# Stable L\'{e}vy diffusion and related model fitting

## TL;DR

This paper introduces a numerical framework for solving stable Lévy stochastic differential equations related to fractional advection-dispersion equations, enabling modeling of heavy-tailed flows with unknown coefficients from observed data.

## Contribution

It proposes a novel numerical method combining finite volume schemes and Levenberg--Marquardt regularization to solve and fit stable SDEs for complex flow data.

## Key findings

- Numerical method effectively solves stable SDEs in general scenarios.
- Levenberg--Marquardt regularization estimates unknown advection and dispersion functions.
- Framework accurately models heavy-tailed flow phenomena.

## Abstract

A fractional advection-dispersion equation (fADE) has been advocated for heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic differential equation (SDE) driven by a stable L\'{e}vy process gives a forward equation that matches the space-fractional advection-dispersion equation and thus gives the stochastic framework of particle tracking for heavy-tailed flows. For constant advection and dispersion coefficient functions, the solution to such SDE itself is a stable process and can be derived easily by least square parameter fitting from the observed flow concentration data. However, in a more generalized scenario, a closed form for the solution to a stable SDE may not exist. We propose a numerical method for solving/generating a stable SDE in a general set-up. The method incorporates a discretized finite volume scheme with the characteristic line to solve the fADE or the forward equation for the Markov process that solves the stable SDE. Then we use a numerical scheme to generate the solution to the governing SDE using the fADE solution. Also, often the functional form of the advection or dispersion coefficients are not known for a given plume concentration data to start with. We use a Levenberg--Marquardt (L-M) regularization method to estimate advection and dispersion coefficient function from the observed data (we present the case for a linear advection) and proceed with the SDE solution construction described above.