Regularized B-spline Wavelet Method For Solving Inverse Boundary Problem Of Laplace Equation

The inverse boundary problem of Laplace equation(LEIP)has important applications in scientific research and engineering.However,the solution of the LEIP is a highly ill-posed problem,that is,the small disturbance of the given data may cause a huge error on the solution.Meanwhile,due to technical and physical limits,some noise is inevitable in the known data.Therefore,in recent years,how to solve the LEIP with noise data input efficiently has attracted more and more attention.During the last 20 years,wavelet methods have raised special attentions,due to its several excellent characteristics,such as the local supports and vanishing moment properties.Based on these good properties,the coefficient matrix in the linear algebraic equations systems are very small compared to the other methods,and hence,can be dropped without significantly affecting the solution.Moreover,the base of wavelets can be obtained by scaling functions and the mother wavelet functions,which allows easy programming and implementation on a computer.Therefore,some scholars paid their attentions to construct various wavelet methods for numerical solution on a basis of the different wavelet basis function.However,there is no research to develop the regularization B-spline wavelet method for the LEIP.In fact,B-spline wavelets methods could lead to higher accuracy among the wavelet families,for approximating the smooth functions.In addition,it is easy to realize on the computer,due to its characteristics of finite support and explicit formula.On the other hand,the regularization technique is essential for the ill-posedness of the coefficient matrix.In this paper,the regularization B-spline wavelet method for solving the LEIP in the presence of noisy data is proposed based on the good properties of B-spline wavelet and Tikhonov regularization technique.And use the regularized B-spline wavelet method to deal with LEIP on different regions(ie rectangular region,circular region and irregular region).By comparing and analyzing the numerical results of the algorithm,it can be concluded that,on the rectangular area,compared with the B-spline wavelet method,the regularized B-spline wavelet method enhances the understanding stability and improves the anti-noise ability.When the noise data interference reaches 20%-30%,the ideal inversion result can still be obtained,compared with the existing algorithm radial basis point method(RBCM)and least squares radial basis point method(LS-RBCM),the regularized B-spline wavelet method has also achieved a high numerical solution accuracy ingeneral.In the circular region,when the noise level is 0.01,the result of the B-spline wavelet method has a large error,and the regularized B-spline wavelet method can still obtain relatively stable ideal results.In the irregular region,the regularized B-spline wavelet method can still obtain ideal inversion results,especially in the spiral boundary region,you can still get better inversion results when the noise level reach 10%.