guard games on graphs a team of mobile agents called guards tries to keep an intruder out of an assigned area by blocking all possible attacks in a graph model for this setting the guards and the intruder are located on the vertices of a graph and they move from node to node via connecting edges the area protected by the guards is an induced subgraph of the given graph we investigate the algorithmic aspects of the guarding problem which is to find the minimum number of guards sufficient to patrol the area we show that the guarding problem is pspace hard and provide a set of approximation algorithms all approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win this variant of the game appears to be a 2 approximation for the guarding problem and for graphs without cycles of length 5 the minimum number of required guards in both games coincides we give a polynomial time algorithm for solving the one step guarding problem in graphs of bounded treewidth and complement this result by showing that the problem is w1 hard parameterized by the treewidth of the input graph we also show that the problem is fixed parameter tractable fpt parameterized by the treewidth and maximum degree of the input graph finally we turn our attention to a large class of sparse graphs including planar graphs and graphs of bounded genus namely apex minor free graphs we prove that the one step guarding problem is fpt and possess a ptas on apex minor free graphs

the complexity of pursuit on a graph abstract a robber and k cops choose starting vertices in a graph and move in alternation from vertex to vertex along the edges of the graph capture occurs if a cop ever shares a vertex with the robber if k is not fixed and if either the graph is directed or initial positions are given we show that the problem is exptime complete similar techniques lead to the pspace and exptime completeness of some other combinatorial games that were previously only known to be np and pspace hard

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

subexponential parameterized algorithms on bounded genus graphs and h minor free graphs we introduce a new framework for designing fixed parameter algorithms with subexponential running time 2o k no 1 our results apply to a broad family of graph problems called bidimensional problems which includes many domination and problems such as vertex cover feedback vertex set minimum maximal matching dominating set edge dominating set disk dimension and many others restricted to bounded genus graphs phrased as bipartite graph problem furthermore it is fairly straightforward to prove that a problem is bidimensional in particular our framework includes as special cases all previously known problems to have such subexponential algorithms previously these algorithms applied to planar graphs single crossing minor free graphs and or map graphs we extend these results to apply to bounded genus graphs as well in a parallel development of combinatorial results we establish an upper bound on the treewidth or branchwidth of a bounded genus graph that excludes some planar graph h as a minor this bound depends linearly on the size v h of the excluded graph h and the genus g g of the graph g and applies and extends the graph minors work of robertson and seymour building on these results we develop subexponential fixed parameter algorithms for dominating set vertex cover and set cover in any class of graphs excluding a fixed graph h as a minor in particular this general category of graphs includes planar graphs bounded genus graphs single crossing minor free graphs and any class of graphs that is closed under taking minors specifically the running time is 2o k nh where h is a constant depending only on h which is polynomial for k o log2 n we introduce a general approach for developing algorithms on h minor free graphs based on structural results about h minor free graphs at the heart of robertson and seymour s graph minors work we believe this approach opens the way to further development on problems in h minor free graphs

the guarding game is e complete the guarding game is a game in which several cops try to guard a region in a directed or undirected graph against robber robber and the cops are placed on the vertices of the graph they take turns in moving to adjacent vertices or staying cops inside the guarded region robber on the remaining vertices the robber region the goal of robber is to enter the guarded region at a vertex with no cop on it the problem is to determine whether for a given graph and given number of cops the cops are able to prevent robber from entering the guarded region fomin et al 2011 7 proved that the problem is np complete when the robber region is restricted to a tree further they prove that is it pspace complete when the robber region is restricted to a directed acyclic graph and they ask about the problem complexity for arbitrary graphs in this paper we prove that the problem is e complete for arbitrary directed graphs

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

recent results and questions in combinatorial game complexities abstract aim present a systematic development of part of the theory of combinatorial games from the ground up approach computational complexity combinatorial games are completely determined the questions of interest are efficiencies of strategies methodology divide and conquer isolate the various difficulties separating hard from easy games and attack them individually presentation informal examples of games sampled from various levels illustrate the theory with emphasis on formulating and motivating new and old research problems

pursuing a fast robber on a graph the cops and robbers game as originally defined independently by quilliot and by nowakowski and winkler in the 1980s has been much studied but very few results pertain to the algorithmic and complexity aspects of it in this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is np hard and that the parameterized version of the problem is w 2 hard the proof extends to the case where the robber moves s time faster than the cops we show that on split graphs the problem is polynomially solvable if s 1 but is np hard if s 2 we further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for email protected 2 finally we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

lattice boltzmann method on unstructured grids further developments we discuss further developments of the finite volume lattice boltzmann formulation on unstructured grids it is shown that the method tolerates significant grid distortions without showing any appreciable numerical viscosity effects at second order in the mesh size a theoretical argument of plausibility for such a property is presented in addition a set of boundary conditions which permit to handle flows with open boundaries is also introduced and numerically demonstrated for the case of channel flows and driven cavity flows

parameterized pursuit evasion games we study the parameterized complexity of four variants of pursuit evasion on graphs seeded pursuit evasion short seeded pursuit evasion directed pursuit evasion and short directed pursuit evasion both seeded pursuit evasion and short seeded pursuit evasion are played on undirected graphs with given starting positions for both the cops and the robber directed pursuit evasion and its short variant are played on directed graphs with the players free to choose their starting positions we show for seeded pursuit evasion and directed pursuit evasion that finding a winning strategy for the cops is aw hard when we parameterize by the number of cops further we show that the short k move variants of these problems short seeded pursuit evasion and short directed pursuit evasion are aw complete when we parameterize by both the number of cops and turns

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

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

the lion and man game on polyhedral surfaces with obstacles abstract we study a geometric version of the cops and robbers game known as the lion and man game in this game a group of lions the pursuers try to capture a man the evader the players have equal speed they can observe each other at all times while the lion and man game is well studied in planar domains such as polygons very little is known about its properties in higher dimensions in this paper we study the game when played on the surface of a polyhedron with or without boundary possibly in the presence of obstacles i e forbidden regions in particular in the absence of obstacles the surface has genus zero and it can be homeomorphic to either a disk or a sphere we assume that the input surface is triangulated we show that three lions with non zero capture distance can capture the man in o a 2 l n 2 2 a 2 l n d steps where a is the area l is the total edge length of the polyhedron n is the number of triangular faces used in the representation of the polyhedron and d is the length of the longest shortest path on the surface

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

spectral element discontinuous galerkin lattice boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator in this paper a spectral element discontinuous galerkin sedg lattice boltzmann discretization and an exponential time marching scheme are used to study the flow field past two circular cylinders in tandem arrangement the basic idea is to discretize the streaming step of the lattice boltzmann equation by using the sedg method to get a system of ordinary differential equations odes whose exact solutions are expressed by using a large matrix exponential the approximate solution of the resulting odes are obtained from a projection method based on a krylov subspace approximation this approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension the exponential time integration scheme is useful especially when computations are carried out at high courant friedrichs lewy cfl numbers where most explicit time marching schemes are inaccurate simulations of flow were carried out for a circular cylinder at re 20 and for two circular cylinders in tandem at re 40 and a spacing of 2 5d where d is the diameter of the cylinders we compare our results with those from a fourth order runge kutta scheme that is restricted by the cfl number in addition important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme these results are compared with results from our simulation using the rk scheme and with existing benchmark results

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

optimization based amp for phase retrieval the impact of initialization and ell_ 2 regularization we consider an ell 2 regularized non convex optimization problem for recovering signals from their noisy phaseless observations we design and study the performance of a message passing algorithm that aims to solve this optimization problem we consider the asymptotic setting m n rightarrow infty m n rightarrow delta and obtain sharp performance bounds where m is the number of measurements and n is the signal dimension we show that for complex signals the algorithm can perform accurate recovery with only m 64 pi 2 4 n approx 2 5n measurements also we provide a sharp analysis on the sensitivity of the algorithm to noise we highlight the following facts about our message passing algorithm 1 adding ell 2 regularization to the non convex loss function can be beneficial and 2 spectral initialization has a marginal impact on the performance of the algorithm the sharp analyses in this paper not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non convex optimization problems

an interpretation and derivation of the lattice boltzmann method using strang splitting the lattice boltzmann space time discretisation as usually derived from integration along characteristics is shown to correspond to a strang splitting between decoupled streaming and collision steps strang splitting offers a second order accurate approximation to evolution under the combination of two non commuting operators here identified with the streaming and collision terms in the discrete boltzmann partial differential equation strang splitting achieves second order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps and the other operator is applied once for a full timestep we show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second order accurate and explicit scheme from an integration of the discrete boltzmann equation along characteristics this approach extends easily to include general matrix collision operators and also body forces finally we show that the validity of the lattice boltzmann discretisation for grid scale reynolds numbers larger than unity depends crucially on the use of a crank nicolson approximation to discretise the collision operator replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive due to the splitting error unless the grid scale reynolds number remains well below unity

a two person game on graphs where each player tries to encircle his opponent s men we present results on a combinatorial game which was proposed to one of the authors by ingo althofer personal communication let g be an undirected finite graph without loops and multiple edges and let k be a positive integer with k 12 g 1 there are two players called white and black both having k men of their color in turn beginning with white the players position their men one at a time on unoccupied vertices of g when all men are placed the players take turns moving a man of their color along an edge to an unoccupied adjacent vertex again beginning with white a player wins if his opponent cannot carry out his next move since none of his men has an unoccupied neighbor if the game does not stop then the outcome is a draw we always assume that both players play optimal among other questions we deal with the following ones 1 1 is it true that for all g and k white cannot win the game 2 2 does there exist a tree t and a positive integer k for which the outcome is a draw let g denote the covering number of g i e g is the minimum number of vertices covering all edges of g we prove that black wins the game if g k we use this result to show that white never wins the game if gis bipartite thus providing a partial answer to the first question we answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw for some k moreover we present results on extremal problems arising in the context of the game we also completely solve the cases when g is a path or a cycle further we completely settle the case k 2 in the proofs of our results matchings and cycles in graphs play a predominant role

a spectral element discontinuous galerkin lattice boltzmann method for nearly incompressible flows we present a spectral element discontinuous galerkin lattice boltzmann method for solving nearly incompressible flows decoupling the collision step from the streaming step offers numerical stability at high reynolds numbers in the streaming step we employ high order spectral element discontinuous galerkin discretizations using a tensor product basis of one dimensional lagrange interpolation polynomials based on gauss lobatto legendre grids our scheme is cost effective with a fully diagonal mass matrix advancing time integration with the fourth order runge kutta method we present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous galerkin approach we show convergence studies for couette flows and demonstrate two benchmark cases with lid driven cavity flows for re 400 5000 and flows around an impulsively started cylinder for re 550 9500 computational results are compared with those of other theoretical and computational work that used a multigrid method a vortex method and a spectral element model

on the finite difference based lattice boltzmann method in curvilinear coordinates the lattice boltzmann method is a microscopic based approach for solving the fluid flow problems at the macroscopic scales the presently popular method uses regularly spaced lattices and cannot handle curved boundaries with desirable flexibility to circumvent such difficulties a finite difference based lattice boltzmann method fdlbm in curvilinear coordinates is explored using body fitted coordinates with non uniform grids several test cases including the impulsively started cylindrical couette flow steady state cylindrical couette flow steady flow over flat plates and steady flow over a circular cylinder are used to examine various issues related to the fdlbm the effect of boundary conditions for the distribution functions on the solution the merits between second order central difference and upwind schemes for advection terms and the effect of the reynolds number are investigated favorable results are obtained using fdlbm in curvilinear coordinates indicating that the method is potentially capable of solving finite reynolds number flow problems in complex geometries

a characteristic galerkin method for discrete boltzmann equation abstract the characteristic galerkin finite element method for the discrete boltzmann equation is presented to simulate fluid flows in complex geometries the inherent geometric flexibility of the finite element method permits the easy use of simple cartesian variables on unstructured meshes and the mesh clustering near large gradients the characteristic galerkin procedure with appropriate boundary condition results in accurate solutions with little numerical diffusion several test cases are conducted including unsteady couette flows lid driven cavity flows and steady flow past a circular cylinder on unstructured meshes the numerical results are in good agreement with previous analytical if applicable numerical and experimental results

least squares finite element scheme for the lattice boltzmann method on an unstructured mesh a numerical model of the lattice boltzmann method lbm utilizing least squares finite element method in space and the crank nicolson method in time is developed this method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite element method while employing accurate least squares optimization fourth order accuracy in space and second order accuracy in time are derived for a pure advection equation on a uniform mesh while high stability is implied from a von neumann linearized stability analysis implemented on unstructured mesh through an innovative element by element approach the proposed method requires fewer grid points and less memory compared to traditional lbm accurate numerical results are presented through two dimensional incompressible poiseuille flow couette flow and flow past a circular cylinder finally the proposed method is applied to estimate the permeability of a randomly generated porous media which further demonstrates its inherent geometric flexibility

cops and robbers from a distance cops and robbers is a pursuit and evasion game played on graphs that has received much attention we consider an extension of cops and robbers distance k cops and robbers where the cops win if at least one of them is of distance at most k from the robber in g the cop number of a graph g is the minimum number of cops needed to capture the robber in g the distance k analogue of the cop number written c k g equals the minimum number of cops needed to win at a given distance k we study the parameter c k from algorithmic structural and probabilistic perspectives we supply a classification result for graphs with bounded c k g values and develop an o n 2 s 3 algorithm for determining if c k g s for s fixed we prove that if s is not fixed then computing c k g is np hard upper and lower bounds are found for c k g in terms of the order of g we prove that nk 1 2 o 1 c k n o nlog 2nk 1 log k 2 k 1 where c k n is the maximum of c k g over all n vertex connected graphs the parameter c k g is investigated asymptotically in random graphs g n p for a wide range of p p n for each k 0 it is shown that c k g as a function of the average degree d n pn forms an intriguing zigzag shape

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

three time integration methods for incompressible flows with discontinuous galerkin boltzmann method abstract this paper presents three time integration methods for incompressible flows with finite element method in solving the lattice bgk boltzmann equation the space discretization is performed using nodal discontinuous galerkin method which employs unstructured meshes with triangular elements and high order approximation degrees the time discretization is performed using three different kinds of time integration methods namely direct decoupling and splitting from the storage cost temporal accuracy numerical stability and time consumption we systematically compare three time integration methods then benchmark fluid flow simulations are performed to highlight efficient time integration methods numerical results are in good agreement with others or exact solutions

finite volume lattice boltzmann method we present a finite volume formulation for the lattice boltzmann method fvlbm based on standard bilinear quadrilateral elements in two dimensions the accuracy of this scheme is demonstrated by comparing the velocity field with the analytical solution of the navier stokes equations for time dependent rotating couette flow and taylor vortex flow to demonstrate the flexibility of the scheme we have also simulated a modified rotating couette flow where the inner cylinder has an elliptical shape the results agree with those obtained from the traditional marker and cell method the fvlbm scheme is applicable to arbitrarily shaped two dimensional regions and thus the range of applicability of the lattice boltzmann method has been significantly extended

viscosity of finite difference lattice boltzmann models two dimensional finite difference lattice boltzmann models for single component fluids are discussed and the corresponding macroscopic equations for mass and momentum conservation are derived by performing a chapman enskog expansion in order to recover the correct mass equation characteristic based finite difference schemes should be associated with the forward euler scheme for the time derivative while the space centered and second order upwind schemes should be associated to second order schemes for the time derivative in the incompressible limit the characteristic based schemes lead to spurious numerical contributions to the apparent value of the kinematic viscosity in addition to the physical value that enters the navier stokes equation formulae for these spurious numerical viscosities are in agreement with results of simulations for the decay of shear waves

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation