MULTIGRID ALGORITHMS FOR INVERSE PROBLEMS WITH LINEAR PARABOLIC PDE CONSTRAINTS
MULTIGRID ALGORITHMS FOR INVERSE PROBLEMS WITH LINEAR PARABOLIC PDE CONSTRAINTS
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DOI：
10.1137/070687426
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发表时间：
2008-01-01
影响因子：
3.1
通讯作者：
Biros, George
中科院分区：
文献类型：
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作者：
Adavani, Santi S.;Biros, George
我们提出了一种用于解决可变线性抛物线抛物线偏微分方程的源识别逆问题解决方案解决方案的多族算法。我们考虑反转变量仅是空间的函数的问题。我们考虑了L-2 Tikhonov正则化的情况。我们的算法的收敛速率是与网格独立的 - 即使在没有正则化的情况下也是如此。此功能使该方法算法对正则化参数的值有鲁棒性，因此对于我们寻求高保真重建的情况。逆问题被提出为PDE受限的优化。我们使用缩小的空间方法，在该方法中我们消除了状态和伴随变量，并使用共轭梯度在反转参数空间中迭代。我们采用V-Cycle Multigrid方案来先进Hessian。 Multigrid Smoooly是一种两步固定的迭代求解器，它通过在高频子空间中（使用高通滤波器）中的迭代而无穷地反转近似的Hessian。我们通过完整的观察结果分析了恒定系数案例的方案的性能；我们通过分析计算降低的Hessian的频谱和Multigrid方案的平滑因子。使用向后的有限差分方案将前进和伴随问题离散化。我们的反转算法的总体复杂性为O（ntn + n log（2）n），其中n是空间中的网格点，n-t是时间步长的数量。我们提供数值实验，以证明该方法对正则化参数的不同扩散系数和值的有效性。我们还提供启发式方法，并对该案例进行数值实验，并具有可变系数和部分观察结果。我们观察到与恒定情况相同的复杂性。最后，我们研究了将缩小的空间求解器用作全空间求解器的预处理的有效性。
We present a multigrid algorithm for the solution of source identification inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space only. We consider the case of L-2 Tikhonov regularization. The convergence rate of our algorithm is mesh-independent-even in the case of no regularization. This feature makes the method algorithmically robust to the value of the regularization parameter, and thus useful for the cases in which we seek high-fidelity reconstructions. The inverse problem is formulated as a PDE-constrained optimization. We use a reduced-space approach in which we eliminate the state and adjoint variables, and we iterate in the inversion parameter space using conjugate gradients. We precondition the Hessian with a V-cycle multigrid scheme. The multigrid smoother is a two-step stationary iterative solver that inexactly inverts an approximate Hessian by iterating exclusively in the high-frequency subspace (using a high-pass filter). We analyze the performance of the scheme for the constant coefficient case with full observations; we analytically calculate the spectrum of the reduced Hessian and the smoothing factor for the multigrid scheme. The forward and adjoint problems are discretized using a backward-Euler finite-difference scheme. The overall complexity of our inversion algorithm is O(NtN + N log(2) N), where N is the number of grid points in space and N-t is the number of time steps. We provide numerical experiments that demonstrate the effectiveness of the method for different diffusion coefficients and values of the regularization parameter. We also provide heuristics, and we conduct numerical experiments for the case with variable coefficients and partial observations. We observe the same complexity as in the constant-coefficient case. Finally, we examine the effectiveness of using the reduced-space solver as a preconditioner for a full-space solver.