# Matrix free finite elements method for visualization in process tomography

I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could multiply regular or matrix free. In general based on special coordinate function. I have a few general question: How popular is this method? Does any another name exist for this method? I am also looking for books and article concentrate on finite element method especially matrix free multiplication and I consider about general books and article. Because many article based on rather complicated examples like conjugate gradient method or something different. As I said in topic name I try to use this method in process tomography visualization.

Matrix-free method is a general name for a class of algorithms, rather than a particular method. For example, consider solving the linear equation $Ax=b.$ If you were to solve this this problem using the Gauss elimination method, for example, then you need to pre-compute all elements of $A$ and keep them during the algorithm. If you were to use Krylov-type iterative solvers, you need to multiply the matrix $A$ by a vector at every step. You can do it differently.

1. Pre-compute $A$ and keep it in memory (as a sparse array, perhaps).
2. Do not store $A,$ but re-compute (some of) its elements every time you need to multiply $A$ by a vector.

The first approach is not a matrix-free method, while the second approach is.