Iterative methods for large continuation problems

Calvetti, Daniela; Reichel, Lothar

doi:10.1016/s0377-0427(00)00405-2

Cited by 17 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This yields a system of ( − 1) 2 are approximations of the function u at the grid points. Numerous techniques for computing w(λ) as λ is increased from, say, λ 0 to λ 1 are available; see, e.g., [1,5,6] and the references therein.…”

Section: Computed Examplesmentioning

confidence: 99%

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Beckermann¹,

Reichel²

2008

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure, which makes it possible to derive a progressive GMRES method. Eigenvalue computation is also considered.

show abstract

Section: Computed Examplesmentioning

confidence: 99%

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Beckermann¹,

Reichel²

2008

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…Bordering algorithms/deflated block elimination: The study of this class of algorithms arose primarily in path following/continuation method problems. Applied to systems of the form when A is nonsingular, these algorithms are typically based on variations of the following theme: Solve

A W = B_{1}^{T}

for W . Solve A w = f for w . Compute S = C − B 2 W . Solve S y = g − B 2 w for y . Compute x = w − W y . …”

Section: The Null Space Methods For Dense Rows and Columnsmentioning

confidence: 99%

Prestructuring sparse matrices with dense rows and columns via null space methods

Howell

2017

Numerical Linear Algebra App

View full text Add to dashboard Cite

Abstract. Several applied problems may produce large sparse matrices with a small number of dense rows and/or columns, which can adversely affect the performance of commonly used direct solvers. By posing the problem as a saddle point system, an unconventional application of a null space method can be employed to eliminate dense rows and columns. The choice of null space basis is critical in retaining the overall sparse structure of the matrix. A one-sided application of the null space method is also presented to eliminate either dense rows or columns. These methods can be considered techniques that modify the nonzero structure of the matrix before employing a direct solver, and may result in improved direct solver performance.

show abstract

“…On a boundary edge e ∈ E B , we set v = v, q = q and [[v]] = vn, with n denoting the unit outward normal vector on the boundary ∂Ω. Furthermore, we introduce, for an edge e ∈ E, the discontinuity penalisation parameter σ by (13) σ| e = p 2 e h e .…”

Section: 12mentioning

confidence: 99%

“…In the one-dimensional setting, an analytical expression for λ c is available, cf. [7,13,17]; for the two-dimensional case, calculations have revealed that λ c = 6.808124423 ( c = 0.146883332) to 9 decimal places, see [17,44,45], and the references cited therein.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

Houston

Wihler

2018

Math. Comp.

View full text Add to dashboard Cite

Abstract. In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

show abstract

Iterative methods for large continuation problems

Cited by 17 publications

References 10 publications

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Prestructuring sparse matrices with dense rows and columns via null space methods

An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

Contact Info

Product

Resources

About