least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

rigidity and energy suppose one holds two sticks in the form of a cross in one hand and places a rubber band in tension around the four ends when it is released it comes to rest in the shape of a convex quadrilateral in a plane it always returns to the same shape no matter how it is distorted as long as the ends do not slip and the rubber band does not break this is a very simple example of a rigid framework of the type we discuss here energy explains why a framework such as the one above is rigid the rubber band deforms in such a way as to minimize the total energy in the framework only in the final deformed shape will the framework have a minimum energy this idea of introducing energy functions is very useful it can be used to prove a key lemma that was used by cauchy 5 in 1813 to show that convex polyhedral surfaces are rigid it can explain why some and perhaps all of r buckminster fuller s tensegrity structures 10 stay up when applied to spider webs it shows why they can only take on certain geometric shapes it also can be used to prove conjecture 6 of branko gri inbaum and g c shephard in their lectures on lost mathematics 12 this conjecture says that if a framework in the shape of a convex polygon with rods sticks on the boundary and cables inside is rigid in the plane so is the framework obtained by reversing the roles of rods and cables it is interesting to compare some of these results with the opening a rm theorem of axel schur 18 see chern 6 also which is a very close smooth analogue to cauchy s lemma schur s theorem says that if a convex planar smooth arc the arm is opened that is it is moved to another position with the same length but with corresponding points having smaller curvature then the two ends are moved apart another amusing application is to show that a regular pentagon in 4 space a pentagon with all 5 sides equal and all 5 angles between the sides equal has its angles bounded between 36 and 108 a comment of o bottema in 4

some least squares galerkin procedures for first order time dependent convection diffusion system abstract four least squares galerkin finite element schemes are formulated to solve first order time dependent convection diffusion systems and the convergence of these schemes is analyzed

asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

a family of h div finite element approximations on polygonal meshes in this paper we present a family of h div compatible finite element spaces on strictly convex n gons whose construction makes use of generalized barycentric coordinates in particular for integers 0 leq k leq2 we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise k th order polynomial normal traces along the element boundary these spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and as such their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence we present a general remedy for restoration of optimal convergence that involves polynomial corrections of vector fields and their divergence at the element level these corrections are consistent with one another and require computation of suitable polynomial projection maps in addition to the theo

exponential convergence of a distributed algorithm for solving linear algebraic equations abstract in a recent paper a distributed algorithm was proposed for solving linear algebraic equations of the form a x b assuming that the equation has at least one solution the equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix a b the current estimates of the equation s solution generated by each of its neighbors and nothing more neighbor relationships are represented by a time dependent directed graph n t whose vertices correspond to agents and whose arcs characterize neighbor relationships sufficient conditions on n t were derived under which the algorithm can cause all agents estimates to converge exponentially fast to the same solution to a x b these conditions were also shown to be necessary for exponential convergence provided the data about a b available to the agents is non redundant the aim of this paper is to relax this non redundant assumption this is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand the conditions are based on a new notion of graph connectivity an improved bound on the convergence rate is also derived

distributed nonconvex constrained optimization over time varying digraphs this paper considers nonconvex distributed constrained optimization over networks modeled as directed possibly time varying graphs we introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex nonseparable function the agent s sum utility plus a difference of convex function with nonsmooth convex part this general formulation arises in many applications from statistical machine learning to engineering the proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push sum consensus mechanism that aims to track locally the gradient of the smooth part of the sum utility sublinear convergence rate is proved when a fixed step size possibly different among the agents is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step size numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

distributed non convex first order optimization and information processing lower complexity bounds and rate optimal algorithms we consider a class of popular distributed non convex optimization problems in which agents connected by a network mathcal g collectively optimize a sum of smooth possibly non convex local objective functions we address the following question if the agents can only access the gradients of local functions what are the fastest rates that any distributed algorithms can achieve and how to achieve those rates first we show that there exist difficult problem instances such that it takes a class of distributed first order methods at least mathcal o 1 sqrt xi mathcal g times bar l epsilon communication rounds to achieve certain epsilon solution where xi mathcal g denotes the spectral gap of the graph laplacian matrix and bar l is some lipschitz constant second we propose near optimal methods whose rates match the developed lower rate bound up to a ploylog factor the key in the algorithm design is to properly embed the classical polynomial filtering techniques into modern first order algorithms to the best of our knowledge this is the first time that lower rate bounds and optimal methods have been developed for distributed non convex optimization problems

communication efficient algorithms for decentralized and stochastic optimization we present a new class of decentralized first order methods for nonsmooth and stochastic optimization problems defined over multiagent networks considering that communication is a major bottleneck in decentralized optimization our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter node communications our major contribution is to present a new class of decentralized primal dual type algorithms namely the decentralized communication sliding dcs methods which can skip the inter node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions by employing dcs agents can find an epsilon solution both in terms of functional optimality gap and feasibility residual in mathcal o 1 epsilon resp mathcal o 1 sqrt epsilon communication rounds for general convex functions resp strongly convex functions while maintaining the mathcal o 1 epsilon 2 resp mathcal o 1 epsilon bound on the total number of intra node subgradient evaluations we also present a stochastic counterpart for these algorithms denoted by sdcs for solving stochastic optimization problems whose objective function cannot be evaluated exactly in comparison with existing results for decentralized nonsmooth and stochastic optimization we can reduce the total number of inter node communication rounds by orders of magnitude while still maintaining the optimal complexity bounds on intra node stochastic subgradient evaluations the bounds on the stochastic subgradient evaluations are actually comparable to those required for centralized nonsmooth and stochastic optimization under certain conditions on the target accuracy

jamming graphs a local approach to global mechanical rigidity we revisit the concept of minimal rigidity as applied to frictionless repulsive soft sphere packings in two dimensions with the introduction of the jamming graph minimal rigidity is a purely combinatorial property encoded via laman s theorem in two dimensions it constrains the global average coordination number of the graph for example however minimal rigidity does not address the geometry of local mechanical stability the jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability we demonstrate how jamming graphs can be constructed using local moves via the henneberg construction such that these graphs fall under the jurisdiction of correlated percolation we then probe how jamming graphs destabilize or become unjammed by deleting a bond and computing the resulting rigid cluster distribution we also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra redundant contacts destabilizes the latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut out or frozen in subsystems

convergence analysis for least squares finite element approximations of second order two point boundary value problems in this paper a c 0 least squares finite element method for second order two point boundary value problems is considered the problem is recast as a first order system standard and improved optimal error estimates in maximum norms are established superconvergence estimates at interelement lobatto and gauss points are developed numerical experiments are given to illustrate theoretical results

the proximal augmented lagrangian method for nonsmooth composite optimization we study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term we introduce an auxiliary variable to separate the objective function components and utilize the moreau envelope of the regularization term to derive the proximal augmented lagrangian a continuously differentiable function obtained by constraining the augmented lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term the continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers mm to compute optimal primal dual pairs by solving a sequence of differentiable problems the mm algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step size update rules than the alternating direction method of multipliers admm these features make it an attractive option for solving structured optimal control problems we also develop an algorithm based on the primal descent dual ascent gradient method and prove global exponential asymptotic stability when the differentiable component of the objective function is strongly convex and the regularization term is convex finally we identify classes of problems for which the primal dual gradient flow dynamics are convenient for distributed implementation and compare contrast our framework to the existing approaches

dsa decentralized double stochastic averaging gradient algorithm this paper considers optimization problems where nodes of a network have access to summands of a global objective each of these local objectives is further assumed to be an average of a finite set of functions the motivation for this setup is to solve large scale machine learning problems where elements of the training set are distributed to multiple computational elements the decentralized double stochastic averaging gradient dsa algorithm is proposed as a solution alternative that relies on i the use of local stochastic averaging gradients ii determination of descent steps as differences of consecutive stochastic averaging gradients strong convexity of local functions and lipschitz continuity of local gradients is shown to guarantee linear convergence of the sequence generated by dsa in expectation local iterates are further shown to approach the optimal argument for almost all realizations the expected linear convergence of dsa is in contrast to the sublinear rate characteristic of existing methods for decentralized stochastic optimization numerical experiments on a logistic regression problem illustrate reductions in convergence time and number of feature vectors processed until convergence relative to these other alternatives

calculating the free energy of nearly jammed hard particle packings using molecular dynamics we present a new event driven molecular dynamics md algorithm for measuring the free energy of nearly jammed packings of spherical and non spherical hard particles this bounding cell molecular dynamics bcmd algorithm exactly calculates the free energy of a single occupancy cell soc model in which each particle is restricted to a neighborhood of its initial position using a hard wall bounding cell our md algorithm generalizes previous ones in the literature by enabling us to study non spherical particles as well as to measure the free energy change during continuous irreversible transformations moreover we make connections to the well studied problem of computing the volume of convex bodies in high dimensions using random walks we test and verify the numerical accuracy of the method by comparing against rigorous asymptotic results for the free energy of jammed and isostatic disordered packings of both hard spheres and ellipsoids for which the free energy can be calculated directly as the volume of a high dimensional simplex we also compare our results to previously published monte carlo results for hard sphere crystals near melting and jamming and find excellent agreement we have successfully used the bcmd algorithm to determine the configurational and free volume contributions to the free energy of glassy states of binary hard disks a donev f h stillinger s torquato do binary hard disks exhibit an ideal glass transition phys rev lett 96 22 2006 225502 the algorithm can also be used to determine phases with locally or globally minimal free energy to calculate the free energy cost of point and extended crystal defects or to calculate the elastic moduli of glassy or crystalline solids among other potential applications

decentralized optimization over time varying directed graphs with row and column stochastic matrices in this article we provide a distributed optimization algorithm termed as tv mathcal ab that minimizes a sum of convex functions over time varying random directed graphs contrary to the existing work the algorithm we propose does not require eigenvector estimation to estimate the non mathbf 1 perron eigenvector of a stochastic matrix instead the proposed approach relies on a novel information mixing approach that exploits both row and column stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed we show that tv mathcal ab converges linearly to the optimal solution when the global objective is smooth and strongly convex and the underlying time varying graphs exhibit bounded connectivity i e a union of every c consecutive graphs is strongly connected we derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument simulations confirm the findings in this article

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles we apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids the theoretical machinery needed to treat such particles including the overlap potentials is developed in full detail we describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids we present performance results for our implementation of the algorithm demonstrating that for dense systems of very aspherical ellipsoids the novel techniques of using neighbor lists and bounding sphere complexes offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event driven molecular dynamics algorithms the practical utility of the algorithm is demonstrated by presenting several interesting physical applications including the generation of jammed packings inside spherical containers the study of contact force chains in jammed packings and melting the densest known equilibrium crystals of prolate spheroids

constrained consensus algorithms with fixed step size for distributed convex optimization over multiagent networks in this technical note we are concerned with constrained consensus algorithms for distributed convex optimization with a sum of convex objective functions subject to local bound and equality constraints in multiagent networks each agent has its own data on objective function and constraints all the agents cooperatively find the minimizer while each agent can only communicate with its neighbors the consensus of multiagent networks with time invariant and undirected graphs is proven by the lyapunov method compared with existing consensus algorithms for distributed optimization with diminishing step sizes the proposed algorithms with fixed step size have better convergence rate simulation results on a numerical example are presented to substantiate the performance and characteristics of the proposed algorithms

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

a proximal gradient algorithm for decentralized composite optimization this paper proposes a decentralized algorithm for solving a consensus optimization problem defined in a static networked multi agent system where the local objective functions have the smooth nonsmooth composite form examples of such problems include decentralized constrained quadratic programming and compressed sensing problems as well as many regularization problems arising in inverse problems signal processing and machine learning which have decentralized applications this paper addresses the need for efficient decentralized algorithms that take advantages of proximal operations for the nonsmooth terms we propose a proximal gradient exact first order algorithm pg extra that utilizes the composite structure and has the best known convergence rate it is a nontrivial extension to the recent algorithm extra at each iteration each agent locally computes a gradient of the smooth part of its objective and a proximal map of the nonsmooth part as well as exchanges information with its neighbors the algorithm is exact in the sense that an exact consensus minimizer can be obtained with a fixed step size whereas most previous methods must use diminishing step sizes when the smooth part has lipschitz gradients pg extra has an ergodic convergence rate of o left 1 over k right in terms of the first order optimality residual when the smooth part vanishes pg extra reduces to p extra an algorithm without the gradients so no g in the name which has a slightly improved convergence rate at o left 1 over k right in a standard non ergodic sense numerical experiments demonstrate effectiveness of pg extra and validate our convergence results

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

goal oriented local a posteriori error estimators for h div least squares finite element methods we propose a goal oriented local a posteriori error estimator for h div least squares ls finite element methods our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest d subset omega the estimator is obtained from the ls functional by scaling residuals with proper weight coefficients the weight coefficients are given in terms of local mesh size h t and a function omega d depending on the distance to d this new error estimator measures the pollution effect from the outside region of d and provides a basis for local refinement in order to efficiently approximate the solution in d numerical experiments show superior performances of our goal oriented a posteriori estimators over the standard ls functional and global error estimators

a fast proximal gradient algorithm for decentralized composite optimization over directed networks abstract this paper proposes a fast decentralized algorithm for solving a consensus optimization problem defined in a directed networked multi agent system where the local objective functions have the smooth nonsmooth composite form and are possibly nonconvex examples of such problems include decentralized compressed sensing and constrained quadratic programming problems as well as many decentralized regularization problems we extend the existing algorithms pg extra and extrapush to a new algorithm pg extrapush for composite consensus optimization over a directed network this algorithm takes advantage of the proximity operator like in pg extra to deal with the nonsmooth term and employs the push sum protocol like in extrapush to tackle the bias introduced by the directed network with a proper step size we show that pg extrapush converges to an optimal solution at a linear rate under some regular assumptions we conduct a series of numerical experiments to show the effectiveness of the proposed algorithm specifically with a proper step size pg extrapush performs linear rates in most of cases even in some nonconvex cases and is significantly faster than subgradient push even if the latter uses a hand optimized step size the established theoretical results are also verified by the numerical results

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles i algorithmic details in this first part of a series of two papers we present in considerable detail a collision driven molecular dynamics algorithm for a system of non spherical particles within a parallelepiped simulation domain under both periodic or hard wall boundary conditions the algorithm extends previous event driven molecular dynamics algorithms for spheres and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size we present a novel partial update near neighbor list nnl algorithm that is superior to previous algorithms at high densities without compromising the correctness of the algorithm this efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes bsc these techniques will be useful in any particle based simulation including monte carlo and time driven molecular dynamics additionally we allow for a non vanishing rate of deformation of the boundary which can be used to model macroscopic strain and also alleviate boundary effects for small systems in the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation demonstrating the practical utility of the algorithm

on different finite element methods for approximating the gradient of the solution to the helmholtz equation abstract we consider the numerical solution of the helmholtz equation by different finite element methods in particular we are interested in finding an efficient method for approximating the gradient of the solution we first approximate the gradient by the standard ritz galerkin method as a second method a two stage method due to aziz and werschulz 1980 is presented it is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case finally a direct method with asymptotic error estimates is given it turns out that the presented direct method is of lowest computational complexity test examples are presented to illustrate the accuracy of the methods

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

distributed linearized alternating direction method of multipliers for composite convex consensus optimization given an undirected graph mathcal g mathcal n mathcal e of agents mathcal n lbrace 1 ldots n rbrace connected with edges in mathcal e we study how to compute an optimal decision on which there is consensus among agents and that minimizes the sum of agent specific private convex composite functions lbrace phi i rbrace i in mathcal n where phi i triangleq xi i f i belongs to agent i assuming only agents connected by an edge can communicate we propose a distributed proximal gradient algorithm dpga for consensus optimization over both unweighted and weighted static undirected communication networks in one iteration each agent i computes the prox map of xi i and gradient of f i and this is followed by local communication with neighboring agents we also study its stochastic gradient variant sdpga which can only access to noisy estimates of nabla f i at each agent i this computational model abstracts a number of applications in distributed sensing machine learning and statistical inference we show ergodic convergence in both suboptimality error and consensus violation for the dpga and sdpga with rates mathcal o 1 t and mathcal o 1 sqrt t respectively

error estimates for least squares finite element methods abstract a least squares finite element scheme for a boundary value problem associated with a second order partial differential equation is considered previous work on this subject is generalized and improved by considering a larger class of equations by working in the natural context without additional smoothness conditions and by deriving error estimates not only in the h1 norm and the hdiv norm but also in the l2 norm some of these estimates are sharpened by using finite element spaces with the grid decomposition property the error estimates are supported by numerical results which extend previous numerical work

a linear programming algorithm to test for jamming in hard sphere packings jamming in hard particle packings has been the subject of considerable interest in recent years in a paper by torquato and stillinger j phys chem b 105 2001 a classification scheme of jammed packings into hierarchical categories of locally collectively and strictly jammed configurations has been proposed they suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements but leave the design of such algorithms as a future task in this work we present a rigorous and practical algorithm to assess whether an ideal hard sphere packing in two or three dimensions is jammed according to the aforementioned categories the algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard wall and periodic boundary conditions if the packing is not jammed the algorithm yields representative multi particle unjamming motions furthermore we extend the jamming categories and the testing algorithm to packings with significant interparticle gaps we describe in detail two variants of the proposed randomized linear programming approach to test for jamming in hard sphere packings the first algorithm treats ideal packings in which particles form perfect contacts another algorithm treats the case of jamming in packings with significant interparticle gaps this extended algorithm allows one to explore more fully the nature of the feasible particle displacements we have implemented the algorithms and applied them to ordered as well as random packings of circular disks and spheres with periodic boundary conditions some representative results for large disordered disk and sphere packings are given but more robust and efficient implementations as well as further applications e g non spherical particles are anticipated for the future

a remark on least squares mixed element methods for reaction diffusion problems in this paper we propose a least squares mixed element procedure for a reaction diffusion problem based on the first order system by selecting the least squares functional properly the resulting procedure can be split into two independent symmetric positive definite schemes one of which is for the unknown variable and the other of which is for the unknown flux variable which lead to the optimal order h 1 w and l 2 w norm error estimates for the primal unknown and optimal h div w norm error estimate for the unknown flux finally we give some numerical examples

a novel least squares finite element method enriched with residual free bubbles for solving convection dominated problems in this paper we devise a novel least squares finite element method lsfem for solving scalar convection dominated convection diffusion problems first we convert a second order convection diffusion problem into a first order system formulation by introducing the gradient mathbf p kappa nabla u as an additional variable the lsfem using continuous piecewise linear elements enriched with residual free bubbles for both variables u and mathbf p is applied to solve the first order mixed problem the residual free bubble functions are assumed to strongly satisfy the associated homogeneous second order convection diffusion equations in the interior of each element up to the contribution of the linear part and vanish on the element boundary to implement this two level least squares approach a stabilized method of galerkin least squares type is used to approximate the residual free bubble functions this enriched lsfem not only inherits the advantages of the primitive lsfem such as the resulting linear system being symmetric and positive definite but also exhibits the characteristics of the residual free bubble method without involving stability parameters several numerical experiments are given to demonstrate the effectiveness of the proposed enriched lsfem the accuracy and computational cost of this enriched lsfem are also compared with those of the primitive lsfem we find that for a small diffusivity kappa the enriched lsfem is much better able to capture the nature of layer structure in the solution than the primitive lsfem even if the primitive lsfem uses a very fine mesh or higher order elements in other words the enriched lsfem provides a significant improvement with a lower computational cost over the primitive lsfem for solving convection dominated problems

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

data dependent convergence for consensus stochastic optimization we study a distributed consensus based stochastic gradient descent sgd algorithm and show that the rate of convergence involves the spectral properties of two matrices the standard spectral gap of a weight matrix from the network topology and a new term depending on the spectral norm of the sample covariance matrix of the data this data dependent convergence rate shows that distributed sgd algorithms perform better on datasets with small spectral norm our analysis method also allows us to find data dependent convergence rates as we limit the amount of communication spreading a fixed amount of data across more nodes slows convergence for asymptotically growing datasets we show that adding more machines can help when minimizing twice differentiable losses

perturbed proximal primal dual algorithm for nonconvex nonsmooth optimization in this paper we propose a perturbed proximal primal dual algorithm pprox pda for an important class of linearly constrained optimization problems whose objective is the sum of smooth possibly nonconvex and convex possibly nonsmooth functions this family of problems can be used to model many statistical and engineering applications such as high dimensional subspace estimation and the distributed machine learning the proposed method is of the uzawa type in which a primal gradient descent step is performed followed by an approximate dual gradient ascent step one distinctive feature of the proposed algorithm is that the primal and dual steps are both perturbed appropriately using past iterates so that a number of asymptotic convergence and rate of convergence results to first order stationary solutions can be obtained finally we conduct extensive numerical experiments to validate the effectiveness of the proposed algorithm

stochastic proximal gradient consensus over random networks we consider solving a convex optimization problem with possibly stochastic gradient and over a randomly time varying multiagent network each agent has access to some local objective function and it only has unbiased estimates of the gradients of the smooth component we develop a dynamic stochastic proximal gradient consensus algorithm with the following key features 1 it works for both the static and certain randomly time varying networks 2 it allows the agents to utilize either the exact or stochastic gradient information 3 it is convergent with provable rate in particular the proposed algorithm converges to a global optimal solution with a rate of mathcal o 1 r resp mathcal o 1 sqrt r when the exact resp stochastic gradient is available where r is the iteration counter interestingly the developed algorithm establishes a close connection among a number of seemingly unrelated distributed algorithms such as the extra the pg extra the ic idc admm the dlm and the classical distributed subgradient method

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

distributed stochastic gradient tracking methods in this paper we study the problem of distributed multi agent optimization over a network where each agent possesses a local cost function that is smooth and strongly convex the global objective is to find a common solution that minimizes the average of all cost functions assuming agents only have access to unbiased estimates of the gradients of their local cost functions we consider a distributed stochastic gradient tracking method dsgt and a gossip like stochastic gradient tracking method gsgt we show that in expectation the iterates generated by each agent are attracted to a neighborhood of the optimal solution where they accumulate exponentially fast under a constant stepsize choice under dsgt the limiting expected error bounds on the distance of the iterates from the optimal solution decrease with the network size n which is a comparable performance to a centralized stochastic gradient algorithm moreover we show that when the network is well connected gsgt incurs lower communication cost than dsgt while maintaining a similar computational cost numerical example further demonstrates the effectiveness of the proposed methods

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

numerical analysis of a least squares finite element method for the time dependent advection diffusion equation a mixed finite element scheme designed for solving the time dependent advection diffusion equations expressed in terms of both the primal unknown and its flux incorporating or not a reaction term is studied once a time discretization of the crank nicholson type is performed the resulting system of equations allows for a stable approximation of both fields by means of classical lagrange continuous piecewise polynomial functions of arbitrary degree in any space dimension convergence in the norm of h 1xh div in space and in appropriate senses in time applying to this pair of fields is demonstrated

on mass conservation in least squares methods we compare three least squares finite element reformulations of the stokes equations paying particular attention to mass conservation the first problem we approximate has a simple analytical solution over a convex region even for this simple problem without special treatment of the conservation of mass term very poor numerical solutions may result sufficiently weighting this term leads to a dramatic improvement in the results over a range of test problems

a least squares virtual element method for second order elliptic problems abstract in this paper a least squares virtual element method is presented for approximating the vector and scalar variables of second order elliptic problems the h div conforming and scalar conforming virtual elements are used to approximate the vector and scalar variables respectively the method allows the use of very general polygonal meshes and leads to a symmetric positive definite system the optimal a priori error estimates are established for the vector variable in h div norm and the scalar variable in h 1 norm a simple a posteriori error estimator is also presented and proved to be reliable and efficient the virtual element method handles the hanging nodes naturally thus the local mesh post processing to remove hanging nodes is not required numerical experiments are conducted to verify the accuracy of the method and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy

on finite element approximation of the gradient for solution of poisson equation a nonconforming mixed finite element method is presented for approximation of w with w f w r 0 convergence of the order left nabla w u h right 0 omega mathcal o h 2 is proved when linear finite elements are used only the standard regularity assumption on triangulations is needed

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

dextra a fast algorithm for optimization over directed graphs this paper develops a fast distributed algorithm termed dextra to solve the optimization problem when n agents reach agreement and collaboratively minimize the sum of their local objective functions over the network where the communication between the agents is described by a directed graph existing algorithms solve the problem restricted to directed graphs with convergence rates of o ln k sqrt k for general convex objective functions and o ln k k when the objective functions are strongly convex where k is the number of iterations we show that with the appropriate step size dextra converges at a linear rate o tau k for 0 given that the objective functions are restricted strongly convex the implementation of dextra requires each agent to know its local out degree simulation examples further illustrate our findings

least squares methods for optimal shape design problems abstract a least squares method for solving an optimal shape design problem which appears in semiconductor device physics is described discretization by finite element methods is used for numerical solving an example and experimental results are presented

a least squares finite element method for the navier stokes equations the navier stokes equations for flow in a plane are reformulated as a first order system in terms of stress and stream functions solutions of this system are obtained by the least squares finite element method a feature of this approach is that the linearised system gives rise to a symmetric and positive definite linear algebra problem at each newton iteration care over handling the incompressibility term is needed to ensure good results are obtained

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

convergence rate of distributed admm over networks we propose a new distributed algorithm based on alternating direction method of multipliers admm to minimize sum of locally known convex functions using communication over a network this optimization problem emerges in many applications in distributed machine learning and statistical estimation our algorithm allows for a general choice of the communication weight matrix which is used to combine the iterates at different nodes we show that when functions are convex both the objective function values and the feasibility violation converge with rate o 1 t where t is the number of iterations we then show that when functions are strongly convex and have lipschitz continuous gradients the sequence generated by our algorithm converges linearly to the optimal solution in particular an epsilon optimal solution can be computed with o left sqrt kappa f log 1 epsilon right iterations where kappa f is the condition number of the problem our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes the smallest nonzero eigenvalue and operator norm of the communication matrix

distributed least squares solver for network linear equations in this paper we study the problem of finding the least square solutions of over determined linear algebraic equations over networks in a distributed manner each node has access to one of the lin

an h 1 galerkin mixed finite element method for an evolution equation with a positive type memory term an h1 galerkin mixed finite element method is analyzed for a class of evolution equations with memory when a classical mixed method is applied to such problems it has not been possible to obtain any estimate for the flux however the proposed approach yields optimal order convergence without the lbb consistency condition and quasi uniformity of the finite element mesh compared to the results proved for one space variable the l2 estimate of the flux is not optimal for problems in two and three space dimensions therefore a modification of the method is proposed and analyzed a maximum norm estimate is also derived in one and two space variables a backward euler approximation of the modified method is also analyzed

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

first order system cl cl fosll scalar elliptic partial differential equations the l2 norm version of first order system least squares fosls attempts to reformulate a given system of partial differential equations so that applying a least squares principle yields a functional whose bilinear part is h1 elliptic this ellipticity means that the minimization process amounts to solving a weakly coupled system of poisson like scalar equations an unfortunate limitation of the l2 norm fosls approach is that this product h1 equivalence generally requires sufficient smoothness of the original problem inverse norm fosls overcomes this limitation but at a substantial loss of real efficiency the fosll approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations this paper provides a theoretical foundation for the fosll methodology and illustrates its performance by applying it numerically to several examples results for the so called two stage approach applied to discontinuous coefficient problems show promising robustness and optimality indeed fosll appears to exhibit the generality of the inverse norm fosls approach while retaining the full efficiency of the l2 norm approach

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

convergence rate analysis of distributed optimization with projected subgradient algorithm abstract in this paper we revisit the consensus based projected subgradient algorithm proposed for a common set constraint we show that the commonly adopted non summable and square summable diminishing step sizes of subgradients can be relaxed to be only non summable if the constrained optimum set is bounded more importantly for a strongly convex aggregate cost with different types of step sizes we provide a systematical analysis to derive the asymptotic upper bound of convergence rates in terms of the optimum residual and select the best step sizes accordingly our result shows that a convergence rate of o 1 k can be achieved with a step size o 1 k

extra an exact first order algorithm for decentralized consensus optimization recently there has been growing interest in solving consensus optimization problems in a multiagent network in this paper we develop a decentralized algorithm for the consensus optimization problem mathrm minimize x in mathbb r p bar f x frac 1 n sum i 1 n f i x which is defined over a connected network of n agents where each function f i is held privately by agent i and encodes the agent s data and objective all the agents shall collaboratively find the minimizer while each agent can only communicate with its neighbors such a computation scheme avoids a data fusion center or long distance communication and offers better load balance to the network this paper proposes a novel decentralized exact first order algorithm abbreviated as extra to solve the consensus optimization problem exact means that it can converge to the exact solution extra uses a fixed large step size which can be determined independently of the network size or topology the local variable of every a

distributed heavy ball a generalization and acceleration of first order methods with gradient tracking we study distributed optimization to minimize a sum of smooth and strongly convex functions recent work on this problem uses gradient tracking to achieve linear convergence to the exact global minimizer however a connection among different approaches has been unclear in this paper we first show that many of the existing first order algorithms are related with a simple state transformation at the heart of which lies a recently introduced algorithm known as mathcal ab we then present distributed heavy ball denoted as mathcal ab m that combines mathcal ab with a momentum term and uses nonidentical local step sizes by simultaneously implementing both row and column stochastic weights mathcal ab m removes the conservatism in the related work due to doubly stochastic weights or eigenvector estimation mathcal ab m thus naturally leads to optimization and average consensus over both undirected and directed graphs we show that mathcal ab m has a global r linear rate when the largest step size and momentum parameter are positive and sufficiently small we numerically show that mathcal ab m achieves acceleration particularly when the objective functions are ill conditioned

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

exponential convergence of distributed primal dual convex optimization algorithm without strong convexity abstract this paper establishes exponential convergence rates for a class of primal dual gradient algorithms in distributed optimization without strong convexity the convergence analysis is based on a carefully constructed lyapunov function by evaluating metric subregularity of the primal dual gradient map we present a general criterion under which the algorithm achieves exponential convergence to facilitate practical applications of this criterion several simplified sufficient conditions are derived we also prove that although these results are developed for the continuous time algorithms they carry over in a parallel manner to the discrete time algorithms constructed by using euler s approximation method

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

pair correlation function characteristics of nearly jammed disordered and ordered hard sphere packings we study the approach to jamming in hard sphere packings and in particular the pair correlation function g2srd around contact both theoretically and computationally our computational data unambiguously separate the narrowing d function contribution to g2 due to emerging interparticle contacts from the background contribution due to near contacts the data also show with unprecedented accuracy that disordered hard sphere packings are strictly isostatic i e the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom once rattlers are removed for such isostatic packings we derive a theoretical connection between the probability distribution of interparticle forces pfsfd which we measure computationally and the contact contribution to g2 we verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state we clearly observe a maximum in pf and a nonzero probability of zero force shedding light on long standing questions in the granular media literature we computationally observe an unusual power law divergence in the near contact contribution to g2 persistent even in the jamming limit with exponent 0 4 clearly distinguishable from previously proposed inversesquare root divergence additionally we present high quality numerical data on the two discontinuities in the split second peak of g2 and use a shared neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features finally we present the computational data on the contact contribution to g2 for vacancy diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings we find that the contact network remains isostatic even when ordering is present unlike previous studies we find that ordering has a significant impact on the shape of pf for small forces

convergence of asynchronous distributed gradient methods over stochastic networks we consider distributed optimization problems in which a number of agents are to seek the global optimum of a sum of cost functions through only local information sharing in this paper we are particularly interested in scenarios where agents are operating asynchronously over stochastic networks subject to random failures most existing algorithms require coordinated and decaying stepsizes to ensure zero gap between the estimated value of each agent and the exact optimum restricting it from asynchronous implementation and resulting in slower convergence results to deal with this issue we develop a new asynchronous distributed gradient method asyndgm based on consensus theory the proposed algorithm not only allows for asynchronous implementation in a completely distributed manner but also most importantly is able to seek the exact optimum even with constant stepsizes we will show that the assumption of boundedness of gradients which is widely used in the literature can be dropped by instead imposing the standard lipschitz continuity condition on gradients moreover we derive an upper bound of stepsize within which the proposed asyndgm can achieve a linear convergence rate for strongly convex functions with lipschitz gradients a canonical example of sensor fusion problems is provided to illustrate the effectiveness of the proposed algorithm