local edge colouring of yao like subgraphs of unit disk graphs the focus of the present paper is on providing a local deterministic algorithm for colouring the edges of yao like subgraphs of unit disk graphs these are geometric graphs such that for some positive integers l k the following property holds at each node v if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v we assume that the nodes are location aware i e they know their cartesian coordinates in the plane the algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant depending on k and l but not on the size of the graph number of hops measured in the original underlying unit disk graph away from it and hence the algorithm terminates in a constant number of steps the number of colours used is 2kl 1 and this is optimal for local algorithms since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm thus showing that in this class of graphs the price for locality is only one additional colour

a governing equation for fluid flow in rough fractures fluid flow in a rock fracture bounded by two irregular surfaces is complex even under a laminar flow regime the major factor causing deviation of predicted fracture flow behavior from the ideal parallel plate theory is the nature of nonparallel and nonsmooth geometry of fracture surfaces important questions on the validity of the cubic law and the reynolds equation for complicated fracture geometries have been studied by many researchers the general conclusion from these efforts is that the cubic law is valid provided that an appropriate average aperture can be defined many average apertures have been proposed and for some cases some work better than others nonetheless to date these efforts have not converged to form a unified definition on the fracture aperture needed in the cubic law which stimulates the current effort to develop a general governing equation for fracture flow from a fundamental consideration in this study a governing equation stemming from the principle of mass conservation and the assumption that the cubic law holds locally is derived for incompressible laminar fluid flow in irregular fractures under steady state conditions the equation is formulated in both local and global coordinates and explicitly incorporates two vectorial variables of fracture geometry true aperture and tortuosity under the assumption of small variations in both tortuosity and aperture the governing equation can be reduced to the reynolds equation two examples are provided to show the importance and generality of the new governing equation in both local and global coordinate systems in a simple fracture with two nonsmooth and nonparallel surfaces the error in permeability estimation can be induced using the reynolds equation with the apparent aperture and can reach 10 for a 25 inclination between the fracture surfaces in a fracture with sinusoidal surfaces the traditional method can cause significant errors in both permeability and pressure calculation

on the convergence of local expansions of layer potentials in a recently developed quadrature method quadrature by expansion or qbx it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off surface points in this paper we derive estimates for the rate of convergence of these local expansions providing the analytic foundation for the qbx method the estimates may also be of mathematical interest particularly for microlocal or asymptotic analysis in potential theory

spectrally accurate fast summation for periodic stokes potentials a spectrally accurate method for the fast evaluation of n particle sums of the periodic stokeslet is presented two different decomposition methods leading to one sum in real space and one in reciprocal space are considered an fft based method is applied to the reciprocal part of the sum invoking the equivalence of multiplications in reciprocal space to convolutions in real space thus using convolutions with a gaussian function to place the point sources on a grid due to the spectral accuracy of the method the grid size needed is low and also in practice for a fixed domain size independent of n the leading cost which is linear in n arises from the to grid and from grid operations combining this fft based method for the reciprocal sum with the direct evaluation of the real space sum a spectrally accurate algorithm with a total complexity of o nlogn is obtained this has been shown numerically as the system is scaled up at constant density

a fast multipole method for the rotne prager yamakawa tensor and its applications we present a fast multipole method fmm for computing sums involving the rotne prager yamakawa tensor the method similar to the approach in tornberg and greengard 2008 26 for the stokeslet decomposes the tensor vector product into a sum of harmonic potentials and fields induced by four different charge and dipole distributions unlike the approach based on the kernel independent fast multipole method ying et al 2004 31 which requires nine scalar fmm calls the method presented here requires only four we discuss its applications to brownian dynamics simulation with hydrodynamic interactions and present some timing results

multiscale temporal integrators for fluctuating hydrodynamics following on our previous work s delong b e griffith e vanden eijnden and a donev phys rev e 87 033302 2013 we develop temporal integrators for solving langevin stochastic differential equations that arise in fluctuating hydrodynamics our simple predictor corrector schemes add fluctuations to standard second order deterministic solvers in a way that maintains second order weak accuracy for linearized fluctuating hydrodynamics we construct a general class of schemes and recommend two specific schemes an explicit midpoint method and an implicit trapezoidal method we also construct predictor corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow time scales we propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics we illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively mixing fluids we first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator we then include the effects of gravity and find that we also need to include the effects of fluid inertia which affects the dynamics of the concentration fluctuations greatly at small wave numbers

numerical quadratures for singular and hypersingular integrals abstract we present a procedure for the design of high order quadrature rules for the numerical evaluation of singular and hypersingular integrals such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions unlike integrals of both smooth and weakly singular functions hypersingular integrals are pseudo differential operators being limits of certain integrals as a result standard quadrature formulae fail for hypersingular integrals on the other hand such expressions are often encountered in mathematical physics see for example 1 and it is desirable to have simple and efficient quadrature formulae for them the algorithm we present constructs high order quadratures for the evaluation of hypersingular integrals the additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals hilbert transforms and integrals involving both smooth and logarithmically singular functions this results in significantly simplified implementations the performance of the procedure is illustrated with several numerical examples

a fluctuating boundary integral method for brownian suspensions abstract we present a fluctuating boundary integral method fbim for overdamped brownian dynamics bd of two dimensional periodic suspensions of rigid particles of complex shape immersed in a stokes fluid we develop a novel approach for generating brownian displacements that arise in response to the thermal fluctuations in the fluid our approach relies on a first kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface with zero mean and covariance proportional to the green s function for stokes flow stokeslet this approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics handles particles of complex shape achieves high order of accuracy and can be generalized to three dimensions and other boundary conditions we show that brownian displacements generated by our method obey the discrete fluctuation dissipation balance relation dfdb based on a recently developed positively split ewald method fiore et al 2017 24 near field contributions to the brownian displacements are efficiently approximated by iterative methods in real space while far field contributions are rapidly generated by fast fourier space methods based on fluctuating hydrodynamics fbim provides the key ingredient for time integration of the overdamped langevin equations for brownian suspensions of rigid particles we demonstrate that fbim obeys dfdb by performing equilibrium bd simulations of suspensions of starfish shaped bodies using a random finite difference temporal integrator

integral equation methods for elastance and mobility problems in two dimensions we present a new unified approach to the solution of the elastance problem in electrostatics and the mobility problem in stokes flow more precisely we construct integral representations that lead to resonance free fredholm integral equations of the second kind and well conditioned linear systems upon discretization by coupling our integral equations with high order quadrature and fast multipole acceleration large scale problems can be solved with only modest computing resources we also discuss some applications of these boundary value problems in applied physics

a fast integral equation method for solid particles in viscous flow using quadrature by expansion boundary integral methods are advantageous when simulating viscous flow around rigid particles due to the reduction in number of unknowns and straightforward handling of the geometry in this work

what can be computed locally the purpose of this paper is a study of computation that can be done locally in a distributed network where locally means within time or distance independent of the size of the network locally checkable labeling lcl problems are considered where the legality of a labeling can be checked locally e g coloring the results include the following there are nontrivial lcl problems that have local algorithms there is a variant of the dining philosophers problem that can be solved locally randomization cannot make an lcl problem local i e if a problem has a local randomized algorithm then it has a local deterministic algorithm it is undecidable in general whether a given lcl has a local algorithm however it is decidable whether a given lcl has an algorithm that operates in a given time t any lcl problem that has a local algorithm has one that is order invariant the algorithm depends only on the order of the processor ids

an integral equation formulation for rigid bodies in stokes flow in three dimensions abstract we present a new derivation of a boundary integral equation bie for simulating the three dimensional dynamics of arbitrarily shaped rigid particles of genus zero immersed in a stokes fluid on which are prescribed forces and torques our method is based on a single layer representation and leads to a simple second kind integral equation it avoids the use of auxiliary sources within each particle that play a role in some classical formulations we use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials so that only a small number of spatial discretization points per particle are required the resulting discrete sums are computed in o n time where n denotes the number of particles using the fast multipole method fmm the particle positions and orientations are updated by a high order time stepping scheme we illustrate the accuracy conditioning and scaling of our solvers with several numerical examples

the fracture flow equation and its perturbation solution 1 this work derives the fracture flow equation from the two dimensional steady form of the navier stokes equation asymptotic solutions are obtained whereby the perturbation parameter is the ratio of the mean width over the length of the fracture segment the perturbation expansion can handle arbitrary variation of the fracture walls as long as the dominant velocity is in the longitudinal direction the effect of the matrix fracture interaction is also taken into account by allowing leakage through the fracture walls the perturbation solution is used to obtain an estimate of the flow rate and the fracture transmissivity as well as the velocity and the pressure distribution in fractures of various geometries the analysis covers eight different configurations of fracture geometry including linear and curvilinear variation as well as sinusoidal variation in the top and bottom walls with varying horizontal alignment and roughness wavelengths the zero order solution yields the reynolds lubrication approximation and the higher order equations provide a correction term to the flow rate in terms of the roughness frequency and the reynolds number for sinusoidal and linear walls the mathematical analysis shows that the zero order flow rate could be expressed in terms of the maximum to minimum width ratio for equal widths at both ends of the fracture the first order correction is zero for sinusoidal fractures the flow rate decreases with increasing reynolds number and with increasing roughness amplitude and frequency the effect of leakage is to create a nonuniform flow distribution in the fracture that deviates significantly from the flow rate estimate for impermeable walls the derived flow expressions can provide a more reliable tool for flow and transport predictions in fractured domain

a silent self stabilizing algorithm for the generalized minimal k dominating set problem abstract we give a silent self stabilizing algorithm for the generalized minimal k dominating set problem in a connected distributed network g given a positive integer k and two sets of processes r d the problem is to find a set d which is minimal subject to the conditions that r d d and that d is k dominating relative to d meaning that every process within distance k of d is also within distance k of d the algorithm is order invariant requires o log n log k space per process works under the distributed unfair scheduler and converges in o n k rounds where n is a given upper bound on the number of processes

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

second kind integral equation formulation of stokes flows past a particle of arbitary shape the problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear fredholm integral equations of the second kind for a

preconditioned krylov subspace methods for sampling multivariate gaussian distributions a common problem in statistics is to compute sample vectors from a multivariate gaussian distribution with zero mean and a given covariance matrix a a canonical approach to the problem is to compute vectors of the form y s z where s is the cholesky factor or square root of a and z is a standard normal vector when a is large such an approach becomes computationally expensive this paper considers preconditioned krylov subspace methods to perform this task the lanczos process provides a means to approximate a 1 2 z for any vector z from an m dimensional krylov subspace the main contribution of this paper is to show how to enhance the convergence of the process via preconditioning both incomplete cholesky preconditioners and approximate inverse preconditioners are discussed it is argued that the latter class of preconditioners has an advantage in the context of sampling numerical tests performed with stationary covariance matrices used to model gaussian processes illustrate

a nystrom method for weakly singular integral operators on surfaces we describe a modified nystrom method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations standard nystrom and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes our method uses appropriately scaled function values in lieu of such representations this results in a scheme which is mathematically equivalent to galerkin discretization in that the resulting matrices are related to those obtained by galerkin methods via conjugation with well conditioned matrices but which avoids the evaluation of double integrals moreover we incorporate a new mechanism for approximating the singular integrals which arise from the discretization of weakly singular integral operators which is considerably more efficient than standard methods we illustrate the performance of our method with numerical experiments

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

spectral ewald acceleration of stokesian dynamics for polydisperse suspensions in this work we develop the spectral ewald accelerated stokesian dynamics seasd a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions seasd is based on the framework of stokesian dynamics sd with extension to compressible solvents and uses the spectral ewald se method lindbo and tornberg 2010 29 for the wave space mobility computation to meet the performance requirement of dynamic simulations we use graphic processing units gpu to evaluate the suspension mobility and achieve an order of magnitude speedup compared to a cpu implementation for further speedup we develop a novel far field block diagonal preconditioner to reduce the far field evaluations in the iterative solver and seasd nf a polydisperse extension of the mean field brownian approximation of banchio and brady 2003 39 we extensively discuss implementation and parameter selection strategies in seasd and demonstrate the spectral accuracy in the mobility evaluation and the overall o n log n computation scaling we present three computational examples to further validate seasd and seasd nf in monodisperse and bidisperse suspensions the short time transport properties the equilibrium osmotic pressure and viscoelastic moduli and the steady shear brownian rheology our validation results show that the agreement between seasd and seasd nf is satisfactory over a wide range of parameters and also provide significant insight into the dynamics of polydisperse colloidal suspensions

deterministic local algorithms unique identifiers and fractional graph colouring in the fractional graph colouring problem the task is to schedule the activities of the nodes so that each node is active for 1 time unit in total and at each point of time the set of active nodes forms an independent set we show that for any 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most 1 in any graph in one synchronous communication round here is the maximum degree of the graph the result is near tight as there are graphs in which the optimal solution has length 1 the result is of course too good to be true the usual definitions of scheduling problems fractional graph colouring fractional domatic partition etc in a distributed setting leave a loophole that can be exploited in the design of distributed algorithms the size of the local output is not bounded our algorithm produces an output that seems to be perfectly good by the usual standards but it is impractical as the schedule of each node consists of a very large number of short periods of activity more generally the algorithm demonstrates that when we study distributed algorithms for scheduling problems we can choose virtually any trade off between the following three parameters t the running time of the algorithm the length of the schedule and the maximum number of periods of activity for any single node here is the objective function of the optimisation problem while captures the subjective quality of the solution if we study for example bounded degree graphs we can trivially keep t and constant at the cost of a large or we can keep and constant at the cost of a large t our algorithm shows that yet another trade off is possible we can keep t and constant at the cost of a large

fluid flow in synthetic rough walled fractures navier stokes stokes and local cubic law simulations 1 the results of three dimensional navier stokes ns and stokes simulations and two dimensional local cubic law lcl simulations of fluid flow through single rough walled fractures are presented synthetic rough walled fractures were created by combining random fields of aperture and the mean wall topography or midsurface which quantifies undulation about the fracture plane a finite volume formulation of the lcl that incorporates geometric corrections for fracture undulation is presented simulations of fluid flow through planar fractures with sinusoidal variation in aperture were compared to published results the rough walled fracture simulations demonstrated that the total flow rates predicted by the corrected lcl were within 10 of those predicted by the stokes equations for all the fractures examined in this work differences between the ns and stokes simulations clearly demonstrated that inertial forces can significantly influence the internal flow field within a fracture and the total flow rate across a fracture by limiting the total flow rate differences between the ns and stokes simulations constraints for three kinematic parameters were determined for all the fractures presented in this work the corrected lcl was determined to be an acceptable approximation to the ns equations provided that the kinematic and geometric constraints were met

spectral accuracy in fast ewald based methods for particle simulations a spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented we follow the established framework of fft based ewald summation but obtain a method with an important decoupling of errors it is shown for the proposed method that the error due to frequency domain truncation can be separated from the approximation error added by the fast method this has the significance that the truncation of the underlying ewald sum prescribes the size of the grid used in the fft based fast method which clearly is the minimal grid both errors are of exponential squared order and the latter can be controlled independently of the grid size we compare numerically to the established spme method by essmann et al and see that the memory required can be reduced by orders of magnitude we also benchmark efficiency i e error as a function of computing time against the spme method which indicates that our method is competitive analytical error estimates are proven and used to select parameters with a great degree of reliability and ease

local computation lower and upper bounds the question of what can be computed and how efficiently is at the core of computer science not surprisingly in distributed systems and networking research an equally fundamental question is what can be computed in a distributed fashion more precisely if nodes of a network must base their decision on information in their local neighborhood only how well can they compute or approximate a global optimization problem in this paper we give the first polylogarithmic lower bound on such local computation for optimization problems including minimum vertex cover minimum connected dominating set maximum matching maximal independent set and maximal matching in addition we present a new distributed algorithm for solving general covering and packing linear programs for some problems this algorithm is tight with the lower bounds whereas for others it is a distributed approximation scheme together our lower and upper bounds establish the local computability and approximability of a large class of problems characterizing how much local information is required to solve these tasks

fluctuating force coupling method for simulations of colloidal suspensions abstract the resolution of brownian motion in simulations of micro particle suspensions can be crucial to reproducing the correct dynamics of individual particles as well as providing an accurate characterisation of suspension properties including these effects in simulations however can be computationally intensive due to the configuration dependent random displacements that would need to be determined at every time step in this paper we introduce the fluctuating force coupling method fcm to overcome this difficulty providing a fast approach to simulate colloidal suspensions at large scale we show explicitly that by forcing the surrounding fluid with a fluctuating stress and employing the fcm framework to obtain the motion of the particles one obtains the random particle velocities and angular velocities that satisfy the fluctuation dissipation theorem this result holds even when higher order multipoles such as stresslets are included in the fcm approximation through several numerical experiments we confirm our analytical results and demonstrate the effectiveness of fluctuating fcm showing also how brownian drift can be resolved by employing the appropriate time integration scheme and conjugate gradient method

lower bounds for local approximation in the study of deterministic distributed algorithms it is commonly assumed that each node has a unique o log n bit identifier we prove that for a general class of graph problems local algorithms constant time distributed algorithms do not need such identifiers a port numbering and orientation is sufficient our result holds for so called simple po checkable graph optimisation problems this includes many classical packing and covering problems such as vertex covers edge covers matchings independent sets dominating sets and edge dominating sets we focus on the case of bounded degree graphs and show that if a local algorithm finds a constant factor approximation of a simple po checkable graph problem with the help of unique identifiers then the same approximation ratio can be achieved on anonymous networks as a corollary of our result we derive a tight lower bound on the local approximability of the minimum edge dominating set problem by prior work there is a deterministic local algorithm that achieves the approximation factor of 4 1 2 in graphs of maximum degree this approximation ratio is known to be optimal in the port numbering model our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier our main technical tool is an algebraic construction of homogeneously ordered graphs we say that a graph is r homogeneous if its nodes are linearly ordered so that an fraction of nodes have pairwise isomorphic radius r neighbourhoods we show that there exists a finite r homogeneous 2k regular graph of girth at least g for any

a fast multipole method for the three dimensional stokes equations many problems in stokes flow and linear elasticity require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions in the fluid mechanics setting these are typically the stokeslet the kernel of the single layer potential or the stresslet the kernel of the double layer potential in this paper we present a simple and efficient method for the rapid evaluation of such fields using a decomposition into a small number of coulombic n body problems following an approach similar to that of fu and rodin y fu g j rodin fast solution methods for three dimensional stokesian many particle problems commun numer meth en 16 2000 145 149 while any fast summation algorithm for coulombic interactions can be employed we present numerical results from a scheme based on the most modern version of the fast multipole method h cheng l greengard v rokhlin a fast adaptive multipole algorithm in three dimensions j comput phys 155 1999 468 498 this approach should be of value in both the solution of boundary integral equations and multiparticle dynamics