symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

an o n log log n time algorithm for triangulating a simple polygon given a simple n vertex polygon the triangulation problem is to partition the interior of the polygon into n 2 triangles by adding n 3 nonintersecting diagonals we propose an o n log log n time algorithm for this problem improving on the previously best bound of o n log n and showing that triangulation is not as hard as sorting improved algorithms for several other computational geometry problems including testing whether a polygon is simple follow from our result

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

solution of high order compact discretized 3d elliptic partial differential equations by an accelerated multigrid method abstract a robust and efficient numerical solver for three dimensional linear elliptic partial differential equations without cross derivative terms but variable coefficients is presented the proposed algorithm is based on a general meshsize fourth order compact finite difference scheme and an accelerated geometric multigrid solver the former not only increases the accuracy of the discretization but also does not affect the bandwidth of the common second order finite difference scheme which is relevant for distributed memory parallel implementations of the multigrid preconditioner the acceleration technique on the other hand allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence furthermore it is found that depending on the coefficients of the elliptic partial differential equation the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence

high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

two level compact implicit schemes for three dimensional parabolic problems we derive a class of two level high order implicit finite difference schemes for solving three dimensional parabolic problems with mixed derivatives the schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of a weighted average parameter m numerical results with m 0 5 are presented to confirm the high accuracy of the derived scheme and to compare it with the standard second order central difference scheme it is shown that the improvement in accuracy does not come at a higher cost of computation and storage since it is possible to choose the grid parameters so that the present scheme requires less work and memory and gives more accuracy than the standard central difference scheme

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

quantitative evaluation of three cortical surface flattening methods during the past decade several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc we report the results of a quantitative comparison of freesurfer circlepack and lscm with respect to measurements of geometric distortion and computational speed our results indicate that freesurfer performs best with respect to a global measurement of metric distortion whereas lscm performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion freesurfer provides more homogeneous distribution of metric distortion across the whole cortex than circlepack and lscm lscm is the most computationally efficient algorithm for generating spherical maps while circlepack is extremely fast for generating planar maps from patches

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

convergence of a compact scheme for the pure streamfunction formulation of the unsteady navier stokes system this paper is devoted to the analysis of a new compact scheme for the navier stokes equations in pure streamfunction formulation numerical results using that scheme have been reported in m ben artzi e j comput phys 205 2005 pp 640 664 the scheme discussed here combines the stephenson scheme for the biharmonic operator and ideas from box scheme methodology consistency and convergence are proved for the full nonlinear system instead of customary periodic conditions the case of boundary conditions is addressed it is shown that in one dimension the truncation error for the biharmonic operator is o h 4 at interior points and o h at near boundary points in two dimensions the truncation error is o h 2 at interior points due to the cross terms and o h at near boundary points hence the scheme is globally of order four in the one dimensional periodic case and of order two in the two dimensional periodic case but of order 3 2 for one and two dimensional nonperiodic boundary conditions we emphasize in particular that there is no special treatment of the boundary thus allowing robust use of the scheme the finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications

multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation a fourth order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three dimensional 3d poisson equation on a cubic domain two multgrid methods are developed to solve the resulting sparse linear systems one is to use the full coarsening multigrid method with plane gauss seidel relaxation which uses line gauss seidel relaxation to compute each planewise solution the other is to construct a partial semi coarsening multigrid method with the traditional point or plane gauss seidel relaxations numerical experiments are conducted to test the computed accuracy of the fourth order compact difference scheme and the computational efficiency of the multigrid methods with the fourth order compact difference scheme

a study of a numerical solution of the steady two dimensions navier stokes equations in a constricted channel problem by a compact fourth order method we present a numerical solution for the steady 2d navier stokes equations using a fourth order compact type method the geometry of the problem is a constricted symmetric channel where the boundary can be varied via a parameter from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular the set of non linear equations is solved by the newton method results have been obtained for reynolds number up to 500 estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method

higher order numerical discretizations for exterior and biharmonic type pdes a higher order numerical discretization technique based on minimum sobolev norm msn interpolation was introduced in our previous work in this article the discretization technique is presented as a tool to solve two hard classes of pdes namely the exterior laplace problem and the biharmonic problem the exterior laplace problem is compactified and the resultant near singular pde is solved using this technique this finite difference type method is then used to discretize and solve biharmonic type pdes a simple book keeping trick of using ghost points is used to obtain a perfectly constrained discrete system numerical results such as discretization error condition number estimate and solution error are presented for both classes of pdes variable coefficient examples on complicated geometries and irregular grids are considered the method is seen to have high order of convergence in all these cases through numerical evidence perhaps for the first time such a systematic higher order procedure for irregular grids and variable coefficient cases is now available though not discussed in the paper the idea seems to be easily generalizable to finite element type techniques as well

evolution equations for phase separation and ordering in binary alloys we explore two phenomenological approaches leading to systems of coupled cahn hilliard and cahn allen equations for describing the dynamics of systems which can undergo first order phase separation and order disorder transitions simultaneously starting from the same discrete lattice free energy function in the first approach a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow in the second approach a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken we demonstrate in the context of bcc fe al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases only then will the two approaches lead to roughly the same continuum descriptions we conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal ton 1 n 2 1 wheren 1 is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andn 2 is the number of chemical components n 1 of these variables are nonconserved and the remaining are conserved and represent the average concentrations for the fe al alloys this implies a description of one conserved order parameter and one nonconserved order parameter the resultant description is given by a cahn hilliard equation coupled to a cahn allen equation via the lower order nonlinear terms the rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations a similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated

a new paradigm for solving navier stokes equations streamfunction velocity formulation in this paper we propose a new paradigm for solving navier stokes equations the proposed methodology is based on a streamfunction velocity formulation of the two dimensional steady state navier stokes equations representing incompressible fluid flows in two dimensional domains similar formulations are also possible for three dimensional fluid flows the main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values especially on solid boundaries encountered when solving the streamfunction vorticity formulations our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity pressure formulations of the navier stokes equations we describe the new formulation of the navier stokes equations and use this formulation to solve a couple of fluid flow problems we use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem it is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem in addition we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature in all cases we obtain high accuracy solutions with little additional cost

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

the surface induced spatial temporal structures in confined binary alloys abstract this paper examines surface induced ordering in confined binary alloys the hyperbolic initial boundary value problem ibvp is used to describe a scenario of spatiotemporal ordering in a disordered phase for concentration of one component of binary alloy and order parameter with non linear dynamic boundary conditions this hyperbolic model consists of two coupled second order differential equations for order parameter and concentration it also takes into account effects of the memory on the ordering of atoms and their densities in the alloy the boundary conditions characterize surface velocities of order parameter and concentration changing which is due to surface super cooling on walls confining the binary alloy it is shown that for large times there are three classes of dynamic non linear boundary conditions which lead to three different types of attractor s elements for the ibvp namely the elements of attractor are the limit periodic simple shock waves with fronts of discontinuities if is finite then the attractor contains spatiotemporal functions of relaxation type if is infinite and countable then we observe the functions of pre turbulent type if is infinite and uncountable then we obtain the functions of turbulent type

implicit brain imaging we describe how implicit surface representations can be used to solve fundamental problems in brain imaging this kind of representation is not only natural following the state of the art segmentation algorithms reported in the literature to extract the different brain tissues but it is also as shown in this paper the most appropriate one from the computational point of view examples are provided for finding constrained special curves on the cortex such as sulcal beds regularizing surface based measures such as cortical thickness and for computing warping fields between surfaces such as the brain cortex all these result from efficiently solving partial differential equations pdes and variational problems on surfaces represented in implicit form the implicit framework avoids the need to construct intermediate mappings between 3 d anatomical surfaces and parametric objects such planes or spheres a complex step that introduces errors and is required by many other cortical processing approaches

compact difference scheme for parabolic and schrodinger type equations with variable coefficients abstract we develop a new compact scheme for the second order pde parabolic and schrodinger type with a variable time independent coefficient it has a higher order and smaller error than classic implicit scheme the dirichlet and neumann boundary problems are considered the relative finite difference operator is almost self adjoint

a fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

a fast solver for the orthogonal spline collocation solution of the biharmonic dirichlet problem on rectangles a fast schur complement algorithm is presented for computing the piecewise hermite bicubic orthogonal spline collocation solution of the biharmonic dirichlet problem on a rectangular region on an n n uniform partition the algorithm which involves the preconditioned conjugate gradient method and fast fourier transforms requires o n2 log2 n arithmetic operations

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

a two step high order compact scheme for the laplacian operator and its implementation in an explicit method for integrating the nonlinear schrodinger equation we describe and test an easy to implement two step high order compact 2shoc scheme for the laplacian operator and its implementation into an explicit finite difference scheme for simulating the nonlinear schrodinger equation nlse our method relies on a compact double differencing which is shown to be computationally equivalent to standard fourth order non compact schemes through numerical simulations of the nlse using fourth order runge kutta we confirm that our scheme shows the desired fourth order accuracy a computation and storage requirement comparison is made between the 2shoc scheme and the non compact equivalent scheme for both the laplacian operator alone as well as when implemented in the nlse simulations stability bounds are also shown in order to get maximum efficiency out of the method we conclude that the modest increase in storage and computation of the 2shoc schemes is well worth the advantages of having the schemes compact and their ease of implementation makes their use very useful for practical implementations

observer independent method for microstructural parcellation of cerebral cortex a quantitative approach to cytoarchitectonics we describe a new observer independent procedure for identifying boundaries between cortical areas the method is useful for images obtained from sections which provide microstructural information on the cortical laminar pattern e g nissl myelin or immunohistochemically stained sections or receptor autoradiographs the laminar pattern is represented by profile curves extending from the cortical surface to the white matter boundary these profiles are constructed from digitized images digitization is based on the grey level index nissl or densitometry myelin immunohistochemistry receptor autoradiography the shapes of neighboring profiles are compared by calculating their distances according to feature vectors extracted from the profiles profiles derived from a homogeneous area can be expected to be similar in shape and hence show low distance values between each other maximum distances can be found between profiles which lie on opposite sides of a structural boundary the mahalanobis distance was found to be more sensitive and to yield greater spatial resolution than other distance measures such as the euclidean distance cell stained sections of the human neocortex were analyzed the method not only verified boundaries which had been defined by visual inspection it also revealed new ones which had not been detected visually the procedure offers an important supplement to the traditional methods based on visual inspection which for the first time is based on quantitative data and therefore offers a new level of reproducibility and observer independence anatomical atlases based on this procedure thus provide a new tool for the interpretation of structural data obtained from functional imaging techniques

cortical cartography using the discrete conformal approach of circle packings cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain metric based approaches are the most common but suffer from high distortions conformal or angle based algorithms are supported by a comprehensive mathematical theory the conformal approach that uses circle packings is versatile in the manipulation and display of results in addition it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods in this paper we begin with a brief description of cortical flat mapping from data acquisition to map displays including a brief review of past flat mapping approaches we then describe the mathematics of conformal geometry and key elements of conformal mapping we introduce the mechanics of circle packing and discuss its connections with conformal geometry using a triangulated surface representing a cortical hemisphere we illustrate several manipulations available using circle packing methods and describe the associated ensemble conformal features ecfs we conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy

multigrid solution of high order discretisation for three dimensional biharmonic equation with dirichlet boundary conditions of second kind abstract in this paper we propose two compact finite difference approximations for three dimensional biharmonic equation with dirichlet boundary conditions of second kind in these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques the unknown solution and its second derivatives are carried as unknowns at grid points we derive second order and fourth order approximations on a 27 point compact stencil classical iteration methods such as gauss seidel and sor for solving the linear system arising from the second order and fourth order discretisation suffer from slow convergence in order to overcome this problem we use multigrid method which exhibit grid independent convergence and solve the linear system of equations in small amount of computer time the fourth order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

near isometric flattening of brain surfaces flattened representations of brain surfaces are often used to visualize and analyze spatial patterns of structural organization and functional activity here we present a set of rigorous criteria and accompanying test cases with which to evaluate flattening algorithms that attempt to preserve shortest path distances on the original surface we also introduce a novel flattening algorithm that is the first to satisfy all of these criteria and demonstrate its ability to produce accurate flat maps of human and macaque visual cortex using this algorithm we have recently obtained results showing a remarkable unexpected degree of consistency in the shape and topographic structure of visual cortical areas within humans and macaques as well as between these two species

quasi isometric parameterization for texture mapping in this paper we present a new 3d triangular mesh parameterization method that is computationally efficient and yields minimized distance errors the method has four steps firstly multidimensional scaling mds is used to flatten each submesh consisting of one vertex and its direct neighbours on the 3d triangular mesh secondly an optimal method is used to compute the linear reconstructing weights of each vertex with respect to its neighbours thirdly a spectral decomposition method is used to obtain initial 2d parameterization coordinates fourthly the initial coordinates are rotated and scaled to minimize the distance errors it is demonstrated that this method can be used for texture mapping analyses and examples show the effectiveness of this parameterization method compared with alternatives

a multi resolution scheme for distortion minimizing mapping between human subcortical structures based on geodesic construction on riemannian manifolds in this paper we deal with a subcortical surface registration problem subcortical structures including hippocampi and caudates have a small number of salient features such as heads and tails unlike cortical surfaces therefore it is hard if not impossible to perform subcortical surface registration with only such features it is also non trivial for neuroanatomical experts to select landmarks consistently for subcortical surfaces of different subjects we therefore present a landmark free approach for subcortical surface registration by measuring the amount of mesh distortion between subcortical surfaces assuming that the surfaces are represented by meshes the input meshes can be constructed using any surface modeling tool available in the public domain since our registration method is independent of a surface modeling process given the source and target surfaces together with their representing meshes the vertex positions of the source mesh are iteratively displaced while preserving the underlying surface shape in order to minimize the distortion to the target mesh by representing each surface mesh as a point on a high dimensional riemannian manifold we define a distance metric on the manifold that measures the amount of distortion from a given source mesh to the target mesh based on the notion of isometry while penalizing triangle flipping under this metric we reduce the distortion minimization problem to the problem of constructing a geodesic curve from the moving source point to the fixed target point on the manifold while satisfying the shape preserving constraint we adopt a multi resolution framework to solve the problem for distortion minimizing mapping between the source and target meshes we validate our registration scheme through several experiments distance metric comparison visual validation using real data robustness test to mesh variations feature alignment using anatomic landmarks consistency with previous clinical findings and comparison with a surface based registration method lddmm surface

a fourth order spatial accurate and practically stable compact scheme for the cahn hilliard equation abstract we present a fourth order spatial accurate and practically stable compact difference scheme for the cahn hilliard equation the compact scheme is derived by combining a compact nine point formula and linearly stabilized splitting scheme the resulting system of discrete equations is solved by a multigrid method numerical experiments are conducted to verify the practical stability and fourth order accuracy of the proposed scheme we also demonstrate that the compact scheme is more robust and efficient than the non compact fourth order scheme by applying to parallel computing and adaptive mesh refinement

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

exponential higher order compact scheme for 3d steady convection diffusion problem abstract in this paper the exponential higher order compact ehoc finite difference schemes proposed by tian and dai 2007 10 for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case the proposed ehoc scheme has the feature that it provides very accurate solution exact in case of constant convection coefficient in 1d of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil the key properties of this scheme are its stability accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes to validate the present ehoc method three test problems mostly with boundary or internal layers are solved comparisons are made between numerical results for the present ehoc scheme and other available methods in the literature

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

exact geodesics and shortest paths on polyhedral surfaces we present two algorithms for computing distances along convex and non convex polyhedral surfaces the first algorithm computes exact minimal geodesic distances and the second algorithm combines these distances to compute exact shortest path distances along the surface both algorithms have been extended to compute the exact minimal geodesic paths and shortest paths these algorithms have been implemented and validated on surfaces for which the correct solutions are known in order to verify the accuracy and to measure the run time performance which is cubic or less for each algorithm the exact distance computations carried out by these algorithms are feasible for large scale surfaces containing tens of thousands of vertices and are a necessary component of near isometric surface flattening methods that accurately transform curved manifolds into flat representations

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

surface fluid registration of conformal representation application to detect disease burden and genetic influence on hippocampus in this paper we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1 forms inverse consistent surface fluid registration and multivariate tensor based morphometry mtbm first we conformally map a surface onto a planar rectangle space with holomorphic 1 forms second we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface third we align the feature image with a chosen template image via the fluid image registration algorithm which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization the inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images this alignment induces a corresponding deformation on the surface we tested the system on alzheimer s disease neuroimaging initiative adni baseline dataset to study ad symptoms on hippocampus in our system by modeling a hippocampus as a 3d parametric surface we nonlinearly registered each surface with a selected template surface then we used mtbm to analyze the morphometry difference between diagnostic groups experimental results show that the new system has better performance than two publicly available subcortical surface registration tools first and spharm we also analyzed the genetic influence of the apolipoprotein e 4 allele apoe4 which is considered as the most prevalent risk factor for ad our work successfully detected statistically significant difference between apoe4 carriers and non carriers in both patients of mild cognitive impairment mci and healthy control subjects the results show evidence that the apoe genotype may be associated with accelerated brain atrophy so that our work provides a new mri analysis tool that may help presymptomatic ad research

the spatial temporal lamellar structures in the confined ideal polymer blends we consider the formation of self organized spatial temporal oscillating structures in symmetric binary polymer blends confined by two flat walls an influence of these walls on the formation of the oscillating volume structures is studied this phenomenon is simulated by an initial boundary value problem for the conserved order parameter or the concentration of one of the components in a binary mixture under a special choice the dynamical puri binder s boundary conditions these structures look like the lamellar structures the behavior of the order parameter is described by the modified cahn hilliard equation which models so called the non fickian diffusion in the symmetric binary polymer blends the nonlinear dynamical boundary conditions correspond to the process of adsorption desorption on the walls as a result these nonlinear surface processes induce into the volume the spatial temporal asymptotically periodic structures of relaxation pre turbulent or turbulent type with finite countable or non countable points of discontinuities on the period correspondingly the frequency of oscillations on the period follows a power law for the relaxation type and increases exponentially in the other cases

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

a cubic hermite finite element continuation method for numerical solutions of the von karman equations abstract we study a cubic hermite finite element method for numerical solutions of the von karman equations defined in a rectangular domain with totally clamped boundary conditions a novel iterative method combined with a predictor corrector continuation algorithm is exploited to trace solution curves of the von karman equations the fourth order finite element approximations compute accurate numerical solutions for the deformation and the airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives in this regard the classical predictor corrector continuation method is interpreted in a different way our numerical results show that the bifurcation scenario of the von karman equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

on the numerical solution of high order multi dimensional elliptic pdes abstract using the idea of differential quadrature two new methods are constructed to approximate the solution of elliptic partial differential equations in higher dimensions to obtain the weighting coefficients of the first method a mixture of grid points and mid points of the uniform partition is used the order of convergence obtained by the first algorithm is non optimal so we mixed the idea with spline collocation to obtain higher order approximations using the weighting coefficients of the non optimal algorithm some new weighting coefficients are obtained which are used to obtain higher accuracy for the first time a sixth order approximation is obtained to the solution of well known high order multi dimensional elliptic pdes such as biharmonic and triharmonic problems we found a way to increase the order of convergence and the accuracy without increasing the cpu runtime the block form of the coefficients matrix for the linear case is presented to have a better vision on the system arised from the method some examples of biharmonic and triharmonic problems are solved to show the good performance and applicability of the proposed algorithms

compact mixed methods for convection diffusion type problems abstract this paper presents a class of fourth order compact finite difference technique for solving two dimensional convection diffusion equation the equation is recasted as a first order mixed system introducing a conservation and flux equations since flux appears explicitly in the mixed formulation we search a fourth order compact approximation of the primary solution field and flux based on taylor series expansion the proposed compact mixed formulation generalizes the work of carey and spotz g f carey w f spotz higher order compact mixed methods commun numer meth eng 13 1997 we show that their fourth order formulation corresponds to a particular case of our presented scheme and we extend their work to variable diffusion and convection coefficients some numerical experiments are performed to demonstrate the fourth order effective convergence rate

a multigrid method for a fourth order diffusion equation with application to image processing in this paper we are concerned with the efficient solution of the elliptic system alpha l nobreak i u nobreak f where l is the biharmonic operator supplemented by higher order boundary conditions this is a core task in image processing arising from the minimization of rotation invariant energy functionals for the purpose of numerical solution we approximate the partial differential equation by finite differences in order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods finally experimental results are reported that demonstrate the effectiveness of the multigrid approach

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

new implementation of elliptic systems method for time dependent diffusion tomography with back reflected and transmitted boundary data abstract a problem of coefficient recovery from incomplete boundary data in inverse problems is considered it uses a new implementation of the elliptic systems method esm in time dependent diffusion tomography the basic formulation of the esm involves solving a system of coupled fourth order partial differential equations with the time variable integrated out using legendre polynomials here we use c 1 bogner fox schmit bi cubic elements over rectangles with a new treatment of boundary conditions in the common case of incomplete boundary data this new method is fourth order accurate for sufficiently smooth functions the new boundary condition approach allows the use of homogeneous natural boundary conditions on parts of the boundary where no measured data is available we will focus on a comparison with three previously published examples using back reflected or transmitted data with one or two inclusions the new implementation gives markedly improved results for inclusion recovery all of which are achieved without use of additional aids such as weight functions which previously have been thought to be essential and is shown to be surprisingly robust with respect to noise we conclude with two examples illustrating the effect of increasing levels of noise

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility