Matrix-free methods

In computational mathematics, a matrix-free method is an algorithm for solving a linear system of equations or an 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. Langville, Amy N.; Meyer, Carl D. (2006), Google's PageRank and beyond: the science of search engine rankings, Princeton University Press, p. 40, ISBN 978-0-691-12202-1
  2. Coppersmith, Don (1993), "Solving linear equations over GF(2): Block Lanczos algorithm", Linear Algebra and its Applications, 192: 33–60, doi:10.1016/0024-3795(93)90235-G
  3. Knyazev, Andrew V. (2001). "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method". SIAM Journal on Scientific Computing. 23 (2): 517–541. CiteSeerX 10.1.1.34.2862. doi:10.1137/S1064827500366124.
  4. Wiedemann, D. (1986), "Solving sparse linear equations over finite fields" (PDF), IEEE Transactions on Information Theory, 32: 54–62, doi:10.1109/TIT.1986.1057137
  5. Lamacchia, B. A.; Odlyzko, A. M. (1991), "Solving Large Sparse Linear Systems Over Finite Fields", Advances in Cryptology-CRYPT0' 90, Lecture Notes in Computer Science, 537, p. 109, doi:10.1007/3-540-38424-3_8, ISBN 978-3-540-54508-8
  6. Kaltofen, E.; Lobo, A. (1996), "Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields", Algorithmica, 24 (3–4), pp. 311–348, CiteSeerX 10.1.1.17.7470, doi:10.1007/PL00008266
  7. Prabhune, Bhagyashree C.; Krishnan, Suresh (4 March 2020). "A fast matrix-free elasto-plastic solver for predicting residual stresses in additive manufacturing". Computer Aided Design. 123: 102829. doi:10.1016/j.cad.2020.102829.


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