MATLAB: What is the best method to solve large linear system equation (Ax=b) (A:million~ by million~)

gpuMATLABsystem matrix

The maxtix A is sparse from finite element analysis.

I have used backslash to solve the equation but I want to reduce the calculation time.

I also tried to use GPU acceleration but there is a memory limitation with Titan xp with 12Gb memory.

Best Answer

You can try one of the Iterative Methods and Preconditioners.

Related Solutions

MATLAB: Could `distributed array` accelerate the speed of solving Linear Equations with Iterative Methods

This is somehow expected: distributed arrays are not meant to increase the speed of execution but to be used for arrays that are so large that they don't fit in the memory of a single computer. The extra cost for communication makes operations on them slower than on regular arrays most of the time. Instead of getting a 'OutOfMemory' error distributed arrays will allow you to run the computation, but you have to keep in mind that this computation is probably quite complex so it is going to take some time.

If you want to accelerate your computation, you may try to use gpuArray instead. Also the efficiency of pcg depends on the preconditioner so you may try to play around with the parameters of ichol ('type' and 'droptol') to find a better preconditioner for your problem.

MATLAB: Is there any way to accelerate the solving of a series of large sparse positive definite linear equations “Ax=b” with same “A” and different “b”

Two things come to my mind:

Why can't you build all the b together then make a single inversion

b = [b1 b2 ... bp]
x = A \ b

this will use direct method by single decomposition of the matrix, so it might help.

You might apply reordering technique before the decomposition/preconditioning.

Preconditioning is like invert the matrix by direct-method. So you need to find a compromise between the cost vs quality to get the best overall runtime.

Best Answer

Related Solutions

MATLAB: Could `distributed array` accelerate the speed of solving Linear Equations with Iterative Methods

MATLAB: Is there any way to accelerate the solving of a series of large sparse positive definite linear equations “Ax=b” with same “A” and different “b”

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