Implementation of the ADMM approach to constrained optimal control problem with a nonlinear time-fractional diffusion equation
- 1 January 2022
- journal article
- research article
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete & Continuous Dynamical Systems - S
Abstract
In this paper, we study the inverse problem of identifying the parameters in a nonlinear subdiffusion model from an observation defined in the given $ \Omega_1 $ subset of $ \Omega $. The nonlinear subdiffusion model involves a Caputo fractional derivative of order $ \alpha\in (0,1) $ in time. To address our model, we first examine the regularity of the solution for the direct problem using the Mittag-Leffler function. To investigate our inverse parameter problem, we reformulate first it in to Least-Squares optimization problem. Then, we establish the existence of the optimal solution and prove the convexity of the considered cost function by using its first derivative. To solve this problem numerically, we adapt a recent method in the literature known as the alternating direction method of multiplier (ADMM) which we establish its convergence. In order to show the effectiveness of the proposed method we present some numerical experiments.
Keywords
This publication has 31 references indexed in Scilit:
- Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimateInverse Problems, 2012
- An inverse problem for a one-dimensional time-fractional diffusion problemInverse Problems, 2012
- Existence and Uniqueness of the Weak Solution of the Space-time Fractional Diffusion Equation and a Spectral Method ApproximationCommunications in Computational Physics, 2010
- Uniqueness in an inverse problem for a one-dimensional fractional diffusion equationInverse Problems, 2009
- Implicit finite difference approximation for time fractional diffusion equationsComputers & Mathematics with Applications, 2008
- Implicit difference approximation for the time fractional diffusion equationJournal of Applied Mathematics and Computing, 2006
- An Abstract Framework for Elliptic Inverse Problems: Part 1. An Output Least-Squares ApproachMathematics and Mechanics of Solids, 2005
- From diffusion to anomalous diffusion: A century after Einstein’s Brownian motionChaos: An Interdisciplinary Journal of Nonlinear Science, 2005
- The fractional‐order governing equation of Lévy MotionWater Resources Research, 2000
- Anomalous transport in laboratory‐scale, heterogeneous porous mediaWater Resources Research, 2000