Matrix-free methods

Source: Wikipedia, the free encyclopedia.

In

eigenvalue problem that does not store the coefficient matrix explicitly, but accesses the matrix by evaluating matrix-vector products.[1] Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computing time, even with the use of methods for sparse matrices. Many iterative methods
allow for a matrix-free implementation, including:

Distributed solutions have also been explored using coarse-grain parallel software systems to achieve homogeneous solutions of linear systems.[6]

It is generally used in solving non-linear equations like Euler's equations in computational fluid dynamics. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver.[7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To avoid this expense, matrix-free methods are employed. In order to remove the need to calculate the Jacobian, the Jacobian vector product is formed instead, which is in fact a vector itself. Manipulating and calculating this vector is easier than working with a large matrix or linear system.

References

  1. ISBN 978-0-691-12202-1
  2. doi:10.1016/0024-3795(93)90235-G
  3. doi:10.1137/S1064827500366124
    .
  4. doi:10.1109/TIT.1986.1057137
  5. ISBN 978-3-540-54508-8
  6. S2CID 13305650
  7. doi:10.1016/j.cad.2020.102829
    .


Stub icon

This applied mathematics-related article is a stub. You can help Wikipedia by expanding it.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Matrix-free_methods&oldid=1169743693"