Reconstructing the Coherent-Mode Structure

Fast Algorithm for Solution of Dirichlet Problem for Laplace Equation

ALEXANDRE GREBENNIKOV Facultad de Ciencias Físico Matemáticas Benemérita Universidad Autónoma de Puebla Av. San Claudio y Río verde, Ciudad Universitaria, CP 72570, Puebla MÉXICO

Abstract-- A new approach for solution the Dirichlet boundary problem for Laplace equation is proposed. It is based on the introduced by author GR-principle and leads to new method that uses the inverse Radon transformation. The fast algorithm for proposed method is constructed, realized as software for MATLAB system and demonstrated with numerical experiments on simulated examples.

Keywords-- fast algorithms, Dirichlet problem, Laplace equation, Radon transformation

finite explicit formulas for the arbitrary convex domain 1. Introduction  . The corresponding fast numerical algorithms are The classic Dirichlet boundary problem of solving the constructed and testified on model numerical ex- Laplace equation with respect to the potential function amples. ux, y inside the plane domain  with a boundary  can be written as the next form: 2. GR–Method and fast algorithm ux, y  0, x, y  ; Application of the GR-principle to the considering (1) problem means to construct an analogue of equation ux, y  f , x, y  ; (1), describing the distribution of the potential function ux, y along of “general rays", which are presented by where  is the Laplace operator, f (x, y) is a given some straight line l with the parameterization: function. There are two main approaches for solving x  p cos  t sin  , y  p sin   t cos . Here p is a this problem in analytical form: Fourier decomposition length of the perpendicular, passed from the centre of and the Green function method. The numerical al- coordinates to the line l ,  is the angle between the gorithms are based on the Finite Differences method, axis x and this perpendicular. Using this parameteriza- Finite Elements method and the Boundary Integral ux, y x, yl Equation method. All methods and algorithms con- tion, we shall define the potential for structed on the bases of these approaches have some as functions ut of variable t . We define also for p  difficulties in realization for the complex geometrical every fixed and potentials u0 p,  u(t0 ), form of the domain  . The Green function method is u p,  u(t ), for parameters t ,t , that corres- the explicit one [1], but it is difficult to construct the 1   1 0 1 Green function for the complex geometry of  . An- ponds to the points of the intersection of the line l and other four approaches lead to solving systems of linear boundary of the domain. We obtain mentioned ana- algebraic equations [2] that requires a lot of computer logue of equation (1) on the line l for every fixed p time and memory. Hence, the development of new fast and  as the next ordinary differential equation algorithms for solution of the considering problem is " very actual. ut t  0 . (2) We consider here a new approach that uses the General Ray (GR) principle, proposed by the author in Suppose that the domain  is a convex one. Then, [3] and successfully used for modelling of the Com- integrating twice equation (2), we obtain for ux, y puter Tomography schemes [3, 4]. This idea gives pos- the next formula sibility to get solution of the Dirichlet problem as the 1 u1 ( p,)  u0 ( p,)  ux, y  R  (t1  t0 ), (3)  2  where R 1 is the inverse Radon transform operator. Formula (3) presents GR-method for the solution of the problem (1). This is the new explicit method using parameters t0 ,t1 and functions u0 p,, u1 p, , which can be constructed on known function f (x, y) and equations for the contour  . Proposed method does not require solving any equations for known parameters t0 ,t1 and functions u0 p,, u1 p, , because the Radon transform can be inversed by explicit formulas. Thus, constructed by author GR–algorithm and corresponding computer software, which realizes this method, are sufficiently fast. We demonstrate it by some numerical experiments, particularly in comparing with PDE toolbox of the MATLAB system.

3. Numerical experiments Fig. 1. Reconstruction time t=0.703 sec. We have constructed the program realization of GR– algorithm in MATLAB system. We used the uniform discretization of variables p [1,1],  [0, ], so as variables x, y , with n nodes. To calculate the inverse Radon transform for discrete data we used iradon program of MATLAB package. We make testes on math- ematically simulated model examples with known ex- act potential functions ux, y . The first part of the tests corresponds to the simple domain  as the unit circle. It was realised for numerical justification of the formula (3) and comparing on the time of calculations with some result of PDE toolbox application. For the unit circle we have very simple formulas for the de- 2 sired parameters t0,1  ∓ 1 p and functions

ui p,  f (xi , yi ), xi  p cos  ti sin;

yi  psin  ti cos; i  0,1.

Results of the reconstruction of the function ux, y for n = 52 are presented in Fig. 1 – 3. In Fig. 4 – 7 it is presented reconstruction of the sufficiently difficult function for different numbers n=52, 202, 502, 1002. these examples confirm good approximating properties Fig. 2. Reconstruction time t= 0.1176 sec. of proposed algorithm. Fig. 3. Fig. 5.

Fig. 4. Fig. 6. Fig. 7. Fig. 8. Reconstruction by proposed algorithm, n=102, t= 1.7120 sec

In Fig.8 the reconstruction by proposed algorithm is presented and in Fig.9 - the reconstruction of the sim- ilar function with the MATLAB PDE toolbox. Ob- tained results demonstrate that for more large number of nodes proposed algorithm has almost twice minor time of calculations. The last numerical experiments correspond to the solution of the problem for the domain  that is re- stricted by the curve  consisting from arcs of two cir- cumferences. It is not difficult for this case to calculate necessary parameters and functions for formula (3). The results presented in Fig.10, Fig.11 confirm the validity of proposed GR-method and fast property of the constructed algorithm. Fig. 9. Reconstruction by PDE toolbox MATLAB, number of nodes 403, t=3.34 sec Fig.11. Reconstruction by proposed algorithm, n=52, t= 0.92 sec 4. Conclusion The new method for the solution of the Dirichlet problem for the Laplace equation is proposed. The fast property of constructed algorithm is justified by numerical experiments. Developed approach can be in principle applied for another boundary problems (for example, Neumann and mixed) and more general equations.

Rreferences: [1] S.L. Sobolev, Equations of Mathematical Physics., Moscow, 1966. [2] A.A. Samarsky, Theory of Difference Schemes, Moscow, 1977. [3] A. I. Grebennikov, “Regularization algorithms for electric tomography images reconstruction,” WSEAS TRANSACTION on SYSTEMS J., Issue 2, Vol. 2, pp. 487 -492, 2003. [4] A. Grebennikov, “Local Regularization Algo- rithms of Solving Coefficient Inverse Prob- lems for Some Differential Equations,” Inverse Fig.10. Reconstruction by proposed algorithm, Problems in Engng , Vol. 11, No 3, 2003. n=32, t= 0.47 sec