Usefulness of Multiple-Precision Arithmetic for Numerical Solution of Inverse Problems
Usefulness of Multiple-Precision Arithmetic for Numerical Solution of Inverse Problems
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多精度算法在反问题数值求解中的用途
DOI：
10.11345/nctam.54.307
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
K. Onishi
中科院分区：
文献类型：
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作者：
K. Iijima;Manabu Nakada;Katsuyoshi Saito;K. Onishi
实数通常在计算机中表示为具有有限位的十六进制浮点数。因此，数值分析往往会产生舍入误差。在逆问题和不适定问题中，舍入误差尤其影响数值解的精度。我们尝试使用一种多精度算法来减小舍入误差。本文通过两维Laplace方程的柯西问题、二维反向热传导问题和三维逆散射形状识别问题三个典型的例子来说明多精度算法的有效性。数值算例表明，对于柯西问题和后向热传导问题，多精度算法与任意网格谱方法相结合，对于反向散射问题，与本征函数展开法相结合，在数值解的分辨率上有很好的效果。
Real numbers are usually represented in the computer as hexadecimal floating point numbers with finite digits. Accordingly the numerical analysis is often suffered from rounding errors. The rounding errors particularly deteriorate the precision of numerical solution in inverse and ill-posed problems. We attempt to use a multiple-precision arithmetic for reducing the rounding error evil. In this paper we try to show effectiveness of the multiple-precision arithmetic by taking three typical examples; the Cauchy problem of the Laplace equation in two dimensions, the backward heat conduction problem in two dimensions, and the shape identification problem by inverse scattering in three dimensions. It is concluded from a few numerical examples that the multiple-precision arithmetic works well on the resolution of those numerical solutions, as it is combined with an arbitrary grid spectral method for the Cauchy problem and the backward heat conduction problem, and with the eigenfunction expansion method for the inverse scattering problem.