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.
Numerical solution by shape optimization method to an inverse shape problem in multi-dimensional advection-diffusion problem with space dependent coefficients
Dynamic financial processes identification using sparse regressive reservoir computers
Center for Innovation in Scientific Computing, Universidad Nacional Aut\'onoma de Honduras, Honduras；Department of Chemical Engineering, Norwegian University of Science；Department of Statistics；Research, National Commission of Banks
From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification
AI Institute in Dynamic Systems, Department of Mechanical Engineering, University of Washington（动态系统人工智能研究所，机械工程系，华盛顿大学）；Department of Mechanical Engineering, University of Hawai‘i at Mānoa（机械工程系，夏威夷大学马诺阿分校）；Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista “Júlio de Mesquita Filho”（数学与计算系，科技学院，保罗斯州立大学“朱利奥·梅斯奎塔·弗洛雷斯”分校）
