[PDF] Top 20 Null-space preconditioners for saddle point systems

... in two steps. When N e 6= N the eigenvalues may not tell us everything about con- vergence [24] although it is commonly observed that tightly clustered eigenvalues do predict convergence of GMRES in non-pathological ... See full document

... Taylor–Hood element) is used, and is the stabilization matrix otherwise. We assume that A is symmetric positive definite, which is the case when a Dirichlet condition is imposed on at least part of the boundary. The ... See full document

... M and N, M( ) ≺ N means that N – M is symmetric positive (semi)definite; ( · ) ∗ denotes the conjugate transpose of either a vector or a matrix; θ ¯ stands for the conjugate of a complex number θ ; Re(θ ) and Im(θ ) stand ... See full document

... sparse systems arising from the (linearized) Navier-Stokes equations is critical to the simulation of incompressible fluid ...Linear systems of equations are typically solved (approximately) by iterative ... See full document

... effective preconditioners are nonsymmetric or symmetric ...certain preconditioners, P −1 A is self-adjoint with respect to a known inner product h·, ... See full document

... solving saddle point problems, including whether the problem has a unique solution, how to discretize a continuous saddle point problem, how to choose an appropriate preconditioner and how to ... See full document

... efficient preconditioners for the reduced linear saddle point system ...linear saddle point sys- tem (), such as the block diagonal preconditioner and the constraint preconditioner [], ... See full document

...  is symmetric positive definite, B  R n m  ( n  m ) has full column rank, and f  R n and g  R m are given vectors. Here and in the sequel, we use ( )  T to denote the transpose,   ( ) the range space and ... See full document

... of systems including molecules in gas phase, molecular crystals and materials, using different target potential energy surfaces (empirical, semiempirical and ab initio) as well as different optimisation tasks ... See full document

... We assume that E and A have dimensions n × n and m × n respectively, with m < n, that E is symmetric positive definite, and that A has rank m. The use of the inverse in the (1,1) block is purely notational, to ... See full document

... singular preconditioners performs rather well as compared with that with nonsingular ...singular preconditioners Q requires the computation of Q † b for a given vector b, which is very time- ... See full document

... is null Mannheim curves from ¨ Oztekin and Erg ¨ut 4. Since a null vector and a nonnull vector are linearly independent in the Minkowski space E 3 1 , they have noticed that the Mannheim partner ... See full document

... A preliminary numerical investigation has been made using this map. It is found that particles are contained in a finite portion of the relevant phase space, that is with finite values of A n and B n , reflecting ... See full document

... the null point as a function of initial wave-pulse length and ...magnetic null point and create a spike of the current density close to the null point, initiating magnetic ... See full document

... Table 4.3 shows the results of the block AISM algorithm compared to point AISM for the matrices tested. The block partitioning was obtained with a parameter τ of the cosine al- gorithm ranging from 0.1 to 0.5 ... See full document

... [r] ... See full document

... The transition state model of intermediate mass fragment emission approximates the emission width in terms of the density of states at the saddle point barrier using a thermal excitation energy obtained by ... See full document

... multibody systems (MBS), such as robots or smart structures, the force imposed on a specific link via re- actions from the motion of other links may need to be con- trolled in an appropriate way to ensure desired ... See full document

... The high-altitude nulls that are not located at similar latitude and longitude in all three data sets are all very close to 69.6 Mm, so slight differences in the global dipolar field due to the lower order harmonics ... See full document

... From Table 2 we can see that the GNSOR-like method requires much less iteration num- ber than NSOR-like [14], SOR-like, MSOR-like [15], so that it costs less CPU time than the others. So, the GNSOR-like method is ... See full document