Efficient numerical algorithm for solving the gravimetry problem of finding a lateral density in a layer: Parallel implementation. (13th January 2020)

• Record Type: Journal Article
• Authors: Akimova, Elena N.; Misilov, Vladimir E.
• Other Names: Simos Theodore guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 13(2020)
• Page Start: 7774
• Page End: 7787
• Publication Date: 2020-01-13
• Subjects: BiCGSTAB method -- density finding problem -- inverse problems -- MPI+OpenMP technology -- parallel algorithms -- Toeplitz matrices; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6206
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : The paper is devoted to developing the new time‐ and memory‐efficient algorithm BiCGSTABmem for solving the inverse gravimetry problem of determination of a variable density in a layer using the gravitational data. The problem is in solving the linear Fredholm integral equation of the first kind. After discretization of the domain and approximation of the integral operator, this problem is reduced to solving a large system of linear algebraic equations. It is shown that the matrix of coefficients is the Toeplitz‐block‐Toeplitz one in the case of the horizontal layer. For calculating and storing the elements of this matrix, we construct an efficient method, which significantly reduces the required memory and time. For the case of the curvilinear layer, we construct a method for approximating the parts of the matrix by a Toeplitz‐block‐Toeplitz one. This allows us to exploit the same efficient method for storing and processing the coefficient matrix in the case of a curvilinear layer. To solve the system of linear equations, we constructed the parallel algorithm on the basis of the stabilized biconjugated gradient method with using the Toeplitz‐block‐Toeplitz structure of the matrix. We implemented the BiCGSTAB and BiCGSTABmem algorithms for the Uran cluster supercomputer using the hybrid MPI + OpenMP technology. A model problem with synthetic data was solved for a large grid. It was shown that the new BiCGSTABmem algorithm reduces the computation time in comparison with the BiCGSTAB. Scalability of the parallel algorithm was studied.