A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations

Abstract

Most current prevalent iterative methods can be classified into the so-called extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrixvector multiplication with a fixed matrix, the newly introduced methods(the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly an accelerative approach(called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behavior. Comparison between benchmark extended Krylov subspace methods(Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

An orthogonally accumulated projection method for symmetric linear system of equations

“…Algorithm 1 (accumulated projection process-AP) The following procedure produces an approximate vector p to the solution vector x which satisfies Ax = b. This algorithm formed the basis of some more efficient solvers for linear system of equations such as SAP and MSAP and APAP methods introduced in [15] and [14]. It is observed that these methods seem to be more efficient than regular Krylov subspace methods in case of large scale systems in some situation.…”

“…These type of methods rely on successive projections over subspaces of R n , which produce a sequence of projection vectors with a monotonically increasing Euclidean norms. Unlike the well-known row-projection technique which can be shown as a traditional stationary iterative methods [7], the AP methods proposed in [14] do not involve matrix-vector multiplications with any fixed matrices and fixed vectors. Equipped with some accelerating technique, the AP methods exhibit some superior behavior than traditional extended Krylov subspace methods [14] in some cases.…”

“…In [14] authors also presents a first non-krylov subspace type methods-The Accumulated Projection Methods. These type of methods rely on successive projections over subspaces of R n , which produce a sequence of projection vectors with a monotonically increasing Euclidean norms.…”

“…Recently all current iterative methods are classified as extended Krylov subspace methods in [14], which are characterized by their major operations: matrixvector multiplications with usually one or two fixed matrices and one or two fixed initial vectors. These includes the most well-known stationary methods such as Jacobi, Gauss Seidal as well as SOR methods with their iterative matrices formed on the base of splitting the coefficient matrices [8], and the row projection methods such as Karcmarz's method and Cimmino's methods where the iterative matrix (not explicitly formed in iterations) are constructed by the successive multiplications of a sequence of projection matrices with a fixed sequence length m(depending on the splitting of the coefficient matrix into m submatrices [4] [7]).…”

“…As a matter of fact, it can be shown that CG can be viewed as a generalized OAP method where the orthogonality between vectors v 1 and v 2 is defined as v 1 Av 2 = 0 instead of v 1 v 2 = 0, thus a restart CG method can also be derived and is also convergent, while successful adoptionof restart strategy( which leads to a convergent iterative scheme) on other classical Krylov subspace methods are hard. It is also possible for us to develop accelerative schemes similar with those presented in [14] [15] for OAP algorithms in case of very large scale computation.…”