential, focusing on inverse scattering problems. Methods include deriving empirical correlations from amplitude measurements of scattered waves and applying reconstruction operators for smooth potentials.
2：Sample Selection and Data Sources:
The potential is a random generalized function supported on a compact set in Rn, with samples or datasets not explicitly specified but assumed to be generated from the probability space.
3：List of Experimental Equipment and Materials:
No specific equipment or materials are mentioned; the work is theoretical and mathematical.
4：Experimental Procedures and Operational Workflow:
Involves computing empirical correlations from scattered wave data, regularizing these correlations, and reconstructing moment maps using Fourier integral operators and Radon transforms.
5：Data Analysis Methods:
Analysis includes the use of Sobolev spaces, wave front sets, and probabilistic measures. Statistical techniques involve expectations and moments of random fields, with proofs based on functional analysis and PDE theory.
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