on positive semidefinite modification schemes for incomplete cholesky factorization incomplete cholesky factorizations have long been important as preconditioners for use in solving large scale symmetric positive definite linear systems in this paper we focus on the relationship between two important positive semidefinite modification schemes that were introduced to avoid factorization breakdown namely the approach of jennings and malik and that of tismenetsky we present a novel view of the relationship between the two schemes and implement them in combination with a limited memory approach we explore their effectiveness using extensive numerical experiments involving a large set of test problems arising from a wide range of practical applications

preconditioned conjugate gradient methods for three dimensional linear elasticity finite element modelling of three dimensional elasticity problems give rise to large sparse matrices various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques incomplete factorizations based on levels of fill drop tolerance and a two level hierarchical basis are developed various techniques for ensuring that the incomplete factors have positive pivots are presented computational tests are carried out for problems generated using unstructured tetrahedral meshes quadratic basis functions are used the performance of the iterative methods is compared to a standard direct sparse matrix solver problems with up to 70 000 degrees of freedom and small 1 element aspect ratio are considered

high performance pcg solvers for fem structural analysis the preconditioned conjugate gradient algorithm is a well known and powerful method used to solve large sparse symmetric positive definite linear systems such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods it is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods

modelling of income distribution in the european union with the fokker planck equation abstract herein we applied statistical physics to study incomes of three low medium and high income society classes instead of the two low and medium income classes studied so far in the frame of the threshold nonlinear langevin dynamics and its threshold fokker planck counterpart we derived a unified formula for description of income of all society classes by way of example of those of the european union in years 2006 and 2008 hence the formula is more general than the well known formula of yakovenko et al that is our formula well describes not only two regions but simultaneously the third region in the plot of the complementary cumulative distribution function vs an annual household income furthermore the known stylised facts concerning this income are well described by our formula namely the formula provides the boltzmann gibbs income distribution function for the low income society class and the weak pareto law for the medium income society class as expected importantly it predicts to satisfactory approximation the zipf law for the high income society class moreover the region of medium income society class is now distinctly reduced because the bottom of high income society class is distinctly lowered this reduction made in fact the medium income society class an intermediate income society class

stable windings on hyperbolic surfaces let be a geometrically finite hyperbolic surface with infinite volume having at least one cusp we obtain the limit law under the patterson sullivan measure on t1 of the windings of the geodesics of around the cusps this limit law is stable with parameter 2 1 where is the hausdorff dimension of the limit set of the subgroup of mobius isometries associated with the normalization is t 1 2 1 for geodesics of length t our method relies on a precise comparison between geodesics and diffusion paths for which we need to approach the patterson sullivan measure mentioned above by measures that are regular along the stable leaves

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

formulation and solution of hierarchical finite element equations the paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems static and eigenvalue linear elastic problems as well as convection diffusion problems are studied the hierarchical formulation is well suited for adaptive procedures for the convection diffusion problem the hierarchical approximation is made in time only different hierarchical functions are proposed for different types of problems both weighted residual and least squares formulations are applied a combination of these two gives a penalty method with a constraint equation corresponding to the least squares method a whole class of time integration formulae is obtained these are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain if a linear discontinuous hierarchical base function is used in the galerkin weak formulation the method so obtained corresponds to the discontinuous galerkin method in time and is especially suited for convection dominated problems the streamline diffusion method is found to be the aforementioned penalty method this paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation properties of these systems arising from static problems are investigated the paper presents some new possibilities for iterative solution of hierarchic element equations and different procedures are compared in a numerical example finally a simple id convection diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow

marcus versus stratonovich for systems with jump noise the famous ito stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise in physics literature this dilemma is often resolved in favour of the stratonovich prescription because of its two characteristic properties valid for systems driven by brownian motion i it allows physicists to treat stochastic integrals in the same way as conventional integrals and ii it appears naturally as a result of a small correlation time limit procedure on the other hand the marcus prescription ieee trans inform theory 24 164 1978 stochastics 4 223 1981 should be used to retain i and ii for systems driven by a poisson process levy flights or more general jump processes in present communication we present an in depth comparison of the ito stratonovich and marcus equations for systems with multiplicative jump noise by the examples of a real valued linear system and a complex oscillator with noisy frequency the kubo anderson oscillator we compare solutions obtained with the three prescriptions

generalized langevin equation with multiplicative noise temporal behavior of the autocorrelation functions the temporal behavior of the mean square displacement and the velocity autocorrelation function of a particle subjected to a periodic force in a harmonic potential well is investigated for viscoelastic media using the generalized langevin equation the interaction with fluctuations of environmental parameters is modeled by a multiplicative white noise by an internal mittag leffler noise with a finite memory time and by an additive external noise it is shown that the presence of a multiplicative noise has a profound effect on the behavior of the autocorrelation functions particularly for correlation functions the model predicts a crossover between two different asymptotic power law regimes moreover a dependence of the correlation function on the frequency of the external periodic forcing occurs that gives a simple criterion to discern the multiplicative noise in future experiments it is established that additive external and internal noises cause qualitatively different dependences of the autocorrelation functions on the external forcing and also on the time lag the influence of the memory time of the internal noise on the dynamics of the system is also discussed

econophysics scaling and its breakdown in finance we discuss recent empirical results obtained by analyzing high frequency data of a stock market index the standard and poor s 500 we focus on the scaling properties and on its breakdown of the index dynamics a simple stochastic model the truncated levy flight is illustrated successes and limitations of this model are presented a discussion about similarities and differences between the scaling properties observed in financial markets and in fully developed turbulence is also provided

parallel preconditioning based on h hierarchical finite elements with application to acoustics abstract in this paper we consider the hierarchical basis preconditioning approach which is closely related to the h version of the hierarchical finite element method in general finite element formulations that employ hierarchical shape functions yield better conditioned matrix problems than those based on the lagrange nodal basis these matrix problems are also better suited to a faster rate of convergence with krylov subspace iterative methods we consider a purely algebraic approach to describe projections between the nodal and the h hierarchical bases functions which are then used to construct the preconditioning operator implementation details of the preconditioner are provided for solving finite element problems on unstructured grids we employ the preconditioning approach in the iterative solution of linear systems that arise from finite element discretization of the exterior helmholtz problem numerical results are presented to examine convergence rates for practical discretizations in acoustics and to illustrate the computational performance of the preconditioning algorithm on serial and data parallel computers

escape driven by α stable white noises we explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the brownian particle is driven by white levy noise in a dynamical regime where inertial effects can safely be neglected the behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise induced discontinuity which is caused by the generalized brownian paths that jump beyond the barrier location without actually hitting it this fact implies that the boundary conditions for the mean first passage time mfpt are no longer determined by the well known local boundary conditions that characterize the case with normal diffusion by numerically implementing properly the set up boundary conditions we investigate the survival probability and the average escape time as a function of the corresponding levy white noise parameters depending on the value of the skewness beta of the levy noise the escape can either become enhanced or suppressed a negative asymmetry parameter beta typically yields a decrease for the escape rate while the rate itself depicts a non monotonic behavior as a function of the stability index alpha that characterizes the jump length distribution of levy noise exhibiting a marked discontinuity at alpha 1 we find that the typical factor of 2 that characterizes for normal diffusion the ratio between the mfpt for well bottom to well bottom and well bottom to barrier top no longer holds true for sufficiently high barriers the survival probabilities assume an exponential behavior versus time distinct non exponential deviations occur however for low barrier heights

element based preconditioners for elasto plastic problems in geotechnical engineering iterative solvers are widely regarded as the most efficient way to solve the very large linear systems arising from finite element models their memory requirements are small compared to those for direct solvers consequently there is a major interest in iterative methods and particularly the preconditioning necessary to achieve rapid convergence in this paper we present new element based preconditioners specifically designed for linear elasticity and elasto plastic problems the study presented here is restricted to simple associated plasticity but should find wide application in other plasticity models used in geotechnics

fluctuations in multiplicative systems with jumps fluctuation properties of the langevin equation including a multiplicative power law noise and a quadratic potential are discussed the noise has the l evy stable distribution if this distribution is truncated the covariance can be derived in the limit of large time it falls exponentially covariance in the stable case studied for the cauchy distribution exhibits a weakly stretched exponential shape and can be approximated by the simple exponential the dependence of that function on system parameters is determined then we consider a dynamics which involves the above process and obeys the generalized langevin equation the same as for the gaussian case the resulting distributions possess power law tails which fall similarly to those for the driving noise whereas central parts can assume the gaussian shape moreover a process with the covariance 1 t at large time is constructed and the corresponding dynamical equation is solved diffusion properties of systems for both covariances are discussed

multiplicative stochastic resonance a multiplicative bistable stochastic system perturbed by a periodic forcing term is shown to exhibit stochastic resonance with increasing intensity of the multiplicative noise such an effect is related to the phenomenon of stochastic stabilization which takes place in the unperturbed system

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

finite element solution of three dimensional incompressible fluid flow problems by a preconditioned conjugate residual method a new algorithm for the solution of the three dimensional navier stokes equation using the mixed formulation and an incomplete factorization technique is presented the linear system obtained at each modified newton raphson iteration is solved by the preconditioned conjugate residual method pcr using an out of core incomplete lu factorization as preconditioning this factorization is performed on a matrix stored by a new static storage method called the packed skyline method the performance of the proposed algorithm is assessed for the wall driven cubic cavity problem using the q p1 enriched element

anomalous diffusion in systems driven by the stable levy noise with a finite noise relaxation time and inertia dynamical systems driven by a general levy stable noise are considered the inertia is included and the noise represented by a generalized ornstein uhlenbeck process has a finite relaxation time a general linear problem the additive noise is solved the resulting distribution converges with time to the distribution for the white noise massless case moreover a multiplicative noise is discussed it can make the distribution steeper and the variance which is finite depends sublinearly on time subdiffusion for a small mass a white noise limit corresponds to the stratonovich interpretation on the other hand the distribution tails agree with the ito interpretation if the inertia is very large an escape time from the potential well is calculated

a coarse fine preconditioner for very ill conditioned finite element problems we consider the problem of applying the conjugate gradient method to solve ill conditioned large algebraic systems of equations resulting from the finite element discretization of some three dimensional boundary value problems we present an effective preconditioner for such systems based on a multigrid technique we assess its performance with examples borrowed from large flexible aerospace structures

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

on the multi grid technique for solving three dimensional boundary value engineering problems the paper proposes an iterative technique with a similar multi grid operator for solving complex spatial bounctary value problems the description of the method and efficient vectorial algorithms based on a simple ctata structure applicable to solving three dimensional engineering problems are presented the discussion of the method s convergence includes theoretical grounds and practical aspects we give a general proof of convergence and consider the model cases to estimate the rate of convergence we also address the histories of convergence derived from the numerical tests and consider two examples of practical problems

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

an h p multigrid method for finite element analysis the finite element method generates solutions to partial differential equations by minimizing a strain energy based functional strain energy based techniques for adaptive mesh refinements are not always effective however the adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h and p finite element methods the multigrid method and a delaunay based mesh generation method the refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

solving systems of elastic bar structures by preconditioned conjugate gradient method abstract for solving systems of linear equations derived from structural analysis by conjugate gradient method a new efficient incomplete factorization preconditioning was published by saint georges et al 1 here we present an algorithm for finding a starting point for the ordering applied in 1 based on which a variant of the spiral ordering due to duff et al 2 for an undirected connected graph is formed we test the solvers in 1 and some of its variants when different orderings are applied and for each ordering some incomplete factorization preconditioners are prepared in the comparison of the considered solvers a remarkable reduction in the number of iterations was found by the presented variant of the spiral ordering with ic 0 preconditioner when systems from elastic bar structures with 3d beam elements were solved 1999 elsevier science ltd all rights reserved

free energy for brownian and geodesic homology existence and analyticity of the free energies associated with the asymptotic homology of brownian paths and geodesics are proved and a simple relation is found between them

stretched gaussian asymptotics of the truncated levy flights for the diffusion in nonhomogeneous media abstract the levy jumping process defined in terms of the jumping size distribution and the waiting time distribution is considered the jumping rate depends on the process value the fractional diffusion equation which contains the variable diffusion coefficient is solved in the diffusion limit that solution resolves itself to the stretched gaussian when the order parameter 2 the truncation of the levy flights in the exponential and power law form is introduced and the corresponding random walk process is simulated by the monte carlo method the stretched gaussian tails are found in both cases the time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter when the cutoff function falls slowly the tail of the distribution appears to be algebraic

anomalous diffusion for a correlated process with long jumps abstract we discuss diffusion properties of a dynamical system which is characterised by long tail distributions and finite correlations the particle velocity has the stable levy distribution it is assumed as a jumping process the kangaroo process with a variable jumping rate both the exponential and the algebraic form of the covariance defined for the truncated distribution are considered it is demonstrated by numerical calculations that the stationary solution of the master equation for the case of power law correlations decays with time but a simple modification of the process makes the tails stable the main result of the paper is a finding that in contrast to the velocity fluctuations the position variance may be finite it rises with time faster than linearly the diffusion is anomalously enhanced on the other hand a process which follows from a superposition of the ornstein uhlenbeck levy processes always leads to position distributions with a divergent variance which means accelerated diffusion

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

an iterative technique for the finite element analysis of near incompressible materials abstract a new preconditioner for the conjugate gradient method applicable to the finite element analysis of near incompressible finite elasticity is presented the boundary value problem is discretized with a displacement pressure mixed formulation the pressure degrees of freedom are eliminated at the element level the problem is linearized using a newton raphson technique each linear system is solved using the preconditioned conjugate gradient the new preconditioner is based on projections onto a subspace associated with the bulk modes of deformation the condition number of the preconditioned stiffness matrix and consequently the rate of convergence of the preconditioned conjugate gradient are totally insensitive to increasing bulk modulus the method has been applied to model problems in linear and nonlinear 2d and 3d using a near incompressible hyperelastic material model the performance of the iterative method compares favorable with direct solvers as well as with conjugate gradient methods for indefinite systems in terms of storage requirements operation count and cpu time

a comparison of direct and preconditioned iterative techniques for sparse unsymmetric systems of linear equations in this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric sparse systems of linear algebraic equations these problems occur in finite difference and finite element simulations of semiconductor devices and fluid flow problems we consider five iterative methods that appear to be the most promising for this class of problems the biconjugate gradient method the conjugate gradient squared method the generalized minimal residual method the generalized conjugate residual method and the method of orthogonal minimization each of these methods was tested using similar preconditioning incomplete lu factorization on a set of large sparse matrices arising from finite element simulation of semiconductor devices results are shown where we compare the computation time and memory requirements for each of these methods against one another as well as against a direct method that uses lu factorization to solve these problems the results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large sparse systems of equations and can offer significant savings in storage and cpu time

multiplicative levy processes ito versus stratonovich interpretation langevin equation with a multiplicative stochastic force is considered that force is uncorrelated it has the levy distribution and the power law intensity the fokker planck equations which correspond both to the ito and stratonovich interpretation are presented they are solved for the case without drift and for the harmonic oscillator potential the variance is evaluated it is always infinite for the ito case whereas for the stratonovich one it can be finite and rise with time slower that linearly which indicates subdiffusion analytical results are compared with numerical simulations

anomalous diffusion in stochastic systems with nonhomogeneously distributed traps the stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed the particle is subjected to a stochastic stimulation obeying a general levy stable statistics and experiences long rests due to nonhomogeneously distributed traps the memory is taken into account by subordination of that process to a random time then the subordination equation is position dependent the problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power law position dependence of the memory in the case of the gaussian statistics the density distribution and moments are derived depending on geometry and memory parameters the system may reveal both the subdiffusion and enhanced diffusion the similar analysis is performed for the levy flights where the finiteness of the variance follows from a variable noise intensity near a boundary two diffusion regimes are found in the bulk and near the surface the anomalous diffusion exponent as a function of the system parameters is derived

fractional fokker planck equation for levy flights in nonhomogeneous environments the fractional fokker planck equation which contains a variable diffusion coefficient is discussed and solved it corresponds to the l evy flights in a nonhomogeneous medium for the case with the linear drift the solution is stationary in the long time limit and it represents the l evy process with a simple scaling the solution for the drift term in the form ensuremath lambda text text sgn x possesses two different scales which correspond to the l evy indexes ensuremath mu and ensuremath mu 1 ensuremath mu l1 the former component of the solution prevails at large distances but it diminishes with time for a given x the fractional moments as a function of time are calculated they rise with time and the rate of this growth increases with ensuremath lambda

the rate function of hypoelliptic diffusions let be a hypoelliptic diffusion operator on a compact manifold m given an a priori smooth reference measure on m we can then rewrite l as the sum of a symmetric part l0 and a first order drift part y the paper investigates the effect of the drift y on the donsker varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion when y is in the linear span of the first and second order lie brackets of the xi s we derive an affine bound relating the rate functions associated with l and l0 as soon as one point exists where y is not in the linear span of the first and second order lie brackets of the xi s we show that such an affine bound is impossible 1994 john wiley sons inc

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

stationary states in two dimensional systems driven by bivariate levy noises systems driven by ensuremath alpha stable noises could be very different from their gaussian counterparts stationary states in single well potentials can be multimodal moreover a potential well needs to be steep enough in order to produce stationary states here it is demonstrated that two dimensional 2d systems driven by bivariate ensuremath alpha stable noises are even more surprising than their 1d analogs in 2d systems intriguing properties of stationary states originate not only due to heavy tails of noise pulses which are distributed according to ensuremath alpha stable densities but also because of properties of spectral measures consequently 2d systems are described by a whole family of langevin and fractional diffusion equations solutions of these equations bear some common properties but also can be very different it is demonstrated that also for 2d systems potential wells need to be steep enough in order to produce bounded states moreover stationary states can have local minima at the origin the shape of stationary states reflects symmetries of the underlying noise i e its spectral measure finally marginal densities in power law potentials also have power law asymptotics

iterative substructuring methods for spectral element discretizations of elliptic systems i compressible linear elasticity an iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed the pure displacement formulation for compressible materials is discretized with the spectral element method the resulting stiffness matrix is symmetric and positive definite the proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse global solver related to the wire basket of the elements as in the scalar case the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree

hybrid v cycle algebraic multilevel preconditioners we consider an algebraic derivation of multilevel preconditioners which are based on a sequence of finite element stiffness matrices they cor respond to a sequence of triangulations obtained by successive refinement and the associated finite element discretizations of second order selfadjoint elliptic boundary value problems the stiffness matrix at a given discretization level is partitioned into a natural hierarchical two level two by two block form then it is factored into block triangular factors the resulting schur complement is then replaced approximated by the stiffness matrix on the preceding coarser level this process is repeated successively for a fixed number ko 1 of steps after each cq steps the preconditioner so derived is corrected by a certain polynomial approximation a properly scaled and shifted chebyshev matrix polynomial which involves the preconditioner and the stiffness matrix at the considered level the hybrid f cycle preconditioner thus derived is shown to be of optimal order of complexity for 2 d and 3 d problem domains the rela tive condition number of the preconditioner is bounded uniformly with respect to the number of levels and with respect to possible jumps of the coefficients of the considered elliptic bilinear form as long as they occur only across edges faces in 3 d of elements from the coarsest triangulation in addition an adap tive implementation of our hybrid k cycle preconditioners is proposed and its practical behavior is demonstrated on a number of test problems

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

nonlinear stochastic equations with multiplicative levy noise the langevin equation with a multiplicative levy white noise is solved the noise amplitude and the drift coefficient have a power law form a validity of ordinary rules of the calculus for the stratonovich interpretation is discussed the solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the stratonovich interpretation the problem of escaping from a potential well is analyzed numerically predictions of different interpretations of the stochastic integral are compared

preconditioning and two level multigrid methods of arbitrary degree of approximation let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem we describe preconditioning methods for two level meshes which for most prob lems solved in practice behave as methods of optimal order in both storage and computa tional complexity namely per mesh point these numbers are bounded above by relatively small constants for all h ho where ho is small enough to cover all but excessively fine meshes we note that in practice multigrid methods are actually solved on a finite often even a fixed number of grid levels in which case also these methods are not asymptotically optimal as h 0 numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems variable coefficients general domains and boundary conditions for all but excessively fine meshes furthermore most of the latter methods have been implemented only for difference schemes of second order of accuracy whereas our methods are applicable to higher order approximations we claim that our scheme could be added fairly easily to many existing finite element codes 1 introduction consider the numerical solution of elliptic boundary value prob lems discretized by finite element methods we assume that the boundary is polygonal or consists of planes we note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first coarse triangulation anyhow if not so in most cases one makes only a few steps of mesh refinement hence the power of multigrid methods their optimal order of computational complexity is most often not achieved fully because optimality requires a large number of recursively defined meshes for details see e g 4 and for further references see 7 hence one might as well consider other methods perhaps simpler and more effective on a fixed mesh but which are not asymptotically optimal here we shall describe a method which uses only a fixed mesh but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for from a practical viewpoint excessively small meshes to be more precise the computational cost per mesh point is bounded by clog n for n no where n is the number of mesh points no is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods as is well known the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels we claim that the new method is more suitable for implementation in existing finite element packages in fact most packages for the multigrid methods are only for second order difference methods

stochastic optimization using the stochastic preconditioned conjugate gradient method the spcg solver has been integrated into nike3d and used to conduct a monte carlo simulation and a stochastic shape optimization of a cantilever beam