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首页> 外文期刊>Journal of Computational Physics >HOMOTOPY, POLYNOMIAL EQUATIONS, GROSS BOUNDARY DATA, AND SMALL HELMHOLTZ SYSTEMS
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HOMOTOPY, POLYNOMIAL EQUATIONS, GROSS BOUNDARY DATA, AND SMALL HELMHOLTZ SYSTEMS

机译：同态，多项式方程，边界数据和小赫尔墨茨系统

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摘要

Inverse problems of the boundary measurement type appear in several geophysical contexts including DC resistivity, electromagnetic induction, and groundwater flow. The objective is to determine a spatially varying coefficient in a partial differential equation from incomplete knowledge of the dependent variable and its normal gradient at the boundary. Equivalent 2D discrete inverse problems based on the Helmholtz or modified Helmholtz equation reduce to systems of polynomial equations indicating that there are only a finite number of exact solutions, excluding certain pathological cases. A homotopy procedure decides whether real, positive solutions exist and, if so, generates the entire list. The computational complexity of the algorithm scales as M(M/2), where M is the number of model parameters to be found. Measurement errors are accommodated by oversampling the boundary data at additional frequencies. For test Helmholtz and modified Helmholtz inverse problems based on (i) perfect and (ii) noisy data I generate the full list of exact solutions. The homotopy approach applies to large scale, multidimensional geophysical inverse problems but at present is practical only for small systems, up to M = 9. Recent advances in homotopy theory should, however, reduce the complexity, making larger problems tractable in the future. (C) 1996 Academic Press, Inc. [References: 18]
机译：边界测量类型的反问题出现在几种地球物理环境中，包括直流电阻率，电磁感应和地下水流。目的是根据对因变量及其边界处的法线梯度的不完全了解，确定偏微分方程中的空间变化系数。基于Helmholtz或改进的Helmholtz方程的等效2D离散逆问题可简化为多项式方程组，表示只有有限数量的精确解，不包括某些病理情况。同态过程决定是否存在真实的肯定解，如果存在，则生成整个列表。该算法的计算复杂度缩放为M（M / 2），其中M是要找到的模型参数的数量。通过在附加频率上对边界数据进行过采样来解决测量误差。对于基于（i）完美和（ii）噪声数据的测试亥姆霍兹和修正的亥姆霍兹反问题，我生成了精确解的完整列表。同伦方法适用于大规模，多维地球物理反问题，但目前仅适用于M = 9的小型系统。但是，同伦理论的最新进展应降低复杂性，从而使较大的问题在将来可解决。 （C）1996 Academic Press，Inc. [参考：18]

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