Numerical analysis of sparse initial data identification for parabolic problems
Numerical analysis of sparse initial data identification for parabolic problems
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DOI：
10.1051/m2an/2019083
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发表时间：
2019-05
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影响因子：
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通讯作者：
D. Leykekhman;B. Vexler;Daniel Walter
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文献类型：
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作者：
D. Leykekhman;B. Vexler;Daniel Walter
本文研究了齐次抛物型方程的初值识别问题。众所周知，由于抛物型方程的强光滑性，这类问题是指数不适定的。我们感兴趣的情况下，我们打算恢复的初始数据是已知的稀疏，即其支持勒贝格措施为零。我们制定的问题作为一个最优控制问题，并将未知的初始数据的稀疏性的信息到结构的目标功能。特别地，我们在正则Borel测度空间中寻找控制变量，并使用相应的范数作为目标泛函中的正则化项。这导致了一个凸但非光滑的优化问题。对于离散化，我们使用连续的分段线性有限元在空间和不连续的伽辽金有限元的任意程度的时间。对于一般情况下，我们建立的状态变量的误差估计。在一定的结构假设下，我们证明了控制变量由有限个Dirac测度的线性组合组成。对于这种情况下，我们得到的狄拉克措施以及相应的系数的位置的误差估计。数值分析的关键是齐次抛物问题的锐光滑型逐点有限元误差估计，这是独立的兴趣。此外，我们讨论了一个有效的算法方法的问题，并显示几个数值实验说明我们的理论结果。
In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.