1. Analysis and numerical solvability of backward-forward conservation laws
- Author
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Liard, Thibault, Zuazua, Enrique, Liard, Thibault, Departamento de Matemáticas [Madrid], and Universidad Autonoma de Madrid (UAM)
- Subjects
Inverse problems ,Entropy solutions ,Wave-front tracking algorithm ,Backward-Forward approach ,Weak-entropy solutions ,Backward-forward method ,[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] ,[MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA] ,Optimal Control Problem ,Non-smooth optimization problem ,Conservation Laws ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Identification problems ,[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP] ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L^2(R) norm. The two main contributions of this work are as follows. -We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. -A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.
- Published
- 2022