the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

a spectral vanishing viscosity for the les of turbulent flows within rotating cavities a spectral vanishing viscosity technique svv is presented for the simulation of 3d turbulent incompressible flows within a rotor stator cavity one characteristic of this technique is that the svv is active only for the short length scales a feature which is reminiscent of large eddy simulation models the spectral vanishing viscosity first introduced by e tadmor for the inviscid burgers equation e tadmor convergence of spectral methods for nonlinear conservation laws siam j numer anal 26 1 1989 30 is incorporated into the cylindrical navier stokes equations written in velocity pressure formulation the second order operator involved in the svv method is implemented in a chebyshev collocation fourier galerkin pseudo spectral code the svv is shown to lead to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization les results are presented here for rotational reynolds numbers ranging from re 7 i 10 4 to re 7 i 10 5 turbulent quantities are shown to compare very favorably with results of direct numerical simulation dns and experimental measurements

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

iterative brinkman penalization for remeshed vortex methods we introduce an iterative brinkman penalization method for the enforcement of the no slip boundary condition in remeshed vortex methods in the proposed method the brinkman penalization is applied iteratively only in the neighborhood of the body this allows for using significantly larger time steps than what is customary in the brinkman penalization thus reducing its computational cost while maintaining the capability of the method to handle complex geometries we demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate we find that the present method enhances significantly the accuracy of the brinkman penalization technique for the simulations of highly unsteady flows past complex geometries

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

an information based neural approach to generic constraint satisfaction a novel artificial neural network heuristic inn for general constraint satisfaction problems is presented extending a recently suggested method restricted to boolean variables in contrast to conventional ann methods it employs a particular type of non polynomial cost function based on the information balance between variables and constraints in a mean field setting implemented as an annealing algorithm the method is numerically explored on a testbed of graph coloring problems the performance is comparable to that of dedicated heuristics and clearly superior to that of conventional mean field annealing

experimental complexity analysis of continuous constraint satisfaction problems continuous constraint satisfaction problems are at the core of many real world applications including planning scheduling control and diagnosis of physical systems such as car planes and factories yet in order to be effective in such resource constrained environments constraint based techniques must take into account the complexity of continuous constrained problems in this paper we study complexity phase transition phenomena of continuous constraint satisfaction problems csps first we analyze three continuous constraint satisfaction formulations based on discrete 3 sat problems which have a strong relation between structure and search cost then we propose a generic benchmarking model for comparing continuous csps and algorithms and present two example problems based on sine functions in our experiments these problems are solved using both local and global search methods besides comparing the complexities of different continuous csps and search algorithms we also connect these back to results from similar studies on sat problems in solving continuous 3 sat and sine based csps we find that the median search cost is characterized by simple parameters such as the constraint to variable ratio and constraint tightness and that discrete search algorithms such as gsat have continuous counter parts with similar behavior regarding local versus global search techniques for constraint solving our results show that local search methods are more efficient for weakly constrained problems whereas global search methods work better on highly constrained problems

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

heuristic sampling a method for predicting the performance of tree searching programs determining the feasibility of a particular search program is important in practical situations especially when the computation involved can easily require days or even years to help make such predictions a simple procedure based on a stratified sampling approach is presented this new method which is called heuristic sampling is a generalization of knuth s original algorithm for estimating the efficiency of backtrack programs with the aid of simple heuristics this method can produce significantly more accurate cost estimates for commonly used tree search algorithms such as depth first breadth first best first and iterative deepening

computation of three dimensional multiphase flow dynamics by fully coupled immersed flow fcif solver abstract this work presents a fully coupled immersed flow fcif solver for the three dimensional simulation of fluid fluid interaction by coupling two distinct flow solvers using an immersed boundary ib method the fcif solver captures dynamic interactions between two fluids with disparate flow properties while retaining the desirable simplicity of non boundary conforming grids for illustration we couple an ib based unsteady reynolds averaged navier stokes urans simulator with a depth integrated long wave solver for the application of slug development with turbulent gas and laminar liquid we perform a series of validations including turbulent laminar flows over prescribed wavy boundaries and freely evolving viscous fluids these confirm the effectiveness and accuracy of both one way and two way coupling in the fcif solver finally we present a simulation example of the evolution from a stratified turbulent laminar flow through the initiation of a slug that nearly bridges the channel the results show both the interfacial wave dynamics excited by the turbulent gas forcing and the influence of the liquid on the gas turbulence these results demonstrate that the fcif solver effectively captures the essential physics of gas liquid interaction and can serve as a useful tool for the mechanistic study of slug generation in two phase gas liquid flows in channels and pipes

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

complexity theoretic models of phase transitions in search problems in recent years numerous studies have observed that many hard combinatorial decision problems exhibit behaviour described as a phase transition this is the phenomenon whereby typical instances of a problem display a dramatic shift in certain characteristics as some parameter of the instances is varied such characteristics include the likelihood of an instance having a solution and the time taken by a search algorithm the apparent pervasiveness of phase transitions in hard combinatorial search problems has led to contrasting claims being advanced concerning to what extent all np complete problems exhibit phase transitions the established importance of exploiting phase transition effects in the design of search heuristics provides a strong motivation for assessing how valid such claims may be in this paper we argue that questions concerning the generality of phase transition phenomena are at present ill defined in order to address this difficulty we propose and examine rigorous complexity theoretic models of the statement the decision problem d has a phase transition within these models it is proved that for certain natural definitions contrasting results about phase transition behaviour can be proved

transitions between streamline topologies of structurally stable hamiltonian flows in multiply connected domains abstract we consider hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two dimensional multiply connected domains one example of such hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow whose hamiltonian is called the stream function here we are concerned with topological structures of the level sets of the hamiltonian which we call streamlines by analogy from incompressible fluid flows classification of structurally stable streamline patterns has been considered in yokoyama and sakajo 2013 where a procedure to assign a unique sequence of words called the maximal word to these patterns is proposed thanks to this procedure we can identify every streamline pattern with its representing sequence of words up to topological equivalence in the present paper based on the theory of word representations we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any hamiltonian although this method cannot deal with topological streamline changes induced by bifurcations it reveals the existence of many non trivial global transitions in a generic sense we also demonstrate how the present theory is applied to fluid flow problems with vortex structures

parallel two step w methods with peer variables a new class of methods for the solution of stiff initial value problems is introduced that is parallel by design it has a two step character and propagates s different peer solution variables with essentially identical characteristics from step to step the main work lies in the solution of s independent linear stage equations which may be solved in parallel convergence of order s 1 and stability for general stepsize sequences are proved conditions for order s and stronger stability criteria are addressed as well promising methods up to order 7 are identified by numerical tests with some widely used stiff test problems some of these are competitive with existing software even in sequential computations

phase transition and landscape statistics of the number partitioning problem the phase transition in the number partitioning problem npp i e the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution is investigated by examining the energy landscape of this classic optimization problem this is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure termed a barrier tree the leaves and internal nodes of which represent respectively the minima and the lowest energy saddles connecting those minima here we apply several measures of shape balance and symmetry as well as of branch lengths barrier heights to the barrier trees that result from the landscape of the npp aiming at identifying traces of the easy hard transition we find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights only the difficulty measure given by the maximum value of the ratio between the barrier height and the energy surplus of local minima succeeded in detecting traces of the phase transition in the tree in addition we show that the barrier trees associated with the npp are very similar to random trees contrasting dramatically with trees associated with the p spin glass and random energy models we also examine critically a recent conjecture on the equivalence between the npp and a truncated random energy model

clustering analysis of the ground state structure of the vertex cover problem vertex cover is one of the classical np complete problems in theoretical computer science a vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset when studied on erd os r enyi random graphs with connectivity c one observes a threshold behavior in the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph recent analytical studies show that on the phase boundary for small connectivities cle the system is replica symmetric while for larger connectivities replica symmetry breaking occurs this change coincides with a change of the typical running time of algorithms from polynomial to exponential to understand the reasons for this behavior and to compare with the analytical results we numerically analyze the structure of the solution landscape for this purpose we have also developed an algorithm which allows the calculation of the backbone without the need to enumerate all solutions we study exact solutions found with a branch and bound algorithm as well as configurations obtained via a monte carlo simulation we analyze the cluster structure of the solution landscape by direct clustering of the states by analyzing the eigenvalue spectrum of correlation matrices and by using a hierarchical clustering method all results are compatible with a change at c e for small connectivities the solutions are collected in a finite small number of clusters while the number of clusters diverges slowly with system size for larger connectivities and replica symmetry breaking but not one step replica symmetry breaking 1 rsb occurs

on network bipartivity systems with two types of agents with a preference for heterophilous interaction produces networks that are more or less close to bipartite we propose two measures quantifying the notion of bipartivity the two measures one well known and natural but computationally intractable one computationally less complex but also less intuitive are examined on model networks that continuously interpolates between bipartite graphs and graphs with many odd circuits we find that the bipartivity measures increase as we tune the control parameters of the test networks to intuitively increase the bipartivity and thus conclude that the measures are quite relevant we also measure and discuss the values of our bipartivity measures for empirical social networks constructed from professional collaborations internet communities and field surveys here we find as expected that networks arising from romantic online interaction have high and professional collaboration networks have low bipartivity values in some other cases probably due to low average degree of the network the bipartivity measures cannot distinguish between romantic and friendship oriented interaction

penalization modeling of a limiter in the tokamak edge plasma an original penalization method is applied to model the interaction of magnetically confined plasma with limiter in the frame of a minimal transport model for ionic density and parallel momentum the limiter is considered as a pure particle sink for the plasma and consequently the density and the momentum are enforced to be zero inside comparisons of the numerical results with one dimensional analytical solutions show a very good agreement in particular the penalization scheme followed in this paper tends to ensure an almost sonic plasma condition at the plasma obstacle interface bohm like criterion with relatively weak dependence on the target mach number profile within the obstacle the new system being solved in a periodic obstacle free domain an efficient pseudo spectral algorithm based on a fast fourier transform is also proposed and associated with an exponential filtering of the unphysical oscillations due to gibbs phenomenon finally the efficiency of the method is illustrated by investigating the flow spreading from the plasma core to the scrape off layer at the wall in a two dimensional system with one then two neighboring limiters

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

bose einstein condensation in satisfiability problems abstract this paper is concerned with the complex behavior arising in satisfiability problems we present a new statistical physics based characterization of the satisfiability problem specifically we design an algorithm that is able to produce graphs starting from a k sat instance in order to analyze them and show whether a bose einstein condensation occurs we observe that analogously to complex networks the networks of k sat instances follow bose statistics and can undergo bose einstein condensation in particular k sat instances move from a fit get rich network to a winner takes all network as the ratio of clauses to variables decreases and the phase transition of k sat approximates the critical temperature for the bose einstein condensation finally we employ the fitness based classification to enhance sat solvers e g chainsat and obtain the consistently highest performing sat solver for cnf formulas and therefore a new class of efficient hardware and software verification tools

search rearrangement backtracking and polynomial average time the average time required for simple search rearrangement backtracking is compared with that for ordinary backtracking when each algorithm is used to find all solutions for random conjunctive normal form predicates the sets of random predicates are characterized by v the number of variables t v the number of clauses and p v the probability that a literal appears in clause for large v if vp v 1n 2 there is a difficult region where the average number of solutions per problem is exponentially small but backtracking requires an exponentially large time the difficult region for search rearrangement backtracking is only slightly smaller than the difficult region for ordinary backtracking it is conjectured that search rearrangement backtracking is exponentially faster than ordinary backtracking for nearly all of the difficult region it is proved that there is no major advantage in using search rearrangement backtracking outside of the difficult region

peer two step methods with embedded sensitivity approximation for parameter dependent odes peer two step methods have been successfully applied to initial value problems for stiff and nonstiff ordinary differential equations odes both on parallel and sequential computers their essential property is the use of several stages per time step with the same accuracy as a new application area these methods are now used for parameter dependent odes where the peer stages approximate the solution also at different places in the parameter space the main interest here is sensitivity data through an approximation of solution derivatives in different parameter directions basic stability and convergence properties are discussed and peer methods of order 2 and 3 in the time stepsize are constructed the computed sensitivity matrix is used in approximate newton and gauss newton methods for shooting in boundary value problems where initial values and or ode parameters are searched for and in parameter identification from partial information on trajectories

easy problems are sometimes hard we present a detailed experimental investigation of the easy hard easy phase transition for randomly generated instances of satisfiability problems problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms this study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard these difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split whilst these hard unsatisfiable problems may have short proofs these appear to be difficult to find and other proofs are long and hard

glass transition seen through asymptotic expansions soft glassy materials exhibit the so called glassy transition which means that the behavior of the model at a low shear rate changes when a certain parameter which we call the glass parameter crosses a critical value this behavior goes from a newtonian behavior to a herschel bulkley behavior through a power law type behavior at the transition point in a previous paper we rigorously proved that the hebraud lequeux model a fokker planck like description of soft glassy material exhibits such a glass transition but the method we used was very specific to the one dimensional setting of the model and as a preparation for generalizing this model to take into account multidimensional situations we look for another technique to study the glass transition of this type of model in this paper we shall use matched asymptotic expansions for such a study the difficulties encountered when using asymptotic expansions for the hebraud lequeux model are that multiple ansaetze have to be used even though the init

relational learning as search in a critical region machine learning strongly relies on the covering test to assess whether a candidate hypothesis covers training examples the present paper investigates learning relational concepts from examples termed relational learning or inductive logic programming in particular it investigates the chances of success and the computational cost of relational learning which appears to be severely affected by the presence of a phase transition in the covering test to this aim three up to date relational learners have been applied to a wide range of artificial fully relational learning problems a first experimental observation is that the phase transition behaves as an attractor for relational learning no matter which region the learning problem belongs to all three learners produce hypotheses lying within or close to the phase transition region second a failure region appears all three learners fail to learn any accurate hypothesis in this region quite surprisingly the probability of failure does not systematically increase with the size of the underlying target concept under some circumstances longer concepts may be easier to accurately approximate than shorter ones some interpretations for these findings are proposed and discussed

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

an empirical study of phase transitions in binary constraint satisfaction problems abstract an empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables domain size and connectivity there is a critical level of constraint tightness at which a phase transition occurs at the phase transition problems change from being soluble to insoluble and the difficulty of problems increases dramatically a theory developed by williams and hogg 44 and independently developed by smith 37 predicts where the hardest problems should occur it is shown that the theory is in close agreement with the empirical results except when constraint graphs are sparse

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

an immersed boundary method using unstructured anisotropic mesh adaptation combined with level sets and penalization techniques the interest on embedded boundary methods increases in computational fluid dynamics cfd because they simplify the mesh generation problem in the case of the navier stokes equations the same simplifications occur for the simulation of multi physics flows the coupling of fluid solid interactions in situation of large motions or deformations to give a few examples nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method in this work the wall boundary conditions are easily taken into account through a penalization technique and the accuracy of the method is recovered using mesh adaptation thanks to the potential of unstructured meshes several classical examples are used to demonstrate that claim

the hardest constraint problems a double phase transition abstract the distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors the first previously recognized is a peak in the median search cost near the connectivity at which half the graphs have solutions this region contains a high proportion of relatively hard problem instances however the hardest instances are in fact concentrated at a second lower transition point near this point most problems are quite easy but there are also a few very hard cases this region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost whose location we also estimate theoretically these behaviors also appear to arise in other constraint problems this work also shows the limitations of simple measures of the cost distribution such as mean or median for identifying outlying cases

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

towards a characterisation of the behaviour of stochastic local search algorithms for sat abstract stochastic local search sls algorithms have been successfully applied to hard combinatorial problems from different domains due to their inherent randomness the run time behaviour of these algorithms is characterised by a random variable the detailed knowledge of the run time distribution provides important information about the behaviour of sls algorithms in this paper we investigate the empirical run time distributions for walksat one of the most powerful sls algorithms for the propositional satisfiability problem sat using statistical analysis techniques we show that on hard random 3 sat problems walksat s run time behaviour can be characterised by exponential distributions this characterisation can be generalised to various sls algorithms for sat and to encoded problems from other domains this result also has a number of consequences which are of theoretical as well as practical interest one of these is the fact that these algorithms can be easily parallelised such that optimal speedup is achieved for hard problem instances

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

an adaptive residual based splitting approach for the penalized navier stokes equations the interest on immersed boundary methods ibm is increasing in computational fluid dynamics as they simplify the mesh generation problem in this work we consider an approach based on the addition of a penalty term to the navier stokes equations to account for the wall boundary conditions to discretize the resulting equations we use a residual distribution approach previously developed by some of the authors to adapt the method to the ibm considered we developed a new formulation of residual distribution based on a strang splitting method in time coupling an implicit asymptotic integration procedure of the penalization ode with a simplified explicit residual distribution for the navier stokes equations the first method provides an operator which is exact up to orders 2 with the penalty parameter assuming values of the order of 10 10 a modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes is also introduced in the paper this guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid we show formally and numerically that the approach proposed is second order accurate for smooth solutions we evaluate its potential for ibm by coupling the resulting method with unstructured mesh adaptation on wall boundaries several steady and time dependent tests are used to show the promising features of the method proposed

modeling and simulation of fish like swimming modeling and simulation of two dimensional flows past deformable bodies are considered the incompressible navier stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method the interface between the solid and the fluid is tracked using a level set description so that it is possible to simulate several bodies freely evolving in the fluid as an illustration of the methods fish like locomotion is analyzed in terms of propulsion efficiency underwater maneuvering and school swimming are also explored

an optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak the penalization method is used to take account of obstacles such as the limiter in a tokamak because of the magnetic confinement of the plasma in a tokamak the transport occurs essentially in the direction parallel to the magnetic field lines we study a 1d nonlinear hyperbolic system as a simplified model of the plasma transport in the area close to the wall a penalization which cuts the flux term of the momentum is studied we show numerically that this penalization creates a dirac measure at the plasma limiter interface which prevents us from defining the transport term in the usual distribution sense hence a new penalty method is proposed for this hyperbolic system for this penalty method an asymptotic expansion and numerical tests give an optimal rate of convergence without spurious boundary layer another two fields penalization has also been implemented and the numerical convergence analysis when the penalization parameter tends to 0 reveals the presence of a boundary layer

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

a new constraint test case generator and the importance of hybrid optimizers in the past decade significant progress has been made in understanding problem complexity of discrete constraint problems in contrast little similar work has been done for constraint problems in the continuous domain in this paper we study the complexity of typical methods for non linear constraint problems and present hybrid solvers with improved performance to facilitate the empirical study we propose a new test case generator for generating non linear constraint satisfaction problems csps and constrained optimization problems cops the optimization methods tested include a sequential quadratic programming sqp method a penalty method with a fixed penalty function a penalty method with a sequence of penalty functions and an augmented lagrangian method for hybrid solvers we focus on the form that combines two or more optimization methods in sequence in the experiments we apply these methods to solve a series of continuous constraint problems with increasing constraint to variable ratios the test problems include artificial benchmark problems from the test case generator and problems derived from controlling a hyper redundant modular manipulator we obtain novel results on complexity phase transition phenomena of the various methods specifically for constraint satisfaction problems the sqp method is the best on weakly constrained problems whereas the augmented lagrangian method is the best on highly constrained ones although the static penalty method performs poorly by itself by combining it with the sqp method we show a hybrid solver that is significantly better than any of the individual methods on problems with moderate to large constraint to variable ratios for constrained optimization problems the hybrid solver obtains much better solutions than sqp while spending comparable amount of time in addition the hybrid solver is flexible and can achieve good results on time bounded applications by setting parameters according to the time limits

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

a brinkman penalization method for compressible flows in complex geometries to simulate flows around solid obstacles of complex geometries various immersed boundary methods had been developed their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non body conformal cartesian grids the brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations its main idea is to model solid obstacles as porous media with porosity f and viscous permeability approaching zero it has the pronounced advantages of mathematical proof of error bound strong convergence and ease of numerical implementation with the volume penalization technique in this paper it is extended to compressible flows the straightforward extension of penalizing momentum and energy equations using brinkman penalization with respective normalized viscous h and thermal h t permeabilities produces unsatisfactory results mostly due to nonphysical wave transmissions into obstacles resulting in considerable energy and mass losses in reflected waves the objective of this paper is to extend the brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media in addition to penalizing momentum and energy equations the continuity equation for porous media is considered inside obstacles in this model the penalized porous region acts as a high impedance medium resulting in negligible wave transmissions the asymptotic analysis reveals that the proposed brinkman penalization technique results in the amplitude and phase errors of order o email protected 1 2 and o h h t 1 email protected 3 4 when the boundary layer within the porous media is respectively resolved or unresolved the proposed method is tested using 1 and 2 d benchmark problems the results of direct numerical simulation are in excellent agreement with the analytical solutions the numerical simulations verify the accuracy and convergence rates

peer methods with improved embedded sensitivities for parameter dependent odes recently an extension of peer two step methods to parameter dependent ivps has been proposed by the author and e kostina where additional embedded stages allow cheap computation of solution derivatives sensitivities with respect to problem parameters for each parameter only one additional satellite stage was used which provides a high order approximation to a neighboring solution however the accuracy of the sensitivities was only first order in the time stepsize h in the present paper we derive implicit peer methods where the sensitivity accuracy is of second order with the same number of stages approximate implementations yield efficient linearly implicit methods for stiff problems and explicit predictor corrector type versions for nonstiff problems tests confirm improved and more robust convergence of inexact newton iterations in shooting for different applications with nonstiff and stiff ivps

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

problem difficulty for tabu search in job shop scheduling tabu search algorithms are among the most effective approaches for solving the job shop scheduling problem jsp yet we have little understanding of why these algorithms work so well and under what conditions we develop a model of problem difficulty for tabu search in the jsp borrowing from similar models developed for sat and other np complete problems we show that the mean distance between random local optima and the nearest optimal solution is highly correlated with the cost of locating optimal solutions to typical random jsps additionally this model accounts for the cost of locating sub optimal solutions and provides an explanation for differences in the relative difficulty of square versus rectangular jsps we also identify two important limitations of our model first model accuracy is inversely correlated with problem difficulty and is exceptionally poor for rare very high cost problem instances second the model is significantly less accurate for structured non random jsps our results are also likely to be useful in future research on difficulty models of local search in sat as local search cost in both sat and the jsp is largely dictated by the same search space features similarly our research represents the first attempt to quantitatively model the cost of tabu search for any np complete problem and may possibly be leveraged in an effort to understand tabu search in problems other than job shop scheduling

minimizing energy below the glass thresholds focusing on the optimization version of the random k satisfiability problem the max k sat problem we study the performance of the finite energy version of the survey propagation algorithm we show that a simple linear time backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states a comparative numerical study on one of the most efficient local search procedures is also given

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

searching for an optimal path in a tree with random costs we consider the problem of finding an optimal path leading from the root of a tree to any of its leaves the tree is known to be uniform binary and of height n and each branch independently may have a cost of 1 or 0 with probability p and 1 p respectively we show that for p 1 2 every algorithm which guarantees finding an exact cheapest path or even a path within a fixed cost ratio of the cheapest must run in exponential average time if however we are willing to accept a near optimal solution almost always then a pruning algorithm exists which finds such a solution in linear expected time the algorithm employs a depth first strategy which stops at regular intervals to appraise its progress and if the progress does not meet a criterion based on domain specific knowledge the current node is irrevocably pruned

configuration landscape analysis and backbone guided local search part i satisfiability and maximum satisfiability boolean satisfiability sat and maximum satisfiability max sat are difficult combinatorial problems that have many important real world applications in this paper we first investigate the configuration landscapes of local minima reached by the walksat local search algorithm one of the most effective algorithms for sat a configuration landscape of a set of local minima is their distribution in terms of quality and structural differences relative to an optimal or a reference solution our experimental results show that local minima from walksat form large clusters and their configuration landscapes constitute big valleys in that high quality local minima typically share large partial structures with optimal solutions inspired by this insight into walksat and the previous research on phase transitions and backbones of combinatorial problems we propose and develop a novel method that exploits the configuration landscapes of such local minima the new method termed as backbone guided search can be embedded in a local search algorithm such as walksat to improve its performance our experimental results show that backbone guided local search is effective on overconstrained random max sat instances moreover on large problem instances from a sat library satlib the backbone guided walksat algorithm finds satisfiable solutions more often than walksat on sat problem instances and obtains better solutions than walksat on max sat problem instances improving solution quality by 20 on average

statistical mechanics perspective on the phase transition in vertex covering offinite connectivity random graphs the vertex cover problem is studied for random graphs gn cn having n vertices and cn edges exact numerical results are obtained by a branch and bound algorithm it is found that a transition in the coverability at a c dependent threshold x xc c appears where xn is the cardinality of the vertex cover this transition coincides with a sharp peak of the typical numerical e1ort which is needed to decide whether there exists a cover with xn vertices or not for small edge concentrations c two less than arrows one inside the other 0 5 a cluster expansion is performed giving very accurate results in this regime these results are extended using methods developed in statistical physics the so called annealed approximation reproduces a rigorous bound on xc c which was known previously the main part of the paper contains an application of the replica method within the replicasymmetric ansatz the threshold xc c and various statistical properties of minimal vertex covers can be calculated for c e 2 the results show an excellent agreement with the numerical findings at average vertex degree 2c e an instability of the simple replica symmetric solution occurs

pairs of sat assignments in random boolean formulae we investigate geometrical properties of the random k satisfiability problem using the notion of x satisfiability a formula is x satisfiable is there exist two sat assignments differing in nx variables we show the existence of a sharp threshold for this property as a function of the clause density for large enough k we prove that there exists a region of clause density below the satisfiability threshold where the landscape of hamming distances between sat assignments experiences a gap pairs of sat assignments exist at small x and around x 12 but they do not exist at intermediate values of x this result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis the cavity method and its algorithmic counterpart the survey propagation algorithm our method uses elementary probabilistic arguments first and second moment methods and might be useful in other problems of computational and physical interest where similar phenomena appear

topology optimization for unsteady flow a computational methodology for optimizing the conceptual layout of unsteady flow problems at low reynolds numbers is presented the geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity the flow is predicted by a stabilized finite element formulation of the incompressible navier stokes equations a brinkman penalization is used to enforce zero velocities in solid material the resulting parameter optimization problem is solved by a non linear programming method the paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems the derivation of the governing equations and the adjoint sensitivity analysis are presented a design dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material the emergence of non physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the brinkman penalization two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady state counterparts copyright 2011 john wiley sons ltd

an adaptive wavelet collocation method for fluid structure interaction at high reynolds numbers two mathematical approaches are combined to calculate high rey nolds number incompressible fluid structure interaction a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion and brinkman penalization to enforce solid boundaries of arbitrary complexity we also implement a wavelet based multilevel solver for the poisson problem for the pressure at each time step the method is applied to two dimensional flow around fixed and moving cylinders for reynolds numbers in the range 3 times 10 1 le re le 10 5 the compression ratios of up to 1000 are achieved for the first time it is demonstrated in actual dynamic simulations that the compression scales like re 1 2 over five orders of magnitude while computational complexity scales like re this represents a significant improvement over the classical complexity estimate of re 9 4 for two dimensional turbulence

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

fourier spectral and wavelet solvers for the incompressible navier stokes equations with volume penalization convergence of a dipole wall collision in this study we use volume penalization to mimic the presence of obstacles in a flow or a domain with no slip boundaries this allows in principle the use of fast fourier spectral methods and coherent vortex simulation techniques based on wavelet decomposition of the flow variables to compute turbulent wall bounded flow or flows around solid obstacles by simply adding one term in the equation convergence checks are reported using a recently revived and unexpectedly difficult dipole wall collision as a benchmark computation several quantities like the vorticity isolines truncation error kinetic energy and enstrophy are inspected for a collision of a dipole with a no slip wall and compared with available benchmark data obtained with a standard chebyshev pseudospectral method we quantify the possible deteriorating effects of the gibbs phenomenon present in the fourier based schemes due to continuity restrictions of the penalized navier stokes equations on the wall it is found that gibbs oscillations have a negligible effect on the flow evolution allowing higher order recovery of the accuracy on a fourier basis by means of postprocessing an advantage of coherent vortex simulations on the other hand is that the degrees of freedom of the flow computation can strongly be reduced in this study we quantify the possible reduction of degrees of freedom while keeping the accuracy for an optimal convergence scenario the penalization parameter has to scale with the number of fourier and wavelet modes in addition an implicit treatment of the darcy drag term in the penalized navier stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements

the centred travelling salesman at high temperature a recently formulated statistical mechanics method is used to study the phase transition occurring in a generalization of the travelling salesman problem tsp known as the centred tsp the method shows that the problem has clear signs of a crossover but is only able to access unscaled finite temperatures above the transition point the solution of the problem using this method displays a curious duality

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

an adaptive newton picard algorithm with subspace iteration for computing periodic solutions this paper is concerned with the efficient computation of periodic orbits in large scale dynamical systems that arise after spatial discretization of partial differential equations pdes a hybrid newton picard scheme based on the shooting method is derived which in its simplest form is the recursive projection method rpm of shroff and keller siam j numer anal 30 1993 pp 1099 1120 and is used to compute and determine the stability of both stable and unstable periodic orbits the number of time integrations needed to obtain a solution is shown to be determined only by the system s dynamics this contrasts with traditional approaches based on newton s method for which the number of time integrations grows with the order of the spatial discretization two test examples are given to show the performance of the methods and to illustrate various theoretical points

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

final states of decaying 2d turbulence in bounded domains influence of the geometry abstract direct numerical simulations of two dimensional decaying turbulence in wall bounded domains are presented the navier stokes equations are solved using a fourier pseudo spectral method with volume penalization starting from random initial conditions we study the influence of the geometry of the domain on the flow dynamics in particular on the long time behaviour circular square triangular and annular domains are considered and we show how the geometry plays a crucial role regarding the decay scenario towards final states three stages can be distinguished formation of coherent vortices from random initial conditions vortex wall interactions and finally relaxation towards a quasi steady structure the eigenvalues estimated from the decay rate of both energy and enstrophy depend on the geometry and agree well with the theoretical eigenvalues based on the stokes mode of the corresponding domain for the final states we find a linear functional relation between vorticity and streamfunction

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd