applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

online searching with turn cost we consider the problem of searching for an object on a line at an unknown distance opt from the original position of the searcher in the presence of a cost of d for each time the searcher changes direction this is a generalization of the well studied linear search problem we describe a strategy that is guaranteed to find the object at a cost of at most 9 opt 2d which has the optimal competitive ratio 9 with respect to opt plus the minimum corresponding additive term our argument for upper and lower bound uses an infinite linear program which we solve by experimental solution of an infinite series of approximating finite linear programs estimating the limits and solving the resulting recurrences for an explicit proof of optimality we feel that this technique is interesting in its own right and should help solve other searching problems in particular we consider the star search or cowpath problem with turn cost where the hidden object is placed on one of m rays emanating from the original position of the searcher for this problem we give a tight bound of 1 2mm m 1 m 1 opt m m m 1 m 1 1 d we also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line

searching in the plane in this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object the model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position this model is meant to give a realistic cost measure for a robot moving in the plane we also examine the effect of decreasing the amount of a priori information given to search problems problems of this type are very simple analogues of non trivial problems on searching an unbounded region processing digitized images and robot navigation we show that for some simple search problems knowing the general direction of the goal is much more informative than knowing the distance to the goal

exponential rosenbrock integrators for option pricing in this paper we are concerned with the time integration of differential equations modeling option pricing in particular we consider the black scholes equation for american options as an alternative to existing methods we present exponential rosenbrock integrators these integrators require the evaluation of the exponential and related functions of the jacobian matrix the resulting methods have good stability properties they are fully explicit and do not require the numerical solution of linear systems in contrast to standard integrators we have implemented some numerical experiments in matlab showing the reliability of the new method

competitive online routing in geometric graphs we consider online routing algorithms for finding paths between the vertices of plane graphs although it has been shown in bose et al internat j comput geom 12 4 2002 283 that there exists no competitive routing scheme that works on all triangulations we show that there exists a simple online o 1 memory c competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property i e the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path our results imply a competitive routing strategy for certain classical triangulations such as the delaunay greedy or minimum weight triangulation since they all possess the diamond property we then generalize our results to show that the o 1 memory c competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

a near tight approximation algorithm for the robot localization problem localization is a fundamental problem in robotics the kidnapped robot possesses a compass and map of its environment it must determine its location at a minimum cost of travel distance the problem is np hard g dudek k romanik and s whitesides siam j comput 27 1998 pp 583 604 even to minimize within factor c log n c tovey and s koenig proceedings of the national conference on artificial intelligence austin tx 2000 pp 819 824 where n is the map size no approximation algorithm has been known we give an o log 3n factor algorithm the key idea is to plan travel in a majority rule map which eliminates uncertainty and permits a link to the frac 1 2 group steiner not group steiner problem the approximation factor is not far from optimal we prove a c log 2 epsilon n lower bound assuming np not subseteq ztime n polylog n for the grid graphs commonly used in practice we also extend the algorithm to polygonal maps by discretizing the problem using novel geometric techniques

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

a decomposition dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems in this paper we shall be concerned with the solution of constrained convex minimization problems the constrained convex minimization problems are proposed to be transformable into a convex additively decomposed and almost separable form e g by decomposition of the objective functional and the restrictions unconstrained dual problems are generated by using fenchel rockafellar duality this decomposition dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable moreover the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return formulas in quadratic programming the decomposition dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byh matrices numerical tests for solving obstacle problems in 1 discretized by using piecewise quadratic finite elements and in 2 by using the five point difference approximation are presented

the weighted 2 server problem we consider a generalization of the 2 server problem in which servers have different costs we prove that in uniform spaces a version of the work function algorithm is 5 competitive and that no better ratio is possible we also give a 5 competitive randomized memoryless algorithm for uniform spaces and a matching lower bound for arbitrary metric spaces in contrast with the non weighted case we prove that there is no memoryless randomized algorithm with finite competitive ratio we also propose a version of the problem in which a request specifies two points to be covered by the servers and the algorithm must decide which server to move to which point for this version we show a 9 competitive algorithm and we prove that no better ratio is possible

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

adaptive finite element methods for hydrodynamic lubrication with cavitation we present an adaptive finite element method for a cavitation model based on reynolds equation a posteriori error estimates and adaptive algorithms are discussed and numerical examples illustrating the theory are supplied

convergence of a penalty finite element approximation for an obstacle problem this study establishes an error estimate for a penalty finite element approximation of the variational inequality obtained by a class of obstacle problems by special identification of the penalty term we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality the rate of convergence of the penalization is where is the penalty parameter to obtain the error of finite element approximation we apply the results obtained by brezzi hager and raviart for the mixed finite element method to the variational inequality

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

linear mixed model selection for false discovery rate control in microarray data analysis in a microarray experiment one experimental design is used to obtain expression measures for all genes one popular analysis method involves fitting the same linear mixed model for each gene obtaining gene specific p values for tests of interest involving fixed effects and then choosing a threshold for significance that is intended to control false discovery rate fdr at a desired level when one or more random factors have zero variance components for some genes the standard practice of fitting the same full linear mixed model for all genes can result in failure to control fdr we propose a new method that combines results from the fit of full and selected linear mixed models to identify differentially expressed genes and provide fdr control at target levels when the true underlying random effects structure varies across genes

multi target ray searching problems abstract we consider the problem of exploring m concurrent rays using a searcher the rays are disjoint with the exception of a single common point and in each ray at most one potential target may be located the objective is to design search strategies for locating t targets with t m while minimizing the search distance traversed this setting generalizes the extensively studied ray search or star search problem in which the searcher seeks a single target we apply two different measures for evaluating the efficiency of the search strategy the first measure is the standard metric in the context of ray search problems and compares the total search cost to the cost of an optimal algorithm that has full information on the targets we present a simple strategy that achieves optimal competitive ratio under this metric our main result pertains to the second measure which is based on a weakening of the optimal cost as proposed by kirkpatrick esa 2009 and mcgregor et al esa 2009 for this model we present an asymptotically optimal strategy that is within a multiplicative factor of log m t from the optimal search cost our results demonstrate that for both problems the problem of locating t targets in m rays is essentially as difficult as the problem of locating a single target in m t 1 rays

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

optimal robot localization in trees the problem of localization that is of a robot finding its position on a map is an important task for autonomous mobile robots it has applications in numerous areas of robotics ranging from aerial photography to autonomous vehicle exploration in this paper we present a new strategy lps localize by placement separation for a robot to find its position on a map where the map is represented as a geometric tree of bounded degree our strategy exploits to a high degree the self similarities that may occur in the environment we use the framework of competitive analysis to analyze the performance of our strategy in particular we show that the distance traveled by the robot is at most o n times longer than the shortest possible route to localize the robot where n is the number of vertices of the tree this is a significant improvement over the best known previous bound of o n2 3 moreover since there is a lower bound of n our strategy is optimal up to a constant factor using the same approach we can also show that the problem of searching for a target in a geometric tree where the robot is given a map of the tree and the location of the target but does not know its own position can be solved by a strategy with a competitive ratio of o n which is again optimal up to a constant factor 2001 elsevier science

regularized model of post touchdown configurations in electrostatic mems equilibrium analysis abstract in canonical models of micro electro mechanical systems mems an event called touchdown whereby the electrical components of the device come into contact is characterized by a blow up in the governing equations and a non physical divergence of the electric field in the present work we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria we employ techniques from variational calculus dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations

adaptive finite elements for elastic bodies in contact to avoid interpenetration of matter under the small strain assumption the linear contact condition is frequently applied where the distance of bodies is controlled only along a certain direction the standard direction is the normal on the surface where interpenetration might occur in this paper we allow other directions as well we address questions such as the correct mathematical model existence of solutions the penalty method for regularization of the variational inequality finite element discretization and a priori and a posteriori error estimates but exclude the error of penalization the computable upper error bound leads to a criterion for automatic mesh refinements within a finite element method numerical simulations of the hertzian contact problem and a supported cantilever beam are included

on line path planning in an unknown polygonal environment we consider the navigation problem of a robot from a starting point s to a target point t inside an unknown rectilinear polygon the robot knows the directions of the points s and t and it can detect the edges of the polygon through its tactile sensors we present a competitive strategy for the robot to reach the target t the competitive ratio of our strategy is 4 5d k 1 k 2 o 1 where k 3 d 2 1 and d is the l 1 distance between s and t we do not assume any structural restriction on the polygon however tbe point s must be on one of the extreme edges of the polygon this is the first strategy to search for a target inside an arbitrary unknown polygon with competitive ratio independent of the number of edges of the polygon

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

an improved approximation ratio for the minimum latency problem networks and matroids given a tour visiting n points in a metric space the latency of one of these points p is the distance traveled in the tour before reaching p the minimum latency problem mlp asks for a tour passing through n given points for which the total latency of the n points is minimum in effect we are seeking the tour with minimum average arrival time this problem has been studied in the operations research literature where it has also been termed the delivery man problem and the traveling repairman problem the approximability of the mlp was first considered by sahni and gonzalez in 1976 however unlike the classical traveling salesman problem tsp it is not easy to give any constant factor approximation algorithm for the mlp recently blum et al a blum p chalasani d coppersimith w pulleyblank p raghavan m sudan proceedings of the 26th acm symposium on the theory of computing 1994 pp 163 171 gave the first such algorithm obtaining an approximation ratio of 144 in this work we develop an algorithm which improves this ratio to 21 55 moreover combining our algorithm with a recent result of garg n garg proceedings of the 37th ieee symposium on foundations of computer science 1996 pp 302 309 provides an approximation ratio of 10 78 the development of our algorithm involves a number of techniques that seem to be of interest from the perspective of the tsp and its variants more generally

a general framework for searching on a line abstract consider the following classical search problem a target is located on a line at distance d from the origin starting at the origin a searcher must find the target with minimum competitive cost the classical competitive cost studied in the literature is the ratio between the distance travelled by the searcher and d note that when no lower bound on d is given no competitive search strategy exists for this problem therefore all competitive search strategies require some form of lower bound on d we develop a general framework that optimally solves several variants of this search problem our framework allows us to achieve optimal competitive search costs for previously studied variants such as 1 where the target is fixed and the searcher s cost at each step is a constant times the length of the step 2 where the target is fixed and the searcher s cost at each step is the length of the step plus a fixed constant often referred to as the turn cost 3 where the target is moving and the searcher s cost at each step is the length of the step our main contribution is that the framework allows us to derive optimal competitive search strategies for variants of this problem that do not have a solution in the literature such as 1 where the target is fixed and the searcher s cost at each step is 1 x 1 for moving distance x away from the origin and 2 x 2 for moving back with constants 1 2 1 2 2 where the target is moving and the searcher s cost at each step is a constant times the length of the step plus a fixed constant turn cost notice that the latter variant can have several interpretations depending on what the turn cost represents for example if the turn cost represents the amount of time for the searcher to turn then this has an impact on the position of the moving target on the other hand the turn cost can represent the amount of fuel needed to make an instantaneous turn thereby not affecting the target s position our framework addresses all of these variations