[Paper Review] Monotonicity-based inversion of the fractional Schr\"odinger equation

Bastian Harrach, Yi‐Hsuan Lin|arXiv (Cornell University)|Nov 15, 2017

Numerical methods in inverse problems46 references25 citations

TL;DR

This paper establishes an if-and-only-if monotonicity relation between positive bounded potentials and their nonlocal Dirichlet-to-Neumann maps for the fractional Schr"odinger equation, enabling a constructive proof of uniqueness for the nonlocal Calder\'{o}n problem. It further introduces a dimension-independent reconstruction method for unknown obstacles using only the background solution.

ABSTRACT

We consider the inverse problems of for the fractional Schr\{o}dinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calder\'{o}n problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schr\{o}dinger equation.

Motivation & Objective

• To establish a rigorous if-and-only-if monotonicity relation between potentials and Dirichlet-to-Neumann maps in the fractional Schr"odinger equation.
• To provide a constructive proof of uniqueness for the nonlocal Calder\'{o}n problem using monotonicity arguments.
• To develop a dimension-independent reconstruction method for unknown obstacles in a domain using only the background solution.
• To ensure the method's applicability in arbitrary dimensions $ n \geq 2 $ without requiring additional assumptions on the domain or potential.

Proposed method

• Derive a monotonicity formula that links the difference of two positive bounded potentials to the difference of their associated nonlocal Dirichlet-to-Neumann maps.
• Use the monotonicity relation to prove that if the Dirichlet-to-Neumann maps are ordered, then the corresponding potentials are ordered, and vice versa.
• Construct a reconstruction algorithm for unknown obstacles by leveraging the monotonicity structure and the background solution of the fractional Schr"odinger equation.
• Ensure the method's validity in any dimension $ n \geq 2 $ by relying only on the solvability and regularity of the background solution.
• Apply variational and nonlocal PDE techniques to analyze the nonlocal Dirichlet-to-Neumann map and its dependence on the potential.
• Use the monotonicity framework to iteratively refine estimates of the unknown obstacle through boundary measurements.

Experimental results

Research questions

• RQ1 Can a monotonicity-based framework establish an if-and-only-if relation between potentials and their nonlocal Dirichlet-to-Neumann maps in the fractional Schr"odinger equation?
• RQ2 Does the monotonicity relation enable a constructive proof of uniqueness for the nonlocal Calder\'{o}n problem?
• RQ3 Can the proposed method reconstruct unknown obstacles in a domain without dependence on the spatial dimension $ n \geq 2 $?
• RQ4 What is the minimal information required to reconstruct obstacles using nonlocal inverse methods?
• RQ5 How does the background solution of the fractional Schr"odinger equation facilitate obstacle reconstruction?

Key findings

• An if-and-only-if monotonicity relation is established between positive bounded potentials and their nonlocal Dirichlet-to-Neumann maps, providing a fundamental link for inverse problems.
• The monotonicity framework yields a constructive proof of uniqueness for the nonlocal Calder\'{o}n problem, resolving a key open question in the field.
• A reconstruction method for unknown obstacles is developed that is valid in all dimensions $ n \geq 2 $, relying only on the background solution.
• The method is independent of the domain's geometry and does not require additional regularity assumptions on the obstacle or potential.
• The approach avoids iterative or optimization-based schemes by using monotonicity to directly infer the obstacle's presence and support.
• The results demonstrate that the nonlocal Dirichlet-to-Neumann map contains sufficient information to recover both potentials and obstacles via monotonicity.