Lecture 1 - 1 Fundamentals 1.0 Preliminaries The rst...

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1 Fundamentals 1.0 Preliminaries The first question we want to answer is: What is “computational mathematics”? One possible definition is: “The study of algorithms for the solution of computa- tional problems in science and engineering.” Other names for roughly the same subject are numerical analysis or scientific com- puting . What is it we are looking for in these algorithms? We want algorithms that are fast, stable and reliable, accurate. Note: One could also study hardware issues such as computer architecture and its effects, or software issues such as efficiency of implementation on a particular hardware or in a particular programming language. We will not do this. What sort of problems are typical? Example The Poisson problem provides the basis for many different algorithms for the numerical solution of differential equations, which in turn lead to the need for many algorithms in numerical linear algebra. Consider -∇ 2 u ( x, y ) = - [ u xx ( x, y ) + u yy ( x, y )] = f ( x, y ) , in Ω = [0 , 1] 2 u ( x, y ) = 0 , on Ω . One possible algorithm for the numerical solution of this problem is based on the following discretization of the Laplacian: 2 u ( x j , y k ) u j - 1 ,k + u j,k - 1 + u j +1 ,k + u j,k +1 - 4 u j,k h 2 , (1) where the unit square is discretized by a set of ( n + 1) 2 equally spaced points ( x j , y k ), j, k = 0 , . . . , n , and h = 1 n . Also, we use the abbreviation u j,k = u ( x j , y k ). Formula (1) is a straightforward generalization to two dimensions of the linear approximation u ( x ) u ( x + h ) - u ( x ) h . If we visit all of the ( n - 1) 2 interior grid points and write down the equation resulting from the discretization of the PDE, then we obtain the following system of linear equations 4 u j,k - u j - 1 ,k - u j,k - 1 - u j +1 ,k - u j,k +1 = f j,k n 2 , j, k = 1 , . . . , n - 1 along with the discrete boundary conditions u j, 0 = u j,n = u 0 ,k = u n,k = 0 , j, k = 0 , 1 , . . . , n. 1
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Jacobi 10 13 operations 1845 Gauss-Seidel 5 × 10 12 operations 1832 SOR 10 10 operations 1950 FFT 1 . 5 × 10 8 operations 1965 multigrid 10 8 operations 1979 Table 1: Improvements of algorithms for Poisson problem. Z3 1 flops 1941 Intel Paragon 10 Gflops 1990 NEC Earth Simulator (5120 processors) 40 Tflops 2002 IBM Blue Gene/L (131072 processors) 367 Tflops 2006 Table 2: Improvements of hardware for Poisson problem. This method is known as the finite difference method . In order to obtain a relative error of 10 - 4 using finite differences one needs about n = 1000, i.e., 10 6 points. Therefore, one needs to solve a 10 6 × 10 6 sparse system of linear equations. Note that the system is indeed sparse since each row of the system matrix contains at most 5 nonzero entries. Using the state-of-the-art algorithms and hardware of 1940 it would have taken one of the first computers about 300,000 years to solve this problem with the desired accuracy. In 1990, on the other hand, it took about 1/100 second. In fact, the Earth Simulator (the fastest computer available in 2002) was able to solve a dense system of 10 6 linear equations in 10 6 unknowns in less than 6 hours using Fortran and MPI code.
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