非线性扩散方程反问题的多重网格-正则化方法
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摘要
考虑非线性扩散方程渗透率估计问题,它在多孔多相介质流渗透率估计中起重要的推动作用。为减少计算量,提出多重网格-正则化方法。在反演过程中,动态调整不同网格上的目标泛函使彼此相容,以满足"最优解是多重网格反演方法固定点"的必要条件。在固定网格反演中,使用快速稳定的正则化-高斯-牛顿法作为基本反演方法。数值模拟验证了多重网格-正则化方法不仅可以提高计算效率、改进反演结果,而且具有较强的抗噪能力。
Consider the problem of estimating the permeability function in a nonlinear diffusion equation,which plays an important role in promoting the permeability estimation within multiphase porous media flow. To reduce amount of calculation,a multigrid-regularization method is applied to solve this inverse problem. In the multigrid inversion process,for making the objective functionals at different grids compatible,they are dynamically adjusted. By this means,the necessary condition of "the optimal solution should be the fixed point of multigrid inversion"can be met. And in every inversion,at different grids,the fast and stable regularized Gauss-Newton method is used. Numerical simulations indicate that the proposed method can not only improve the computational efficiency and inversion results,but also have the anti-noise ability.
Consider the problem of estimating the permeability function in a nonlinear diffusion equation,which plays an important role in promoting the permeability estimation within multiphase porous media flow. To reduce amount of calculation,a multigrid-regularization method is applied to solve this inverse problem. In the multigrid inversion process,for making the objective functionals at different grids compatible,they are dynamically adjusted. By this means,the necessary condition of "the optimal solution should be the fixed point of multigrid inversion"can be met. And in every inversion,at different grids,the fast and stable regularized Gauss-Newton method is used. Numerical simulations indicate that the proposed method can not only improve the computational efficiency and inversion results,but also have the anti-noise ability.
引文
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[2]ZHAO J J,LIU T,LIU S S.An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation[J].Inverse Problems in Science and Engineering,2013,21(4):585-604.
[3]HAN B,FENG G F,LIU J Q.A widely convergent generalized pulse-spectrum technique for the inversion of two-dimensional acoustic wave equation[J].Applied Mathematics and Computation,2006,172(1):406-420.
[4]OH S,MILSTEIN A B,BOUMAN C A,et al.A general framework for nonlinear multigrid inversion[J].IEEE Transactions on Image Processing,2005,14(1):125-140.
[5]YE J C,BOUMAN C A,WEBB K J,et al.Nonlinear multigrid algorithms for Bayesian optical diffusion tomography[J].IEEE Transactions on Image Processing,2001,10(6):909-922.
[6]NASH S G.A multigrid approach to discretized optimization problems[J].Optimization Methods and Software,2000,14(1-2):99-116.
[7]MCCORMICK S F,WADE J G.Multigrid solution of a linearized,regularized least-squares problem in electrical impedance tomography[J].Inverse Problems,1993,9(6):697-7.