constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

infinite dimensional integration and the multivariate decomposition method abstract we further develop the multivariate decomposition method mdm for the lebesgue integration of functions of infinitely many variables x 1 x 2 x 3 with respect to a corresponding product of a one dimensional probability measure the method is designed for functions that admit a dominantly convergent decomposition f u f u where u runs over all finite subsets of positive integers and for each u i 1 i k the function f u depends only on x i 1 x i k although a number of concepts of infinite dimensional integrals have been used in the literature questions of uniqueness and compatibility have mostly not been studied we show that under appropriate convergence conditions the lebesgue integral equals the anchored integral independently of the anchor for approximating the integral the mdm assumes that point values of f u are available for important subsets u at some known cost in this paper we introduce a new setting in which it is assumed that each f u belongs to a normed space f u and that bounds b u on f u f u are known this contrasts with the assumption in many papers that weights u appearing in the norm of the infinite dimensional function space are somehow known often such weights u were determined by minimizing an error bound depending on the b u the u and the chosen algorithm resulting in weights that depend on the algorithm in contrast in this paper only the bounds b u are assumed to be known we give two examples in which we specialize the mdm in the first case f u is the u fold tensor product of an anchored reproducing kernel hilbert space in the second case it is a particular non hilbert space for integration over an unbounded domain

rank and regularity for averages over submanifolds abstract this paper establishes endpoint l p l q and sobolev mapping properties of radon like operators which satisfy a homogeneity condition similar to semiquasihomogeneity and a condition on the rank of a matrix related to rotational curvature for highly degenerate operators the rank condition is generically satisfied for algebraic reasons similar to an observation of greenleaf pramanik and tang a greenleaf m pramanik w tang oscillatory integral operators with homogeneous polynomial phases in several variables j funct anal 244 2 2007 444 487 concerning oscillatory integral operators

random pairwise gossip on text cat kappa metric spaces in the context of sensor networks gossip algorithms are a popular well established technique for achieving consensus when sensor data are encoded in linear spaces gossip algorithms also have several extensions to non linear data spaces most of these extensions deal with riemannian manifolds and use riemannian gradient descent this paper instead exhibits a very simple metric property that does not rely on any differential structure this property strongly suggests that gossip algorithms could be studied on a broader family than riemannian manifolds and it turns out that indeed local convergence is guaranteed as soon as the data space is a mere text cat kappa metric space we also study convergence speed in this setting and establish linear rates for text cat 0 spaces and local linear rates for text cat kappa spaces with kappa 0 numerical simulations on several scenarii with corresponding state spaces that are either riemannian manifolds as in the problem of positive definite matrices consensus or bare metric spaces as in the problem of arms consensus validate the results this shows that our metric approach not only allows for a simpler and more general mathematical analysis but also paves the way for new kinds of applications that go beyond the riemannian setting

riemannian consensus for manifolds with bounded curvature consensus algorithms are popular distributed algorithms for computing aggregate quantities such as averages in ad hoc wireless networks however existing algorithms mostly address the case where the measurements lie in euclidean space in this work we propose riemannian consensus a natural extension of existing averaging consensus algorithms to the case of riemannian manifolds unlike previous generalizations our algorithm is intrinsic and in principle can be applied to any complete riemannian manifold we give sufficient convergence conditions on riemannian manifolds with bounded curvature and we analyze the differences with respect to the euclidean case we test the proposed algorithms on synthetic data sampled from the space of rotations the sphere and the grassmann manifold

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

a reduced fast component by component construction of lattice points for integration in weighted spaces with fast decreasing weights lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the s dimensional unit cube since no explicit constructions of such quadrature methods are known for dimensions s 2 one usually has to resort to computer search algorithms the fast component by component approach is a useful algorithm for finding suitable quadrature rules we present a modification of the fast component by component algorithm which yields savings of the construction cost for polynomial lattice rules in weighted function spaces the idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large weights we analyze tractability conditions of the resulting qmc rules numerical results demonstrate the effectiveness of our method

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

computer experiments with both qualitative and quantitative variables computer experiments have received a great deal of attention in many fields of science and technology most literature assumes that all the input variables are quantitative however researchers often encounter computer experiments involving both qualitative and quantitative variables bqqv in this article a new interface on design and analysis for computer experiments with bqqv is proposed the new designs are one kind of sliced latin hypercube designs with points clustered in the design region and possess good uniformity for each slice for computer experiments with bqqv such designs help to measure the similarities among responses of different level combinations in the qualitative variables an adaptive analysis strategy intended for the proposed designs is developed the proposed strategy allows us to automatically extract information from useful auxiliary responses to increase the precision of prediction for the target response the interface between the proposed design and the analysis strategy

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

space filling fractional factorial designs fractional factorial designs are widely used in various scientific investigations and industrial applications level permutation of factors could alter their geometrical structures and statistical properties this article studies space filling properties of fractional factorial designs under two commonly used space filling measures discrepancy and maximin distance when all possible level permutations are considered the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel generalized minimum aberration designs are shown to have good space filling properties on average in terms of both discrepancy and distance several novel relationships between distance distribution and generalized word length pattern are derived it is also shown that level permutations can improve space filling properties for many existing saturated designs a two step construction pro

how to solve path integrals in quantum mechanics a systematic classification of feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years including of course the main contributions since the invention of the path integral by feynman in 1942 an outline of the general theory is given which will serve as a quick reference for solving path integrals explicit formulas for the so called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

optimal randomized multilevel algorithms for infinite dimensional integration on function spaces with anova type decomposition in this paper we consider the infinite dimensional integration problem on weighted reproducing kernel hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type the weights model the relative importance of different groups of variables we present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error furthermore we provide in this setting the first nontrivial lower error bounds for general randomized algorithms which in particular may be adaptive or nonlinear these lower bounds show that our multilevel algorithms are optimal our analysis refines and extends the analysis provided in f j hickernell t muller gronbach b niu and k ritter j complexity 26 2010 pp 229 254 and our error bounds improve substantially on the error bounds presented there as an illustrative example we discuss the unanchored sobolev space and employ randomized quasi monte carlo multilevel a

infinite dimensional integration on weighted hilbert spaces we study the numerical integration problem for functions with infinitely many variables the functions we want to integrate are from a reproducing kernel hilbert space which is endowed with a weighted norm we study the worst case e complexity which is defined as the minimal cost among all algorithms whose worst case error over the hilbert space unit ball is at most e here we assume that the cost of evaluating a function depends polynomially on the number of active variables the infinite dimensional integration problem is polynomially tractable if the e complexity is bounded by a constant times a power of 1 e the smallest such power is called the exponent of tractability first we study finite order weights we provide improved lower bounds for the exponent of tractability for general finite order weights and improved upper bounds for three newly defined classes of finite order weights the constructive upper bounds are obtained by multilevel algorithms that use for each level quasimonte carlo integration points whose projections onto specific sets of coordinates exhibit a small discrepancy the newly defined finite intersection weights model the situation where each group of variables interacts with at most other groups of variables where is some fixed number for these weights we obtain a sharp upper bound this is the first class of weights for which the exact exponent of tractability is known for any possible decay of the weights and for any polynomial degree of the cost function for the other two classes of finite order weights our upper bounds are sharp if e g the decay of the weights is fast or slow enough we extend our analysis to the case of arbitrary weights in particular from our results for finite order weights we conclude a lower bound on the exponent of tractability for arbitrary weights and a constructive upper bound for product weights

uniform designs limit aliasing summary when fitting a linear regression model to data aliasing can adversely affect the estimates of the model coefficients and the decision of whether or not a term is significant optimal experimental designs give efficient estimators assuming that the true form of the model is known while robust experimental designs guard against inaccurate estimates caused by model misspecification although it is rare for a single design to be both maximally efficient and robust it is shown here that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together aberration and resolution measure how well fractional factorial designs guard against the effects of aliasing here it is shown that the definitions of aberration and resolution may be generalised to other types of design using the discrepancy

on tractability of path integration many applications require approximate values of path integrals a typical approach is to approximate the path integral by a high dimensional integral and apply a monte carlo randomized algorithm however monte carlo algorithm requires roughly e 2 integrand evaluations to provide an e approximation moreover the error bound of e is guaranteed only in a stochastic sense do we really need to use randomized algorithms for path integrals perhaps we can find a deterministic algorithm that is more effective even in the worst case setting to answer this question we study the worst case complexity of path integration which roughly speaking is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e we consider path integration with respect to a gaussian measure and for various classes of integrands tractability of path integration means that the complexity depends polynomially on 1 e we show that for the class of r times frechet differentiab

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

a path integration formulation of stochastic lagrangian models of turbulent flow abstract in analogy with the feynman kac path integration formulation of quantum mechanics a general path integration formulation of stochastic lagrangian models of turbulent flow is developed with respect to the equivalence with the usual eulerian and stochastic lagrangian descriptions we then discuss efficient numerical methods for the realisation of the lagrangian solutions via path integration which provide an effective alternative to the simpler monte carlo approach

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

sharp l p boundedness of oscillatory integral operators with polynomial phases in this paper we shall prove the l p endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases s in mathbb r times mathbb r as a consequence sharp l p decay estimates are also obtained when polynomial phases have the form s x m 1 y m 2 with m 1 and m 2 being positive integers

strong tractability of integration using scrambled niederreiter points we study the randomized worst case error and the randomized error of scrambled quasi monte carlo qmc quadrature as proposed by owen the function spaces considered in this article are the weighted hilbert spaces generated by haar like wavelets and the weighted sobolev hilbert spaces conditions are found under which multivariate integration is strongly tractable in the randomized worst case setting and the randomized setting respectively the e exponents of strong tractability are found for the scrambled niederreiter nets and sequences the sufficient conditions for strong tractability for sobolev spaces are more lenient for scrambled qmc quadratures than those for deterministic qmc net quadratures

why are high dimensional finance problems often of low effective dimension many problems in mathematical finance can be formulated as high dimensional integrals where the large number of dimensions arises from small time steps in time discretization and or a large number of state variables quasi monte carlo qmc methods have been successfully used for approximating such integrals to understand this success this paper focuses on investigating the special features of some typical high dimensional finance problems namely option pricing and bond valuation we provide new insight into the connection between the effective dimension and the efficiency of qmc and we present methods to analyze the dimension structure of a function we confirm the observation of caflisch morokoff and owen that functions from finance are often of low effective dimension in the sense that they can be well approximated by their low order anova analysis of variance terms usually just the order 1 and order 2 terms we explore why the effective dimension is small for many integrals from finance by deriving explicit forms of the anova terms in simple cases we find that the importance of each dimension is naturally weighted by certain hidden weights these weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher order anova terms we study the variance ratios captured by low order anova terms and their asymptotic properties as the dimension tends to infinity and we show that with the increase of dimension the lower order terms continue to play a significant role and the higher order terms tend to be negligible this provides some insight into high dimensional problems from finance and explains why qmc algorithms are efficient for problems of this kind

coordinated motion design on lie groups the present paper proposes a unified geometric framework for coordinated motion on lie groups it first gives a general problem formulation and analyzes ensuing conditions for coordinated motion then it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics it thereby shows that coordination can be studied in a systematic way once the lie group geometry of the configuration space is well characterized applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds a link with brockett s double bracket flows is also made the concepts are illustrated on so 3 se 2 and se 3

stochastic gradient descent on riemannian manifolds stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise in this paper we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a riemannian manifold we prove that as in the euclidian case the gradient descent algorithm converges to a critical point of the cost function the algorithm has numerous potential applications and is illustrated here by four examples in particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically

sharp l estimates for one dimensional oscillatory integral operators with c phase in this paper sharp l 2 estimates are proved for general local one dimensional oscillatory integral operators with smooth phase the proof involves a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of d h phong and e m stein

construction of maximin distance latin squares and related latin hypercube designs summarymaximin distance latin hypercube designs are widely used in computer experiments yet their construction is challenging based on number theory and finite fields we propose three algebraic methods to construct maximin distance latin squares as special latin hypercube designs we develop lower bounds on their minimum distances the resulting latin squares and related latin hypercube designs have larger minimum distances than existing ones and are especially appealing for high dimensional applications

numerical path integral techniques for long time dynamics of quantum dissipative systems recent progress in numerical methods for evaluating the real time path integral in dissipative harmonic environments is reviewed quasi adiabatic propagators constructed numerically allow convergence of the path integral with large time increments integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path with an influence functional that describes nonadiabatic corrections the resulting quasi adiabatic propagator path integral is evaluated by efficient system specific quadratures in most regimes of parameter space although some cases are handled by grid monte carlo sampling exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths no uncontrolled approximations are introduced and the resulting methodology converges to the exact quantum result with modest

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

oscillatory integral operators with homogeneous polynomial phases in several variables abstract we obtain l 2 decay estimates in for oscillatory integral operators t whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions the decay rates obtained are optimal in the case of 2 2 dimensions for any m while in higher dimensions the result is sharp for m sufficiently large the proof for large m follows from essentially algebraic considerations for cubics in 2 2 dimensions the proof involves decomposing the operator near the conic zero variety of the determinant of the hessian of the phase function using an elaboration of the general approach of phong and stein d h phong e m stein models of degenerate fourier integral operators and radon transforms ann of math 2 140 1994 703 722

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

consensus optimization on manifolds the present paper considers distributed consensus algorithms that involve n agents evolving on a connected compact homogeneous manifold the agents track no external reference and communicate their relative state according to a communication graph the consensus problem is formulated in terms of the extrema of a cost function this leads to efficient gradient algorithms to synchronize i e maximizing the consensus or balance i e minimizing the consensus the agents a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time varying the cost function is linked to a specific centroid definition on manifolds introduced here as the induced arithmetic mean that is easily computable in closed form and may be of independent interest for a number of manifolds the special orthogonal group so n and the grassmann manifold text it grass p n are treated as original examples a link is also drawn with the many existing results on the circle

a new algorithm and worst case complexity for feynman kac path integration abstract we study algorithms for approximation of feynman kac path integrals in the worst case setting the algorithms use a finite number of samples of the initial condition and potential functions we present a new algorithm and an explicit bound on its cost to compute an e approximation to the feynman kac path integral we also establish bounds on the worst case complexity of feynman kac path integration the upper bound is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem the lower bound is given in terms of the complexity of a certain weighted integration problem for some classes of functions these two bounds coincide modulo a multiplicative factor in this case the new algorithm is almost optimal the new algorithm requires precomputation of some real coefficients that are combinations of multivariate integrals with special weights this precomputation is difficult and limits the application of the new algorithm we report the results of the precomputation for specific cases

efficient algorithms for computing the l 2 discrepancy the l 2 discrepancy is a quantitative measure of precision for multivariate quadrature rules it can be computed explicitly previously known algorithms needed o m 2 operations where m is the number of nodes in this paper we present algorithms which require o m log m d operations

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

consensus in ad hoc wsns with noisy links part i distributed estimation of deterministic signals we deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks wsns we cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation different from existing alternatives our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages thus allowing for decentralized computation even of nonlinear estimators including maximum likelihood estimators mle in nonlinear and non gaussian data models we prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal furthermore our decentralized algorithms exhibit resilience in the presence of receiver and or quantization noise in particular we introduce a decentralized scheme for least squares and best linear unbiased estimation blue and establish its convergence in the presence of communication noise our algorithms also exhibit potential for higher convergence rate with respect to existing schemes corroborating simulations demonstrate the merits of the novel distributed estimation algorithms