The purpose of the present paper is to show a new numeric and symbolic algorithm for inverting a general nonsingular k-heptadiagonal matrix.This work is based on Doolitle LU factorization of the matrix.We obtain a ser...The purpose of the present paper is to show a new numeric and symbolic algorithm for inverting a general nonsingular k-heptadiagonal matrix.This work is based on Doolitle LU factorization of the matrix.We obtain a series of recursive relationships then we use them for constructing a novel algorithm for inverting a k-heptadiagonal matrix.The computational cost of the algorithm is calculated.Some illustrative examples are given to demonstrate the effectiveness of the proposed method.展开更多
According to the least square criterion of minimizing the misfit between modeled and observed data, this paper provides a preconditioned gradient method to invert the visco-acoustic velocity structure on the basis of ...According to the least square criterion of minimizing the misfit between modeled and observed data, this paper provides a preconditioned gradient method to invert the visco-acoustic velocity structure on the basis of using sparse matrix LU factorization technique to directly solve the visco-acoustic wave forward problem in space-frequency domain. Numerical results obtained in an inclusion model inversion and a layered homogeneous model inversion demonstrate that different scale media have their own frequency responses, and the strategy of using low-frequency inverted result as the starting model in the high-frequency inversion can greatly reduce the non-tmiqueness of their solutions. It can also be observed in the experiments that the fast convergence of the algorithm can be achieved by using diagonal elements of Hessian matrix as the preconditioned operator, which fully incorporates the advantage of quadratic convergence of Gauss-Newton method.展开更多
A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex...A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.展开更多
LU and Cholesky factorizations for dense matrices are one of the most fundamental building blocks in a number of numerical applications.Because of the O(n^(3))complexity,they may be the most time consuming basic kerne...LU and Cholesky factorizations for dense matrices are one of the most fundamental building blocks in a number of numerical applications.Because of the O(n^(3))complexity,they may be the most time consuming basic kernels in numerical linear algebra.For this reason,accelerating them on a variety of modern parallel processors received much attention.We in this paper implement LU and Cholesky factorizations on novel massively parallel artificial intelligence(AI)accelerators originally developed for deep neural network applications.We explore data parallelism of the matrix factorizations,and exploit neural compute units and on-chip scratchpad memories of modern AI chips for accelerating them.The experimental results show that our various optimization methods bring performance improvements and can provide up to 41.54 and 19.77 GFlop/s performance using single precision data type and 78.37 and 33.85 GFlop/s performance using half precision data type for LU and Cholesky factorizations on a Cambricon AI accelerator,respectively.展开更多
文摘The purpose of the present paper is to show a new numeric and symbolic algorithm for inverting a general nonsingular k-heptadiagonal matrix.This work is based on Doolitle LU factorization of the matrix.We obtain a series of recursive relationships then we use them for constructing a novel algorithm for inverting a k-heptadiagonal matrix.The computational cost of the algorithm is calculated.Some illustrative examples are given to demonstrate the effectiveness of the proposed method.
文摘According to the least square criterion of minimizing the misfit between modeled and observed data, this paper provides a preconditioned gradient method to invert the visco-acoustic velocity structure on the basis of using sparse matrix LU factorization technique to directly solve the visco-acoustic wave forward problem in space-frequency domain. Numerical results obtained in an inclusion model inversion and a layered homogeneous model inversion demonstrate that different scale media have their own frequency responses, and the strategy of using low-frequency inverted result as the starting model in the high-frequency inversion can greatly reduce the non-tmiqueness of their solutions. It can also be observed in the experiments that the fast convergence of the algorithm can be achieved by using diagonal elements of Hessian matrix as the preconditioned operator, which fully incorporates the advantage of quadratic convergence of Gauss-Newton method.
文摘A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising in complex computational problems in three space dimensions. The proposed class of approximate inverse is chosen as the basis to yield systems on which classic and preconditioned iterative methods are explicitly applied. Optimized versions of the proposed approximate inverse are presented using special storage (k-sweep) techniques leading to economical forms of the approximate inverses. Application of the adaptive algorithmic methodologies on a characteristic nonlinear boundary value problem is discussed and numerical results are given.
基金supported by the National Natural Science Foundation of China under Grant No.61972415the Science Foundation of China University of Petroleum,Beijing under Grant Nos.2462019YJRC004,2462020XKJS03.
文摘LU and Cholesky factorizations for dense matrices are one of the most fundamental building blocks in a number of numerical applications.Because of the O(n^(3))complexity,they may be the most time consuming basic kernels in numerical linear algebra.For this reason,accelerating them on a variety of modern parallel processors received much attention.We in this paper implement LU and Cholesky factorizations on novel massively parallel artificial intelligence(AI)accelerators originally developed for deep neural network applications.We explore data parallelism of the matrix factorizations,and exploit neural compute units and on-chip scratchpad memories of modern AI chips for accelerating them.The experimental results show that our various optimization methods bring performance improvements and can provide up to 41.54 and 19.77 GFlop/s performance using single precision data type and 78.37 and 33.85 GFlop/s performance using half precision data type for LU and Cholesky factorizations on a Cambricon AI accelerator,respectively.
