on the stability of relaxed incomplete lu factorizations when solving large linear systems of equations arising from the discretization of elliptic boundary value problems a combination of iterative methods and preconditioners based on incomplete lu factorizations is frequently used given a model problem with variable coefficients we investigate a class of incomplete lu factorizations depending on a relaxation parameter we show that the associated preconditioner and the factorization itself both are numerically stable the theoretical results are complemented by numerical experiments

behavior of solutions of burgers equation with nonlocal boundary conditions in this paper we discuss the long time behavior of positive solutions of burgers equation ut uxx uux 0 0 t 0 with the nonlocal boundary condition u 0 t 0 ux 1 t 12 u2 1 t aup 1 t 10u x t dx q where 0 p 0 q criteria for stability are given blowup in finite time for some solutions is shown general results are discussed

blow up of solutions of semilinear euler poisson darboux equations with nonlocal boundary conditions this article studies the hyperbolic initial nonlocal boundary value problem u t t k t u t u x x u 0 0 u x 0 u 0 x u t x 0 0 0 0 u a t 0 an y u y t q dy t 0 where k is a real number p and q are nonnegative constants and u 0 m and n are given functions criteria for u to blow up in finite time are given

rilu preconditioning a computational study abstract we present a computational study of the rilu preconditioning strategy applied to finite difference discretizations of self adjoint elliptic boundary value problems with highly discontinuous coefficients the behaviour of the eigenvalues of the preconditioned system the numerical stability of the factorization process and the efficiency of the preconditioning strategy are discussed

singular boundary value problems on a half line singular boundary value problems of nonlinear second order differential equations on a half line are considered the main methods used are modified upper and lower solution techniques necessary and sufficient conditions for the existence of solutions to a class of sublinear singular boundary value problems are obtained

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

improving multifrontal methods by means of block low rank representations matrices coming from elliptic partial differential equations pdes have been shown to have a low rank property well defined off diagonal blocks of their schur complements can be approximated by low rank products given a suitable ordering of the matrix which gives to the blocks a geometrical meaning such approximations can be computed using an svd or a rank revealing qr factorization the resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations several strategies have been proposed to exploit this property we propose a low rank format called block low rank blr and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method we present experimental results that show how the blr format delivers gains that are comparable to those obtained with hierarchical formats such as hierarchical matrices h matrices and hierarchically semi separable hss matrices but provides much greater flexibility and ease of use which are essential in the context of a general purpose algebraic solver

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

three point boundary value problems for second order ordinary differential equations in banach spaces in this paper by using the sadovskii fixed point theorem we study the existence of at least one solution for the second order three point boundary value problem u t f t u t u t email protected 0

solutions in weighted spaces of singular boundary value problems on the half line in this paper we establish the existence of an unbounded solution to a second order boundary value problem on the half line also the existence of multiple unbounded positive solutions is discussed using the theory of fixed point index

parallel accelerated cyclic reduction preconditioner for three dimensional elliptic pdes with variable coefficients abstract we present a robust and scalable preconditioner for the solution of large scale linear systems that arise from the discretization of elliptic pdes amenable to rank compression the preconditioner is based on hierarchical low rank approximations and the cyclic reduction method the setup and application phases of the preconditioner achieve log linear complexity in memory footprint and number of operations and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment numerical experiments with linear systems that feature symmetry and nonsymmetry definiteness and indefiniteness constant and variable coefficients demonstrate the preconditioner applicability and robustness furthermore it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations and the tuning of the admissibility condition parameter together these parameters allow for optimization of the memory requirements and performance of the preconditioner

a parallel sweeping preconditioner for heterogeneous 3d helmholtz equations a parallelization of a sweeping preconditioner for three dimensional helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models the setup and application costs of the sequential preconditioner are shown to be o gamma 2 n 4 3 and o gamma n log n where gamma omega denotes the modestly frequency dependent number of grid points per perfectly matched layer several computational and memory improvements are introduced relative to using black box sparse direct solvers for the auxiliary problems and competitive runtimes and iteration counts are reported for high frequency problems distributed over thousands of cores two open source packages are released along with this paper parallel sweeping preconditioner psp and the underlying distributed multifrontal solver clique

blow up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition abstract in this paper we consider a semilinear heat equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove global existence theorem for max p l 1 some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

unbounded solutions for singular boundary value problems on the semi infinite interval upper and lower solutions and multiplicity this paper presents a lower and upper solution technique for singular second order boundary value problems on the half line and discusses sufficient and necessary conditions for the existence of positive solutions also multiplicity is discussed using the fixed point index

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

blow up problem for semilinear heat equation with absorption and a nonlocal boundary condition abstract in this paper we consider a semilinear parabolic equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove some global existence results criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given

a cyclic reduction approach to the numerical solution of boundary value odes a parallel algorithm for solving linear systems that arise from the discretization of boundary value odes is described it is a modification of a cyclic reduction algorithm that takes advantage of the almost block diagonal structure of the linear system a stable modification of the original algorithm is also proposed numerical results on a distributed memory parallel computer with 32 processors are presented and discussed

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

fourier analysis of iterative methods for elliptic boundary value problems his paper presents a fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems as in the von neumann stability analysis of hyperbolic and parabolic problems the approach is easier to apply reveals more details about convergence properties than about standard techniques and can be applied in a systematic way to a wide class of numerical methods although the analysis is applicable only to periodic problems the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions such as the dirichlet problem in addition they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the dirichlet case

hierarchical matrix approximations for space fractional diffusion equations abstract space fractional diffusion models generally lead to dense discrete matrix operators which lead to substantial computational challenges when the system size becomes large for a state of size n full representation of a fractional diffusion matrix would require o n 2 memory storage requirement with a similar estimate for matrix vector products in this work we present h 2 matrix representation and algorithms that are amenable to efficient implementation on gpus and that can reduce the cost of storing these operators to o n asymptotically matrix vector multiplications can be performed in asymptotically linear time as well performance of the algorithms is assessed in light of 2d simulations of space fractional diffusion equation with constant diffusivity attention is focused on smooth particle approximation of the governing equations which lead to discrete operators involving explicit radial kernels the algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain and then for the steady forced fractional diffusion equation in a bounded domain both matrix inverse and pseudo transient solution approaches are considered in the latter case our experiments show that the construction of the fractional diffusion matrix the matrix vector multiplication and the generation of an approximate inverse pre conditioner all perform very well on a single gpu on 2d problems with n in the range 1 0 5 1 0 6 in addition the tests also showed that for the entire range of parameters and fractional orders considered results obtained using the h 2 approximations were in close agreement with results obtained using dense operators and exhibited the same spatial order of convergence overall the present experiences showed that the h 2 matrix framework promises to provide practical means to handle large scale space fractional diffusion models in several space dimensions at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

on the approximate cyclic reduction preconditioner we present a preconditioning method for the iterative solution of large sparse systems of equations the preconditioner is based on ideas both from ilu preconditioning and from multigrid the resulting preconditioning technique requires the matrix only a multilevel structure is obtained by using maximal independent sets for graph coarsening a schur complement approximation is constructed using a sequence of point gaussian elimination steps the resulting preconditioner has a transparent modular structure similar to the algorithmic structure of a multigrid v cycle