an optimization based approach for the design of pde solution algorithms we develop and analyze an optimization based approach for the robust and efficient solution of pde problems consisting of multiple physics operators with fundamentally different mathematical properties our approach relies on three essential steps decomposition of the original problem into subproblems for which robust solution algorithms are available integration of the subproblems into an equivalent pde constrained optimization problem and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the pde constraints this strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components

invariant analytic orthonormalization procedure with an application to coherent states we discuss a general strategy which produces an orthonormal set of vectors stable under the action of a given set of unitary operators aj j 1 2 n starting from a fixed normalized vector in h and from a set of unitary operators we discuss several examples of this procedure and in particular we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done

a rational function preconditioner for indefinite sparse linear systems this paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices a by resorting to rational functions of a the algorithm decomposes the spectrum of a into two disjoint regions and approximates the restriction of a 1 on these regions separately we show a systematic way to construct these rational functions so that they can be applied stably and inexpensively an attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism moreover the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form a ci which can be useful in some applications the efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems including problems arising from the helmholtz equation in three dimensions

convergence conditions for splitting iteration methods for non hermitian linear systems abstract necessary and sufficient convergence conditions are studied for splitting iteration methods for non hermitian system of linear equations when the coefficient matrix is nonsingular when this theory is specialized to the generalized saddle point problem we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods which include the uzawa and the inexact uzawa methods as special cases moreover we apply this theory to the two stage iteration methods for non hermitian positive definite linear systems and obtain sufficient conditions for guaranteeing the convergence of these methods

a direct proof and a generalization for a kantorovich type inequality a direct proof for a kantorovich type inequality due to bauer and householder is presented a generalization of the inequality is also established by the theory of compound matrices

limit laws of erdos renyi shepp type on obtient la forme complete des theoremes d erdos renyi 1970 et shepp 1964 les conditions etendent celles de deheuvels devroye et lynch 1986 pour couvrir une classe de distributions plus grande

comparisons between multiplicative and additive schwarz iterations in domain decomposition methods we consider the multiplicative and additive schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence moreover we compare the multiplicative schwarz method with the weighted restricted additive schwarz method we prove that the multiplicative schwarz method is the fastest method among these three our comparisons can be done in the case of exact and inexact subspaces solves in addition we analyse two ways of adding a coarse grid correction multiplicatively or additively

inexact preconditioned conjugate gradient method with inner outer iteration an important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner outer iterations g h golub and m l overton numerical analysis lecture notes in math 912 springer berlin new york 1982 where the preconditioner is solved by an inner iteration to a prescribed precision in this paper we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive definite system and analyze its convergence property we establish a linear convergence result using a local relation of residual norms we also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to high accuracy the analysis is in agreement with observed numerical behavior of the algorithm in particular it suggests a heuristic choice of the stopping threshold for the inner iteration numerical examples are given to show the effectiveness of this choice and to compare the convergence bound

weighted and flexible versions of block cmrh method for solving nonsymmetric linear systems with multiple right hand sides abstract block krylov subspace methods are the most popular algorithms for solving large nonsymmetric linear systems with multiple right hand sides one of them is the block cmrh method this method generates a non orthogonal basis of the krylov subspace through the block hessenberg process to accelerate the convergence of the block cmrh method we will introduce two new methods first we present the block cmrh method with weighting strategy in this method the block cmrh method uses a different product at each restart second we introduce a flexible version of the block cmrh algorithm that allows varying preconditioning at every step of the algorithm numerical experiments illustrate the benefits of the presented methods

a law of the logarithm for kernel density estimators in this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function extensions to higher derivatives are included the results are applied to get optimal window widths with respect to almost sure uniform convergence

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

flexible global generalized hessenberg methods for linear systems with multiple right hand sides a variant of the global generalized hessenberg method is presented which allows varying preconditioning at each restart theoretical results that relate the residual norm of this new method with its original version are developed as two special variants the flexible global gmres method and the flexible global cmrh method are investigated both theoretically and experimentally numerical examples are conducted to illustrate the performance of these two flexible global methods in comparison with both the original global methods and weighted global methods

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

analysis of the finite precision bi conjugate gradient algorithm for nonsymmetric linear systems in this paper we analyze the bi conjugate gradient algorithm in finite precision arithmetic and suggest reasons for its often observed robustness by using a tridiagonal structure which is preserved by the finite precision bi conjugate gradient iteration we are able to bound its residual norm by a minimum polynomial of a perturbed matrix i e the residual norm of the exact gmres applied to a perturbed matrix multiplied by an amplification factor this shows that occurrence of near breakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded numerical examples are given to gain insights into these bounds

scaling to the stars a linearly scaling elliptic solver for p multigrid abstract high order methods gain increased attention in computational fluid dynamics however due to the time step restrictions arising from the semi implicit time stepping for the incompressible case the potential advantage of these methods depends critically on efficient elliptic solvers due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom n dof the runtime of the best available multigrid solvers scales with o p n dof this scaling with p significantly lowers the applicability of high order methods to high orders while the operators for residual evaluation can be linearized when using static condensation schwarz type smoothers require their inverses on fixed subdomains no explicit inverse is known in the condensed case and matrix matrix multiplications scale with p n dof this paper derives a matrix free explicit inverse for the static condensed operator in a cuboidal cartesian subdomain it scales with p 3 per element i e n dof globally and allows for a linearly scaling additive schwarz smoother yielding a p multigrid cycle with an operation count of o n dof the resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n dof for polynomial degrees at least up to 48 furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a cpu leading to time stepping for the incompressible navier stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers

flexible bicg and flexible bi cgstab for nonsymmetric linear systems flexible versions of the bicg and bi cgstab algorithms are presented which allow for the use of a different preconditioner at each step of the algorithm in particular inexact solutions of the preconditioned equations are allowed as well as the use of some inner iterative method as a preconditioner flexible bicg is investigated theoretically and a theorem is presented comparing the residual norms of flexible bicg and flexible qmr that generalizes a well known relation between the residual norms of bicg and qmr numerical experiments are conducted for flexible bi cgstab for several matrices including some indefinite and highly nonsymmetric matrices these experiments illustrate the applicability of the flexible method

an algebraic convergence theory for restricted additive schwarz methods using weighted max norms convergence results for the restrictive additive schwarz ras method of cai and sarkis siam j sci comput 21 1999 pp 792 797 for the solution of linear systems of the form ax b are provided using an algebraic view of additive schwarz methods and the theory of multisplittings the linear systems studied are usually discretizations of partial differential equations in two or three dimensions it is shown that in the case of a symmetric positive definite the projections defined by the methods are not orthogonal with respect to the inner product defined by a and therefore the standard analysis cannot be used here the convergence results presented are for the class of m matrices and more generally for h matrices using weighted max norms comparison between different versions of the ras method are given in terms of these norms a comparison theorem with respect to the classical additive schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory several ras variants are considered including new ones and two level schemes

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

electromagnetic induction in a generalized 3d anisotropic earth part 2 the lin preconditioner a practical limitation in the use of generalized 3d forward modeling algorithms for inversion of electromagnetic data is the high computational cost of solving large ill conditioned systems of linear equations arising from the discretization of the governing maxwell equations to address this problem a new class of preconditioners has recently been proposed which is based on a helmholtz decomposition of the electric field in the low induction number lin regime this paper further develops that idea and introduces a lin preconditioner which can be applied to problems characterized by a fully generalized anisotropic medium included are sample calculations demonstrating a reduction by two orders of magnitude in the number of quasi minimal residual iterates and a speedup by a factor of approximately four in the solution time for one forward calculation also included are results previously unobtainable by standard jacobi preconditioning for simulating multicomponent induction sonde response in a horizo

the effect of non optimal bases on the convergence of krylov subspace methods there are many examples where non orthogonality of a basis for krylov subspace methods arises naturally these methods usually require less storage or computational effort per iteration than methods using an orthonormal basis optimal methods but the convergence may be delayed truncated krylov subspace methods and other examples of non optimal methods have been shown to converge in many situations often with small delay but not in others we explore the question of what is the effect of having a non optimal basis we prove certain identities for the relative residual gap i e the relative difference between the residuals of the optimal and non optimal methods these identities and related bounds provide insight into when the delay is small and convergence is achieved further understanding is gained by using a general theory of superlinear convergence recently developed our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important only the need to maintain linear independence is numerical examples illustrate our theoretical results

an invariant analytic orthonormalization procedure with applications we apply the orthonormalization procedure previously introduced by bagarello and triolo j math phys 48 043505 2007 and adopted in connection with coherent states to gabor frames and other examples for instance for gabor frames we show how to construct g x l2 r in such a way the functions gn x eian1xg x an2 n z2 and a some positive real number are mutually orthogonal we discuss in some detail the role of the lattice naturally associated with the procedure in this analysis

relations between multiresolution analysis and quantum mechanics we discuss a procedure to construct multiresolution analyses mra of l2 r starting from a given seed function h s which should satisfy some conditions our method originally related to the quantum mechanical hamiltonian of the fractional quantum hall effect is shown to be model independent the role of a canonical map between certain canonically conjugate operators is discussed this clarifies our previous procedure and makes much easier most of the original formulas producing a convenient framework to produce examples of mra

fast iterative solution of elliptic control problems in wavelet discretization we investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional sobolev norms of the control and the state starting point is the formulation of the infinite dimensional control problem in terms of boundary adapted biorthogonal spline wavelets involving only l2 norms of wavelet expansion coefficients where different norms are realized by a diagonal scaling together with a riesz map and constraints in form of an l2 isomorphism the coupled system of equations resulting from optimization is solved by an inexact conjugate gradient cg method for the control which involves the approximate inversion of the primal and the adjoint operator using again cg iterations starting from a coarse discretization level we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level the resulting inexact cg scheme is a fast solver it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid a consequence of grid independent condition numbers of the linear operators in wavelet coordinates in the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution moreover for different situations the performance of the fully iterative inexact cg scheme is investigated confirming the theoretical results

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

laws of the iterated logarithm for nonparametric density estimators we establish a law of the iterated logarithm for a triangular array of independent random variables and apply it to obtain laws for a large class of nonparametric density estimators we consider the case of rosenblatt parzen kernel estimators trigonometric series estimators and orthogonal polynomial estimators in detail and point out that our technique has wider application

gabor like systems in l2 rd and extensions to wavelets in this paper we show how to construct a certain class of orthonormal bases in l 2 r d starting from one or more gabor orthonormal bases in l 2 r each such basis can be obtained acting on a single function x l 2 r d with a set of unitary operators which operate as translation and modulation operators in suitable variables the same procedure is also extended to frames and wavelets many examples are discussed