preconditioning by fast direct methods for nonself adjoint nonseparable elliptic equations we consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself adjoint elliptic boundary value problems we derive bounds on convergence rates that are independent of discretization mesh size for two dimensional problems on rectangular domains discretized on an n times n grid these bounds lead to asymptotic operation counts of o n 2 log n log varepsilon 1 to achieve relative error varepsilon and o n 2 log n 2 to reach truncation error

a variance minimization problem for a markov decision process abstract in the steady state of a discrete time markov decision process we consider the problem to find an optimal randomized policy that minimizes the variance of the reward in a transition among the policies which give the mean not less than a specified value the problem is solved by introducing a parametric markov decision process with average cost criterion it is shown that there exists an optimal policy which is a mixture of at most two pure policies as an application the toymaker s problem is discussed

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

new conditions for non stagnation of minimal residual methods in the solution of large linear systems a condition guaranteeing that a minimal residual krylov subspace method makes some progress i e that it does not stagnate is that the symmetric part of the coefficient matrix be positive definite this condition results in a well established worst case bound for the convergence rate of the iterative method due to elman this bound has been extensively used e g when the linear system comes from discretized partial differential equations to show that the convergence of gmres is independent of the underlying mesh size in this paper we introduce more general non stagnation conditions which do not require the symmetric part of the coefficient matrix to be positive definite and that guarantee for example the non stagnation of restarted gmres for certain values of the restarting parameter

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

a generalized conjugate gradient least square method a generalizeds term truncated conjugate gradient method of least square type proposed in 1a b is extended to a form more suitable for proving when the truncated version is identical to the full term version advantages with keeping a control term in the truncated version is pointed out a computationally efficient new algorithm based on a special inner product with a small demand of storage is also presented we also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans term recursion identical to the full term version important earlier results on this latter topic are found in 15 16 8 and 11

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

preconditioned iterative methods for the nine point approximation to the convection diffusion equation iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth order finite difference schemes for convection diffusion problems simple recurrences for implementing the ilu 0 factorization of the nine point scheme are derived different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices in terms of algebraic properties of the preconditioned matrices our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses comparison of the convergence rates of gmres applied to the preconditioned linear systems is done with respect to the field of values ritz and harmonic ritz values of the preconditioned matrices numerical results show that the gmres residual norm decreases rapidly when the difference between the ritz and harmonic ritz values becomes small we also describe the results of experiments when some well known krylov subspace methods are used to solve the linear system arising from the compact fourth order discretizations

mean variance tradeoffs in an undiscounted mdp a stationary policy and an initial state in an mdp markov decision process induce a stationary probability distribution of the reward the problem analyzed here is generating the pareto optima in the sense of high mean and low variance of the stationary distribution in the unichain case pareto optima can be computed either with policy improvement or with a linear program having the same number of variables and one more constraint than the formulation for gain rate optimization the same linear program suffices in the multichain case if the ergodic class is an element of choice

mean variance criteria in an undiscounted markov decision process abstract in the steady state of an undiscounted markov decision process we consider the problem of finding an optimal stationary probability distribution that minimizes the variance of the reward in a transition among the stationary probability distributions which give a mean not less than a specified value the problem consists of a mathematical program with linear constraints and a non linear objective the solution technique replaces the non linear part of the objective with a constant inserts the constant as a constraint and then parametrically analyzes the resulting linear program three numerical examples are discussed

discounted mdp s distribution functions and exponential utility maximization the present value of the rewards associated with a discrete time markov process has a probability distribution which depends on the initial state the first part of the paper applies fixed point theory to a system of equations for the distribution functions of the present value the second part of the paper expands the model to a markov decision process mdp and considers the maximization of the expected utility of the present value when the utility function is exponential

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

robust preconditioning estimates for convection dominated elliptic problems via a streamline poincare friedrichs inequality this paper is devoted to the streamline diffusion finite element method combined with equivalent preconditioning for solving convection dominated elliptic problems the preconditioner is obtained from the streamline diffusion inner product it is proved that the obtained convergence is robust i e bounded independently of the perturbation parameter varepsilon for proper convection vector fields the key to the estimates is an improved streamline poincare friedrichs inequality

zeta functions for curves and log canonical models the topological zeta function and igusa s local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p adic field when f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so called log canonical model of f 1 0 in bbb a 2 this result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions also in the formula for igusa s local zeta function appears a remarkable non symmetric q deformation of the intersection matrix of the minimal resolution of a hirzebruch jung singularity 1991 mathematics subject classification 32s50 11s80 14e30 14g20

existence of log canonical closures let f x u be a projective morphism of normal varieties and x a dlt pair we prove that if there is an open set u 0 u such that x u u 0 has a good minimal model over u 0 and the images of all the non klt centers intersect u 0 then x has a good minimal model over u as consequences we show the existence of log canonical compactifications for open log canonical pairs and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

contribution of jumping numbers by exceptional divisors abstract we investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension inspired by the results for curves on surfaces by smith and thompson 17 and tucker 18 in particular we construct an example of an exceptional divisor that is not contracted in the log canonical model and does not contribute any jumping numbers

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

variance penalized markov decision processes we consider a markov decision process with both the expected limiting average and the discounted total return criteria appropriately modified to include a penalty for the variability in the stream of rewards in both cases we formulate appropriate nonlinear programs in the space of state action frequencies averaged or discounted whose optimal solutions are shown to be related to the optimal policies in the corresponding variance penalized mdp the analysis of one of the discounted cases is facilitated by the introduction of a cartesian product of two independent mdps

field of values analysis of preconditioned iterative methods for nonsymmetric elliptic problems the convergence rate of krylov subspace methods for the solution of nonsymmetric systems of linear equations such as gmres or fom is studied bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse estimates for these quantities are available during the iteration from the underlying arnoldi process it is shown how these bounds can be used to study the convergence properties in particular the dependence on the mesh size and on the size of the skew symmetric part for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems this is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance