Matrixfree

Matrix-free refers to computational approaches that avoid forming or storing a matrix explicitly. Instead, the action of the matrix on a vector, typically denoted Ax, is provided by a routine or operator. This allows solving very large or expensive problems where the matrix is difficult to assemble or prohibitively large to store, such as discretized partial differential equations or complex operators.

The approach is most common with iterative Krylov subspace methods, including conjugate gradient, GMRES, BiCGStab, and related solvers. These methods rely on matrix-vector products rather than explicit matrices and often require a compatible preconditioner defined in terms of operations on vectors rather than matrix entries. In nonlinear problems, matrix-free techniques appear in Jacobian-free Newton–Krylov (JFNK) methods, where the Jacobian is not formed explicitly; instead, Jacobian-vector products are computed (via finite differences, automatic differentiation, or operator-based means) to advance Newton iterations.

Key considerations include having an efficient and robust operator that computes Ax, memory savings from not storing a full matrix, and the design of effective preconditioners that can be applied without an explicit matrix. Parallelism and locality of the operator are important for performance. Matrix-free methods are particularly suited to problems from finite element and finite volume discretizations, computational fluid dynamics, structural mechanics, electromagnetics, large-scale eigenvalue problems, and certain optimization and machine learning contexts.

Advantages include reduced memory usage and improved scalability for large systems; limitations include the challenge of designing good preconditioners and potential slower convergence when the operator is ill-conditioned.