gravity interpretation using the mellin transform the mellin transform of the gravity effect of a buried sphere and two dimensional horizontal circular cylinder and the first horizontal derivative of the gravity effect of a two dimensional thin fault layer are derived the transformed functions are bounded by two asymptotes they are analyzed and procedures are formulated excluding the asymptotic regions for the extraction of the body parameters the application of the mellin transform is tested on simulated models as well as on two field examples 1 the humble dome gravity anomaly near houston usa and 2 the louga gravity anomaly usa

co volume methods for degenerate parabolic problems a complementary volume co volume technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems such as the stefan problem it is shown that these algorithms give rise to a discrete semigroup theory that parallels the continuous problem in particular the discrete stefan problem gives rise to nonlinear semigroups in both the discretel 1 andh 1 spaces

an implicit immersed boundary method for three dimensional fluid membrane interactions we present an implicit immersed boundary method for the incompressible navier stokes equations capable of handling three dimensional membrane fluid flow interactions the goal of our approach is to greatly improve the time step by using the jacobian free newton krylov method jfnk to advance the location of the elastic membrane implicitly the most attractive feature of this jacobian free approach is newton like nonlinear convergence without the cost of forming and storing the true jacobian the generalized minimal residual method gmres which is a widely used krylov subspace iterative method is used to update the search direction required for each newton iteration each gmres iteration only requires the action of the jacobian in the form of matrix vector products and therefore avoids the need of forming and storing the jacobian matrix explicitly once the location of the boundary is obtained the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model we then use the immersed boundary method to calculate the hydrodynamic effects and fluid structure interaction effects such as membrane deformation the present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force

some properties of the nonlinear semigroup for the problem ut δf u 0 le probleme est de trouver u t l 1 donnee par l equation du d lf u 0 u 0 u 0 r n est un domaine borne a frontiere lisse f r r est une fonction continue de lifschitz non decroissante avec f 0 0

analysis of depth resolution in potential field inversion we study the inversion of potential fields and evaluate the degree of depth resolution achievable for a given problem to this end we introduce a powerful new tool the depth resolution plot drp the drp allows a theoretical study of how much the depth resolution in a potential field inversion is influenced by the way the problem is discretized and regularized the drp also allows a careful study of the influence of various kinds of ambiguities such as those from data errors or of a purely algebraic nature the achievable depth resolution is related to the given discretization regularization and data noise level we compute drp by means of singular value decomposition svd or its generalization gsvd depending on the particular regularization method chosen to illustrate the use of the drp we assume a source volume of specified depth and horizontal extent in which the solution is piecewise constant within a 3d grid of blocks we consider various linear regularization terms in a tikhonov damped least squares formulation some based on using higher order derivatives in the objective function drps are illustrated for both synthetic and real data our analysis shows that if the algebraic ambiguity is not too large and a suitable smoothing norm is used some depth resolution can be obtained without resorting to any subjective choice of depth weighting

advantages of using the vertical gradient of gravity for 3 d interpretation gravity gradiometric data or gravity data transformed into vertical gradient can be efficiently processed in three dimensions for delineating density discontinuities model studies performed with the combined use of maxima of analytic signal and of horizontal gradient and the euler deconvolution techniques on the gravity field and its vertical gradient demonstrate the superiority of the latter in locating density contrasts particularly in the case of interfering anomalies where the use of gravity alone fails the gravity gradient is able to provide useful information with satisfactory accuracy

an accurate and efficient method for the incompressible navier stokes equations using the projection method as a preconditioner the projection method is a widely used fractional step algorithm for solving the incompressible navier stokes equations despite numerous improvements to the methodology however imposing physical boundary conditions with projection based fluid solvers remains difficult and obtaining high order accuracy may not be possible for some choices of boundary conditions in this work we present an unsplit linearly implicit discretization of the incompressible navier stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations since our scheme is not a fractional step algorithm it is straightforward to specify general physical boundary conditions accurately however this capability comes at the price of having to solve the time dependent incompressible stokes equations at each timestep to solve this linear system efficiently we employ a krylov subspace method preconditioned by the projection method in our implementation the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning the accuracy of the scheme is demonstrated for several problems including forced and unforced analytic test cases and lid driven cavity flows these tests consider a variety of physical boundary conditions with reynolds numbers ranging from 1 to 30000 the effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the schur complement for the time dependent incompressible stokes operator the projection method is found to be a more efficient preconditioner in most cases considered in the present work

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

magnetic interpretation in three dimensions using euler deconvolution magnetic survey data in grid form may be interpreted rapidly for source positions and depths by deconvolution using euler s homogeneity relation the method employs gradients either measured or calculated data need not be pole reduced so that remanence is not an interfering factor geologic constraints are imposed by use of a structural index model studies show that the method can locate or outline confined sources vertical pipes dikes and contacts with remarkable accuracy a field example using data from an intensively studied area of onshore britain shows that the method works well on real data from structurally complex areas and provides a series of depth labeled euler trends which mark magnetic edges notably faults with good precision

a comparison of implicit time integration methods for nonlinear relaxation and diffusion several time integration methods for nonlinear systems are compared all of the time discretizations are based on the method but differ in their treatment of the implicit nonlinear terms one method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent nc newton s method or its approximation newton krylov is used to converge the nonlinearities the other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem the linearized one step method that uses the full jacobian is shown to have similar accuracy as nc methods the lagged linearization method and an extension that is second order accurate are also studied a trunction error analysis complements the numerical results for the relaxation problem it is shown that each of the second order accurate linearized methods may be more accurate than an nc method depending on the degree of nonlinearity in the problem for the radiation diffusion problem in general the nc method is most accurate and allows a larger time step however the linearized methods perfom surprisingly well

iterative linear solvers in a 2d radiation hydrodynamics code computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion icf phenomena a crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations we present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation hydrodynamics problems the linear solvers used are diagonally scaled conjugate gradient gmres with incomplete lu preconditioning conjugate gradient with incomplete cholesky preconditioning multigrid and multigrid preconditioned conjugate gradient these problems involve shock propagation opacities varying over 5 6 orders of magnitude tabular equations of state and dynamic ale arbitrary lagrangian eulerian meshes we perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1000 to 100 000 zones the fundamental question we address in this paper is is it more efficient to invert the matrix in many inexpensive steps like diagonally scaled conjugate gradient or in fewer expensive steps like multigrid in addition what is the answer to this question as a function of problem size and is the answer problem dependent we find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size increasing in both iteration count and overall cpu time with the size of the problem and also increasing for larger time steps for all problems considered the multigrid algorithms scale almost perfectly i e the iteration count is approximately independent of problem size and problem time step for pure radiation flow problems i e no hydrodynamics we see speedups in cpu time of factors of 15 30 for the largest problems when comparing the multigrid solvers relative to diagonal scaled conjugate gradient for the incomplete factorization preconditioners we see a weak dependence of iteration count on problem size the speedups observed for pure radiation flow are typically on the order of 10 relative to diagonal scaled conjugate gradient for radiation hydrodynamics problems we again see multigrid scaling perfectly however for the problems considered we see speedups relative to diagonal scaled conjugate gradient of no more than 10 with incomplete cholesky in fact either equaling or outperforming multigrid we trace these observations to the time step control and the feature of ale to relax distorted zones

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

a fully second order implicit explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems we present a fully second order implicit explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems the idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations in this scope the hydrodynamics equations are discretized explicitly making use of the capability of well understood explicit schemes on the other hand the nonlinear heat conduction is solved implicitly such methods are often referred to as imex methods 2 1 3 the jacobian free newton krylov jfnk method e g 10 9 is applied to the problem in such a way as to render a nonlinearly iterated imex method we solve three test problems in order to validate the numerical order of the scheme for each test we established second order time convergence we support these numerical results with a modified equation analysis mea 21 20 the set of equations studied here constitute a base model for radiation hydrodynamics

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

linearized least squares method for interpretation of potential field data from sources of simple geometry we present a new method for interpreting isolated potential field gravity and magnetic anomaly data a linear equation involving a symmetric anomalous field and its horizontal gradient is derived to provide both the depth and nature of the buried sources in many currently available methods either higher order derivatives or postprocessing is necessary to extract both pieces of information therefore data must be of very high quality in contrast for gravity work with our method only a first order horizontal derivative is needed and the traditional data quality is sufficient our proposed method is similar to the euler technique it uses a shape factor instead of a structural index to characterize the buried sources the method is tested using theoretical anomaly data with and without random noise in all cases the method adequately estimates the location and the approximate shape of the source the practical utility of the method is demonstrated using gravity and magnetic field examples from the united states and zimbabwe

on a least squares approach to depth determination from gravity data discussion and reply in his paper gupta was able to transform the problem of depth estimation of buried structures into a problem of finding a solution of a nonlinear equation in the form of f z 0 gupta also indicated that such a numerical approach is found to be capable of determining optimum depths particularly from residual anomaly profiles even if small segments of the gravity profiles are observed no doubt this numerical approach has its point of view both in theory and practice over any other depth estimation techniques such as those defined by the half g max rule nettleton 1940 1942 telford et al 1976 however gupta s technique would be much more effective if applied not to residuals but to derivative anomalies particularly when the regional field has few extrema in it this is obviously due to the following

calabi yau manifolds motivations and constructions the possible ways of compacitification of thee8 e8 superstring theory to four dimensions are reviewed the phenomenological need forn 1 supersymmetry is argued on quite general grounds to favour the choice of a calabi yau manifold for the compact internal manifold the massless spectrum after compactification is derived in full detail revealing beside the usual particles others that may have great phenomenological impact the technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given a big family of such constructions giving rise to many new calabi yau manifolds is presented and its relevance to the search of a phenomenologically acceptable solution is discussed

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

a least squares minimization approach to depth determination from numerical horizontal gravity gradients the sphere and the horizontal cylinder models can be very useful in quantitative interpretation of gravity data measured in a small area over buried structures several graphical and numerical methods have been developed by many workers for interpreting the residual gravity anomalies caused by these models to find the depth of most geologic structures excellent reviews are given in saxov and nygaard 1953 and bowin et al 1986 the numerical approaches odegard and berg 1965 gupta 1983 sharma and geldart 1968 lines and treitel 1984 and shaw and agarwal 1990 may have advantages in theory and practice over graphical depth estimation techniques pick et al 1973 nettleton 1976 telford et al 1976 however effective quantitative interpretation procedures using the least squares method based on the analytical expression of simple numerical horizontal gravity gradient anomalies are yet to be developed

adaptive regularization linearization and discretization and a posteriori error control for the two phase stefan problem we consider in this paper the time dependent two phase stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward euler temporal discretizations regularization of the enthalpy temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual as well as on the l2 l2 error of the temperature and the l2 h1 error of the enthalpy moreover they allow to distinguish the space time regularization and linearization error components an adaptive algorithm is proposed which ensures computational savings through the online choice of a sufficient regularization parameter a stopping criterion for the linearization iterations local space mesh refinement time step adjustment and equilibration of the spatial and temporal errors we also prove the efficiency of our estimate our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization as an example we apply it to the vertex centered finite volume finite element with mass lumping and quadrature and newton methods numerical results illustrate the effiectiveness of our estimates and the performance of the adaptive algorithm

development of a 2 d algorithm to simulate convection and phase transition efficiently we develop a jacobian free newton krylov jfnk method for the solution of a two dimensional convection phase change model using the incompressible navier stokes equation set and enthalpy as the energy conservation variable the simple algorithm acts as a physics based preconditioner to jfnk this combined algorithm is compared to solutions using simple as the main solver algorithm performance is assessed for two benchmark problems of phase change convection of a pure material one melting and one freezing the jfnk simple method is shown to be more efficient per time step and more robust at larger time steps overall cpu savings of more than an order of magnitude are realized

on the least squares residual anomaly determination this paper discusses an approach to determine the least squares optimum order of the regional surface which when subtracted from the bouguer gravity anomaly data minimizes distortion of the residual component of the field the least squares method was applied to theoretical composite gravity fields each consisting of a constant residual component sphere or vertical cylinder and a regional component of different order using successively increasing orders of polynomial regionals for residual determination the overall similarity between each two successive residual maps was determined by computing the correlation factor between the mapped variables similarity between residual maps of the lowest orders verified by good correlation may generally be considered a criterion for determining the optimum order of the regional surface and consequently the least distorted residual component the residual map of the lower order in this well correlated doublet is considered the most plausible one and may be used

on the uniqueness of the inverse potential problem for homogeneous polyhedrons the uniqueness of a homogeneous polyhedron as determined from its exterior newtonian potential has been studied as a result of the special apparatus developed in this paper the two dimensional research methods could be applied to study three dimensional cases

a least squares minimization approach to shape determination from gravity data the gravity anomaly expression produced by most geologic structures can be represented by a continuous function of both shape shape factor and depth related variables with an amplitude coefficient related to mass abdelrahman and el araby 1993 few methods have been developed to determine the shape of the buried geologic structure from residual gravity anomaly profiles these methods include a walsh transform approach shaw and agarwal 1990 and the employment of a correlation factor between successive least squares residuals abdelrahman and el araby 1993 in the present note a least squares minimization approach to shape factor determination from a residual gravity anomaly profile is presented the problem of the shape factor determination is transformed into the problem of finding a solution of a nonlinear equation of the form f q 0

on balanced approximations for time integration of multiple time scale systems the effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis mea first mea is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation ode and in a linear partial differential equation pde several time discretizations of the ode and pde are considered and the resulting truncation terms are compared analytically and numerically it is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing fast time scales is employed many of the issues uncovered on the simple problems are shown to persist in more realistic applications specifically several differencing schemes using linearization and or time splitting are applied to problems in nonequilibrium radiation diffusion magnetohydrodynamics and shallow water flow and their solutions are compared to those using balanced time integration methods

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

weighted inner products for gmres and gmres dr the convergence of the restarted gmres method can be significantly improved for some problems by using a weighted inner product that changes at each restart how does this weighting affect convergence and when is it useful we show that weighted inner products can help in two distinct ways when the coefficient matrix has localized eigenvectors weighting can allow restarted gmres to focus on eigenvalues that otherwise cause slow convergence for general problems weighting can break the cyclic convergence pattern into which restarted gmres often settles the eigenvectors of matrices derived from differential equations are often not localized thus limiting the impact of weighting for such problems incorporating the discrete cosine transform into the inner product can significantly improve gmres convergence giving a method we call w gmres dct integrating weighting with eigenvalue deflation via gmres dr can also give effective solutions

a least squares minimization approach to depth determination from moving average residual gravity anomalies we have developed a least squares minimization method to estimate the depth of a buried structure from moving average residual gravity anomalies the method involves fitting simple models convolved with the same moving average filter as applied to the observed gravity data as a result our method can be applied not only to residuals but also to the bouguer gravity data of a short profile length the method is applied to synthetic data with and without random errors the validity of the method is tested in detail on two field examples from the united states and senegal

a new method for shape and depth determinations from gravity data we have developed a simple method to determine simultaneously the shape and depth of a buried structure from residualized gravity data using filters of successive window lengths the method is similar to euler deconvolution but it solves for shape and depth independently the method involves using a relationship between the shape factor and the depth to the source and a combination of windowed observations the relationship represents a parametric family of curves window curves for a fixed window length the depth is determined for each shape factor the computed depths are plotted against the shape factors representing a continuous monotonically increasing curve the solution for the shape and depth of the buried structure is read at the common intersection of the window curves this method can be applied to residuals as well as to the bouguer gravity data of a short or long profile length the method is applied to theoretical data with and without random errors and is tested on a known field example from the united states in all cases the shape and depth solutions obtained are in good agreement with the actual ones

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

interpreting potential field anomaly of an isolated source of regular geometry revisited abstracti have developed an improved practical method for interpreting a symmetrical shaped potential field anomaly due to an isolated source body of regular geometric configuration the method uses the first order horizontal derivative of the logarithmically transformed absolute value of the anomaly in estimating the source body parameters such as the location depth of burial and shape factor to tackle noise in data a regularization technique is designed which ensures a robust estimate of the first order derivative of logarithmically transformed data the regularization technique uses an optimal value of regularization parameter that although noise dependent requires no a priori knowledge of the noise level in the data a graphical method is designed to determine an optimal value of the regularization parameter from the position of the local minimum of a specially defined functional with respect to the regularization parameters numerical tests have been conducted on the noise contaminated synthe

gravity inversion using open reject and shape of anomaly fill criteria a gravity inversion method is developed by iteratively applying open reject and fill o r f criteria within a model space comprising a great many rectangular prisms each prism is assigned a density contrast the modeling procedure consists of filling some prisms while leaving others empty only one element is filled for each pass generally elements are added only to the periphery of the growing model models can be allowed to grow in any combination of directions or in all directions by application of a shape of anomaly fill criterion the model rapidly attains a form which yields gravity approximating the given gravity scaled down by some constant factor as the model continues to grow this scale factor approaches unity the method readily yields inverse models comprising several thousand individual prisms examples presented here give applications to 2 d problems the method is readily applicable to 212 d and 3 d problems as well overhanging elements are obtained by appropriate use of model

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

evolution of phase transitions in methane hydrate abstract we consider a simplified model of methane hydrates which we cast as a nonlinear evolution problem for its well posedness we extend the existing theory to cover the case in which the problem involves a measurable family of graphs we represent the nonlinearity as a subgradient and prove a useful comparison principle thus optimal regularity results follow for the numerical solution we apply a fully implicit scheme without regularization and use the semismooth newton algorithm for a solver and the graph is realized as a complementarity constraint cc the algorithm is very robust and we extend it to define an easy and superlinearly convergent fully implicit scheme for the stefan problem and other multivalued examples

the tangent bundle of a calabi yau manifold deformations and restriction to rational curves the tangent bundle x of a calabi yau threefoldx is the only known example of a stable bundle with non trivial restriction to any rational curve onx by deforming the direct sum of x and the trivial line bundle one can try to obtain new examples we use algebro geometric techniques to derive results in this direction the relation to the finiteness of rational curves on calabi yau threefolds is discussed

polynomial deformations and cohomology of calabi yau manifolds we study the method of polynomial deformations that is used in the physics literature to determine the hodge numbers of calabi yau manifolds as well as the related yukawa couplings we show that the argument generally presented in the literature in support of these computations is seriously misleading give a correct proof which applies to all the cases we found in the literature and present examples which show that the method is not universally valid we present a general analysis which applies to all calabi yau manifolds embedded as complete intersections in products of complex projective spaces yields sufficient conditions for the validity of the polynomial deformation method and provides an alternative computation of all the hodge numbers in many cases in which the polynomial method fails

new physics based preconditioning of implicit methods for non equilibrium radiation diffusion this note presents an extension of previous work on physics based preconditioning of the non equilibrium radiation diffusion equations the new physics based preconditioner presented in this manuscript is a minor modification to the operator split preconditioner presented previously results show that the new preconditioner is more effective on test problems that are more nonlinear

the application of walsh transforms to interpret gravity anomalies due to some simple geometrically shaped causative sources a feasibility study walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform in contrast to sinusoidal waveforms whose amplitudes may assume any value between 1 to 1 walsh functions assume only discrete amplitudes of or 1 which form the kernel function of the walsh transform because of this special nature of the kernel computation of the walsh transform of a given signal is simpler and faster than that of the fourier transform the properties of the fourier transform in linear time are similar to those of the walsh transform in dyadic time the fourier transform has been widely used in interpretation of geophysical problems considering various aspects of the walsh transform an attempt has been made to apply it to some gravity data a procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources viz a sphere a horizontal cylinder and a 2 d vertical prism of large depth extent the technique has been applied to data from the published literature to evaluate its applicability and the results are in good agreement with the more conventional ones

practical applications of uniqueness theorems in gravimetry part i constructing sound interpretation methods to obtain a unique and stable solution to the gravity inverse problem a priori information reflecting geological attributes of the gravity source must be used mathematical conditions to obtain stable solutions are established in tikhonov s regularization method where the a priori information is introduced via a stabilizing functional which may be suitably designed to incorporate some relevant geological information however there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem rather there are many theorems usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution in part i of these companion papers we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inv

time step size selection for radiation diffusion calculations the purpose of this note is to describe a time step control technique as applied to radiation diffusion standard practice only provides a heuristic criteria related to the relative change in the dependent variables the authors propose an alternative based on relatively simple physical principles this time step control applies to methods of solution that are unconditionally stable and converges nonlinearities within a time step in the governing equations commonly nonlinearities in the governing equations are evaluated using existing old time data the authors refer to this as the semi implicit si method when a method converges nonlinearities within a time step the entire governing equation including all nonlinearities is self consistently evaluated using advance time data with appropriate time centering for accuracy

an efficient physics based preconditioner for the fully implicit solution of small scale thermally driven atmospheric flows in atmospheric flow situations typical of a small scale atmospheric thermal a separation of time scales exists between the fast sound wave time scale and the advective time scale atmospheric models have been designed to take advantage of this disparity of time scales with numerical approaches such as the semi implicit or split explicit approach being used to efficiently step over the fast sound waves some of these numerical approaches are first order in time to improve accuracy over these methods a fully implicit and nonlinearly consistent inc flow solver has been developed for the navier stokes equation set in our inc method the equation set is solved by use of the jacobian free newton krylov jfnk method an efficient preconditioner has been developed which uses the semi implicit method to solve the governing equations being that this preconditioner was designed to attack the fastest waves in the system and not other features in the implicit system such as advection or turbulent diffusion the preconditioning technique is labeled as a physics based preconditioner a variety of linear solvers including ssor krylov methods and or multigrid approaches are used to approximately invert the pressure matrix in the semi implicit algorithm a suite of simulations will be conducted utilizing different linear solvers for the simple problem of the bouyant rise of a warm bubble the problem will also document the ability of the inc approach to achieve second order in time accuracy

a monolithic fem multigrid solver for non isothermal incompressible flow on general meshes we present special numerical simulation methods for non isothermal incompressible viscous fluids which are based on lbb stable fem discretization techniques together with monolithic multigrid solvers for time discretization we apply the fully implicit crank nicolson scheme of 2nd order accuracy while we utilize the high order q 2 p 1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes to treat the nonlinearities in each time step as well as for direct steady approaches the resulting discrete systems are solved via a newton method based on divided differences to calculate explicitly the jacobian matrices in each nonlinear step the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure schur complement smoothers of vanka type for validation and evaluation of the presented methodology we perform the mit benchmark 2001 m a christon p m gresho s b sutton computational predictability of natural convection flows in enclosures in first mit conference on computational fluid and solid mechanics vol 40 elsevier 2001 pp 1465 1468 of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency additionally we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation moreover we discuss how these fem multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

the relationship between local wavenumber and analytic signal in magnetic interpretation recently a number of automated methods for source location from 2d profile magnetic data have been developed based on either the local wavenumber thurston and smith 1997 smith et al 1998 thurston et al 2002 salem et al 2005 smith and salem 2005 or the amplitude of the analytic signal hsu et al 1996 bastani and pedersen 2001 salem and ravat 2003 salem et al 2004 salem 2005 the main advantage of using derived quantities such as the local wavenumber lw and amplitude of the analytic signal as is that they are generally independent of source magnetization and dip effects therefore allowing positional parameters such as depth and horizontal location to be determined more directly than from the magnetic field

gravity interpretation using correlation factors between successive least squares residual anomalies the correlation factors between successive least squares residual or regional gravity anomalies from a buried sphere a two dimensional 2 d horizontal cylinder and a vertical cylinder and the first horizontal derivative of the gravity from a 2 d thin faulted layer are computed correlation values are used to determine the depth to the center of the buried structure and the radius of the sphere or the cylinder and the thickness of the fault are estimated the method can be applied not only to residuals but also to the bouguer anomaly profile consisting of the combined effect of a residual component due to a purely local structure and a regional component represented by a polynomial of any order the method is easy to apply and may be automated if desired it can also be applied to the derivative anomalies of the gravity field the validity of the method is tested on two field examples from the united states and denmark

practical applications of uniqueness theorems in gravimetry part ii pragmatic incorporation of concrete geologic information we illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute we also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth independent density distributions the two part guideline is 1 to introduce a priori information favoring uniqueness either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous simply connected polygons or polyhedra with known density displaying no fancy shapes and no curling apophyses at their borders and 2 to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data following the guideline we apply different methods to gravity data using interp

magnetic source parameters of two dimensional structures using extended euler deconvolution the euler homogeneity relation expresses how a homogeneous function transforms under scaling when implemented it helps to determine source location for particular potential field anomalies in this paper we introduce an additional relation that expresses the transformation of homogeneous functions under rotation the combined implementation of the two equations called here extended euler deconvolution for 2 d structures gives a more complete source parameter estimation that allows the determination of susceptibility contrast and dip in the cases of contact and thin sheet sources this allows for the structural index to be correctly chosen on the basis of a priori knowledge about susceptibility and dip the pattern of spray solutions emanating from a single source anomaly can be attributed to interfering sources which have their greatest effect on the flanks of the anomaly these sprays follow different paths when using either conventional euler deconvolution or extended euler deconvolution the paths of these spray solutions cross and cluster close to the true source location this intersection of spray paths is used as a discriminant between poor and well constrained solutions allowing poor solutions to be eliminated extended euler deconvolution has been tested successfully on 2 d model and real magnetic profile data over contacts and thin dikes

three field block preconditioners for models of coupled magma mantle dynamics for a prescribed porosity the coupled magma mantle flow equations can be formulated as a two field system of equations with velocity and pressure as unknowns previous work has shown that while optimal preconditioners for the two field formulation can be obtained the construction of preconditioners that are uniform with respect to model parameters is difficult this limits the applicability of two field preconditioners in certain regimes of practical interest we address this issue by reformulating the governing equations as a three field problem which removes a term that was problematic in the two field formulation in favor of an additional equation for a pressure like field for the three field problem we develop and analyze new preconditioners and we show numerically that they are optimal in terms of problem size and less sensitive to model parameters compared to the two field preconditioner this extends the applicability of optimal preconditioners for coupled mantle magma dynamics into parameter

splitting based block preconditioning methods for block two by two matrices of real square blocks recently bai proposed rotated block preconditioners for block two by two matrices of real square blocks these rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two by two triangular matrix theoretical and numerical results have shown the superiority of these rotated block preconditioners see bai 2013 10 in this paper inspired by the efficiency and the special structure of the rotated block preconditioners we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises we construct block jacobi and block gauss seidel splitting iteration methods based on the coefficient matrix of the new linear system the convergence of these splitting iterations is also demonstrated then by utilizing the proposed block jacobi and block gauss seidel splittings we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two by two diagonal or a block two by two triangular matrix spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate krylov subspace iteration methods such as gmres for solving the block two by two liner systems

three least squares minimization approaches to depth shape and amplitude coefficient determination from gravity data three different least squares approaches are developed to determine successively the depth shape shape factor and amplitude coefficient related to the radius and density contrast of a buried structure from the residual gravity anomaly by defining the anomaly value g max at the origin on the profile the problem of depth determination is transformed into the problem of solving a nonlinear equation f z 0 formulas are derived for spheres and cylinders knowing the depth and applying the least squares method the shape factor and the amplitude coefficient are determined using two simple linear equations in this way the depth shape and amplitude coefficient are determined individually from all observed gravity data a procedure is developed for automated interpretation of gravity anomalies attributable to simple geometrical causative sources the method is applied to synthetic data with and without random errors in all the cases examined the maximum error in depth shape and amplitude coeffic

self adaptive time integration of flux conservative equations with sources we present a novel flux conserving asynchronous method for the explicit time integration of multi scale flux conservative partial differential equations with source terms unlike the conventional explicit and implicit integration schemes it is based on a discrete event simulation paradigm which describes time advance in terms of increments to physical quantities and causality rules rather than time stepping this method exerts self adaptive control over local update rates of solution by predicting and correcting changes to simulation variables in accordance with local physical scales the discrete event simulation paradigm is independent of the underlying spatial mesh and thus can be incorporated into block structured and unstructured mesh refinement techniques the effectiveness and robustness of the new method is demonstrated on a number of one dimensional uniform mesh models based on diffusion convection reaction equations the event driven integration reduces numerical approximation errors due to large local time derivatives prevents explosive numerical instabilities in locally super courant calculations and automatically reduces the cpu overhead associated with stiff terms and inactive regions in computation space

endomorphism valued cohomology and gauge neutral matter for several physically interesting calabi yau manifolds we count and parametrize gauge neutral matter particles occurring in corresponding superstring compactifications to this end we use the technique of exact and spectral sequences and then describe and discuss our results in the more familiar tensor notation

shape and depth solutions from gravity data using correlation factors between successive least squares residuals the gravity anomaly expression produced by most geologic structures can be represented by a continuous function in both shape shape factor and depth variables with an amplitude coefficient related to the mass correlation factors between successive least squares residual gravity anomalies from a buried vertical cylinder horizontal cylinder and sphere are used to determine the shape and depth of the buried geologic structure for each shape factor value the depth is determined automatically from the correlation value the computed depths are plotted against the shape factor representing a continuous correlation curve the solution for the shape and depth of the buried structure is read at the common intersection of correlation curves this method can be applied to a bouguer anomaly profile consisting of a residual component caused by local structure and a regional component this is a powerful technique for automatically separating the bouguer data into residual and regional polynomial components this method is tested on theoretical examples and a field example in both cases the results obtained are in good agreement with drilling results

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

on preconditioning newton krylov methods in solidifying flow applications solidifying flow equations can be used to model industrial metallurgical processes such as casting and welding and material science applications such as crystal growth these flow equations contain locally stiff nonlinearities at the moving phase change interface we are developing a three dimensional parallel simulation tool for such problems using a jacobian free newton krylov solver and unstructured finite volume methods a segregated distributed block triangular preconditioning strategy is being developed for the newton krylov solver in this preconditioning approach we are only required to approximately invert matrices coming from a single field variable not matrices arising from a coupled system additionally simple linearizations are used in constructing our preconditioning operators the preconditioning strategy is presented along with the performance of the methods we consider problems in phase change heat transfer and the thermally driven incompressible navier stokes equations separately this is a required intermediate step toward developing a successful preconditioning strategy for the fully coupled physics problem

calculating endomorphism valued cohomology singlet spectrum in superstring models we present a computational strategy based on exact sequences for determining the dimension of endomorphism valued cohomology groups for complete intersections in complex projective space this cohomology group gives rise to part of the gauge singlet spectrum in superstring compactifications we establish the underlying justification for the known deformation theoretical algorithm and by comparison with the exact sequence method indicate its limitations

the preconditioned jacobian free newton krylov methods for nonequilibrium radiation diffusion equations in general it is difficult to use the newton krylov methods to solve the large scale multi variable nonequilibrium reaction diffusion systems in this paper by employing two new semi implicit discretization schemes to construct the preconditioners the preconditioned newton krylov methods are presented to solve the multidimensional problems these methods cannot only improve the number of iterations but also speed up the convergence of solutions numerical results are given to illustrate the effectiveness

newton s method for the navier stokes equations with finite element initial guess of stokes equations abstract it is shown that finite element solutions of stokes equations may be chosen as the initial guess for the quadratic convergence of newton s algorithm applied to navier stokes equations provided there are sufficiently small mesh size h and the moderate reynold s number we provide also a mixed convergence analysis in terms of iterations and finite error estimates of the initial guess with a regularity estimate and error analysis for each newton s step

a least squares approach to depth determination from gravity data the present paper deals with a numerical approach to determine the depth of a buried structure from the residual anomaly the problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of the form f z 0 formulas have been derived for a sphere vertical and horizontal cylinders and for a vertical fault thin plate approximation the procedure is applied to synthetic data with and without random errors finally a field example is presented in which the depth to a fault is estimated at 3 8 km and verified from drilling results

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa

a multigrid preconditioner and automatic differentiation for non equilibrium radiation diffusion problems we study the efficient solution of non equilibrium radiation diffusion problems an implicit time discretization leads to the solution of systems of non linear equations which couple radiation energy and material temperature we consider the implicit euler method the mid point scheme the two step backward differentiation formula and a two stage implicit runge kutta method for time discretization we employ a newton krylov method in the solution of arising non linear problems we describe the computation of the jacobian matrix for newton s method using automatic differentiation based on the operator overloading in fortran 90 for gmres iterations we propose a simple multigrid preconditioner applied directly to the coupled linearized problems we demonstrate the efficiency and scalability of the proposed solution procedure by solving one dimensional and two dimensional model problems