Информационная система "Конференции"

Abstracts

Parallel numerical algorithms

The gravimetry inverse problem of reconstructing the boundary surface between two media with constant densities is reduсed to the two-dimensional nonlinear equation with integral operator $$ A[z]equiv Delta sigma %int!!!int_{-infty}^{+infty} int_{a}^{b}!!!int_{c}^{d} left{ frac{1}{[(x-x^{prime})^2+(y-y^{prime})^2+ z_i^2(x^{prime},y^{prime}]^{1/2}}- right. $$ begin{equation} left. -frac{1}{[(x-x^{prime})^2+(y-y^{prime})^2+ z_0^2(x^{prime},y^{prime}]^{1/2}} right} dx^{prime} dy^{prime}=F(x,y), end{equation} noindent where $z_0(x,y)$ is some given function. For solving the nonlinear integral equation the iterative regularizing Newton method is used begin{equation} u^{k+1}=u^k-[A^{prime}(u^k)+alpha_kI]^{-1}(A(u^k)+alpha_ku^k-F), end{equation} noindent where $I$ is the identity operator, $A^{prime}(u)$ is the Frechet derivative, $alpha_k$ is a sequence of the positive parameters. After discretizating the surface and using the quadrature formulas, the problem (1)---(2) may be written out in the form of the system of linear equations with completed matrix $B$ and may be solved Gaussian elimination algorithm for every iteration. The parallel realization of the Gauss method for $m$ processors is based on the dividing the vector $F$ in $m$ parts and the matrix $B$ by the horizontal lines in the $m$ blocks, respectively. %After that each processor eliminates its own part of the unknowns. Parallelization of the basic algorithm and its realization on the Parallel Computer System MBC--1000 are implemented. The analysis of the efficiency of parallelization of the iterative algorithm with different number of processors is carried out. The work was supported by the RFBR, grant no.~00-01-00325.

Note. Abstracts are published in author's edition

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