insight - Computational Complexity - # Numerical Reconstruction of Potential in Fractional Calderón Problem

Numerical Method for Reconstructing Potential in Fractional Calderón Problem with Single Measurement

Q: How can the numerical method be extended to handle more general fractional Schrödinger equations, such as those with nonlinear or time-dependent terms?

The numerical method developed for the fractional Calderón problem can be extended to handle more general fractional Schrödinger equations by adapting the finite difference scheme and the variational framework to accommodate nonlinear and time-dependent terms. For nonlinear fractional Schrödinger equations, one approach is to employ fixed-point iteration or Newton's method to linearize the nonlinear terms at each iteration. This involves solving a linearized version of the equation at each step, which can be achieved using the existing conjugate gradient method framework. For time-dependent fractional Schrödinger equations, the method can be adapted by discretizing the time variable using techniques such as the method of lines or implicit-explicit schemes. This would involve treating the spatial discretization as previously described while introducing a time-stepping algorithm to evolve the solution over time. The stability and convergence properties of the numerical method would need to be re-evaluated in this context, particularly focusing on the interaction between the fractional Laplacian and the time-dependent terms.

Q: What are the potential applications of the developed numerical framework in real-world problems involving fractional partial differential equations?

The developed numerical framework for the fractional Calderón problem has several potential applications across various fields. In quantum mechanics, the fractional Schrödinger equation can model quantum systems with anomalous diffusion, providing insights into particle behavior in complex media. In environmental science, the framework can be applied to groundwater solute transport, where fractional models better capture the non-local effects of contaminant spread. In finance, fractional partial differential equations can be used to model asset prices and option pricing under fractional Brownian motion, allowing for more accurate risk assessments. Additionally, in materials science, the framework can be utilized to study phenomena such as phase transitions and diffusion processes in heterogeneous materials, where fractional derivatives can account for memory effects and spatial heterogeneity. The versatility of the numerical method makes it a valuable tool for tackling inverse problems in these diverse applications.

Q: Can the stability and convergence analysis be further improved by incorporating additional a priori information about the potential or the measurement data?

Yes, the stability and convergence analysis can be significantly improved by incorporating additional a priori information about the potential or the measurement data. By utilizing prior knowledge, such as bounds on the potential or smoothness assumptions, one can refine the regularization parameter selection, leading to enhanced stability in the reconstruction process. Incorporating a priori information can also help in formulating more robust Tikhonov regularization functionals that account for the specific characteristics of the potential, such as sparsity or boundedness. This can lead to better convergence rates and reduced sensitivity to noise in the measurement data. Furthermore, advanced techniques such as Bayesian inference can be employed to integrate prior distributions of the potential, allowing for a probabilistic interpretation of the reconstruction process. This approach not only improves the stability and convergence of the numerical method but also provides a framework for quantifying uncertainty in the reconstructed potential.

Core Concepts

A numerical method is developed to efficiently reconstruct the potential in one and two-dimensional fractional Calderón problems using a single measurement.

Abstract

The paper presents a numerical method for determining the potential in one and two-dimensional fractional Calderón problems with a single measurement. The key highlights are:

A finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution.
Conjugate gradient method is utilized to solve the variational problem. A suggestion is provided to choose the regularization parameter.
Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.
For the one-dimensional case, a logarithmic stability estimate is derived for the potential reconstruction. A similar result is obtained for the two-dimensional case.
The numerical schemes leverage the Toeplitz matrix structure for efficient computations using fast Fourier transform techniques.

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Stats

The following key metrics are used to support the analysis:

Fractional Laplacian constant c_n,s defined in equation (1.3)
Truncation parameter R for the computational domain
Discretization error bound O(h^κ) where κ depends on the solution regularity
Truncation error bound O(R^(-2s) max_{r∈∂[-R,R]} f(r))

Quotes

"Fractional partial differential equations have attracted more and more attentions owing to successful applications in various fields such as quantum mechanics, ground-water solute transport, finance, and stochastic dynamics."
"Research on numerical methods of fractional Calderón problem and its corresponding generalizations have not been paid much attention."

Key Insights Distilled From

A numerical method for reconstructing the potential in fractional Calder\'{o}n problem with a single measurement

by Xinyan Li at arxiv.org 09-26-2024

https://arxiv.org/pdf/2409.16711.pdf

$A numerical method for reconstructing the potential in fractional Calder\'{o}n problem with a single measurement$

Deeper Inquiries

How can the numerical method be extended to handle more general fractional Schrödinger equations, such as those with nonlinear or time-dependent terms?

The numerical method developed for the fractional Calderón problem can be extended to handle more general fractional Schrödinger equations by adapting the finite difference scheme and the variational framework to accommodate nonlinear and time-dependent terms. For nonlinear fractional Schrödinger equations, one approach is to employ fixed-point iteration or Newton's method to linearize the nonlinear terms at each iteration. This involves solving a linearized version of the equation at each step, which can be achieved using the existing conjugate gradient method framework.
For time-dependent fractional Schrödinger equations, the method can be adapted by discretizing the time variable using techniques such as the method of lines or implicit-explicit schemes. This would involve treating the spatial discretization as previously described while introducing a time-stepping algorithm to evolve the solution over time. The stability and convergence properties of the numerical method would need to be re-evaluated in this context, particularly focusing on the interaction between the fractional Laplacian and the time-dependent terms.

What are the potential applications of the developed numerical framework in real-world problems involving fractional partial differential equations?

The developed numerical framework for the fractional Calderón problem has several potential applications across various fields. In quantum mechanics, the fractional Schrödinger equation can model quantum systems with anomalous diffusion, providing insights into particle behavior in complex media. In environmental science, the framework can be applied to groundwater solute transport, where fractional models better capture the non-local effects of contaminant spread.
In finance, fractional partial differential equations can be used to model asset prices and option pricing under fractional Brownian motion, allowing for more accurate risk assessments. Additionally, in materials science, the framework can be utilized to study phenomena such as phase transitions and diffusion processes in heterogeneous materials, where fractional derivatives can account for memory effects and spatial heterogeneity. The versatility of the numerical method makes it a valuable tool for tackling inverse problems in these diverse applications.

Can the stability and convergence analysis be further improved by incorporating additional a priori information about the potential or the measurement data?

Yes, the stability and convergence analysis can be significantly improved by incorporating additional a priori information about the potential or the measurement data. By utilizing prior knowledge, such as bounds on the potential or smoothness assumptions, one can refine the regularization parameter selection, leading to enhanced stability in the reconstruction process.
Incorporating a priori information can also help in formulating more robust Tikhonov regularization functionals that account for the specific characteristics of the potential, such as sparsity or boundedness. This can lead to better convergence rates and reduced sensitivity to noise in the measurement data. Furthermore, advanced techniques such as Bayesian inference can be employed to integrate prior distributions of the potential, allowing for a probabilistic interpretation of the reconstruction process. This approach not only improves the stability and convergence of the numerical method but also provides a framework for quantifying uncertainty in the reconstructed potential.