A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation

Biccari U, Song Y, Yuan X, Zuazua Iriondo E (2023)

Publication Language: English

Publication Status: Published

Publication Type: Journal article

Future Publication Type: Journal article

Publication year: 2023

Journal

Inverse Problems Institute of Physics: Hybrid Open Access

Edition: IP-103963

URI: https://arxiv.org/abs/2202.01589

DOI: 10.48550/arXiv.2202.01589

Open Access Link: https://arxiv.org/abs/2202.01589

Abstract

We consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion-advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.

Authors with CRIS profile

Yongcun Song Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

University of Hong Kong (HKU) / 香港大學

Hong Kong (HK) University of Deusto / Universidad de Deusto

Spain (ES)

How to cite

APA:

Biccari, U., Song, Y., Yuan, X., & Zuazua Iriondo, E. (2023). A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation. Inverse Problems. https://dx.doi.org/10.48550/arXiv.2202.01589

MLA:

Biccari, Umberto, et al. "A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation." Inverse Problems (2023).

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