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|Title: ||ON THE MODIFIED TIKHONOV’S REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF THE LAPLACE EQUATION|
|Authors: ||Jiang-Ren Chang;Weichung Yeih;Min-Harng Shieh|
|Keywords: ||Tikhonov’s regularization method;singular value decomposition, L-curve;modified Tikhonov’s regularization method;Cauchy problem|
|Issue Date: ||2017-02-15T08:15:43Z
|Publisher: ||Journal of Marine Science and Technology|
|Abstract: ||Abstract:In this paper, an inverse problem of the Laplace equation with
Cauchy data is examined. Due to the ill-posed behavior of this inverse
problem, the Tikhonov’s regularization technique is employed and
the L-curve concept is adopted to determine the optimal regularization
parameter. Also, the singular value decomposition method is
used in conjunction with the L-curve concept for the same problem.
Numerical results show that neither the traditional Tikhonov’s regularization
method nor the singular value decomposition method can
yield acceptable results when the influence matrix is highly ill-posed.
A modified regularization method, which combines the singular value
decomposition method and regularization method, is thus proposed,
and this new method shows that it is a better way to treat this kind of
inverse problems comparing with the other two traditional methods.
Numerical results also show that the inverse problem with Cauchy
data is better to formulate by the singular integral equation than by the
hypersingular integral equation for the constant element scheme. The
inverted boundary data becomes closer to the exact solution when the
number of elements increases, and numerical experiments show that
the rate of convergence is higher for the formulation using the singular
integral equation. Numerical experiments are made to examine how
the boundary Cauchy data affect the inverted process. It is concluded
that the inversion of unknown boundary data is more effective when
the Cauchy data are given more precisely and are distributed on the
whole boundary more diversely.
|Appears in Collections:||[系統工程暨造船學系] 期刊論文|
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