![]() Please note that I had already copied (A,x,b) to GPU via CuArray constructs and fed these to CUSOLVER.csrlsvqr!, so the HtoD memcpy is confusing to me. The only thing I found was CUSOLVER.csrlsvqr! which takes a staggering 4seconds, which I profiled and noticed that 49% of time is spent on “void csqr_leftLooking” and 47% on “CUDA memcpy HtoD”. Using CUDA.jl I am unable to use the backslash operator as it does not work for SparseMatrixCSR. I want to parallelize this in hopes of getting a faster solve time. Strangely (or maybe I am missing something) when I change the values in t_num_threads I see no difference in the timings reported by macro. It takes around 0.05 seconds for the ldiv! step. I use LU decomposition for A and pass this to ldiv!. A is constant throughout my simulation, while b changes at every step in my loop, and I have recalculate x once b changes. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially the elliptic type.I want to solve Ax = b where A is a sparse matrix of size 10^5 x 10^5 (around 18 non-zero entries per row), x and b are vectors. The conjugate gradient method was also invented in the 1950s, with independent developments by Cornelius Lanczos, Magnus Hestenes and Eduard Stiefel, but its nature and applicability were misunderstood at the time. The theory of stationary iterative methods was solidly established with the work of D.M. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest. Jamshīd al-Kāshī used iterative methods to calculate the sine of 1° and π in The Treatise of Chord and Sine to high precision.Īn early iterative method for solving a linear system appeared in a letter of Gauss to a student of his. The construction of preconditioners is a large research area. The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The analysis of these methods is hard, depending on a complicated function of the spectrum of the operator. However, in the presence of rounding errors this statement does not hold moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. Since these methods form a basis, it is evident that the method converges in N iterations, where N is the system size. In the case of non-symmetric matrices, methods such as the generalized minimal residual method (GMRES) and the biconjugate gradient method (BiCG) have been derived.Ĭonvergence of Krylov subspace methods In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A x = b one works with the minimal residual method (MINRES). In contrast, direct methods attempt to solve the problem by a finite sequence of operations. A mathematically rigorous convergence analysis of an iterative method is usually performed however, heuristic-based iterative methods are also common. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A specific implementation of an iterative method like gradient descent or hill climbing, including any termination criteria, is an algorithm of the iterative method. In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. Algorithm in which each approximation of the solution is derived from prior approximations
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