adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

a new surrogate modeling technique combining kriging and polynomial chaos expansions application to uncertainty analysis in computational dosimetry in numerical dosimetry the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate sar characterizing the human exposure to electromagnetic waves however this procedure remains time consuming and a single simulation can request several hours as a consequence the influence of uncertain input parameters on the sar cannot be analyzed using crude monte carlo simulation the solution presented here to perform such an analysis is surrogate modeling this paper proposes a novel approach to build such a surrogate model from a design of experiments considering a sparse representation of the polynomial chaos expansions using least angle regression as a selection algorithm to retain the most influential polynomials this paper proposes to use the selected polynomials as regression functions for the universal kriging model the leave one out cross validation is used to select the optimal number of polynomials in the deterministic part of the kriging model the proposed approach called lars kriging pc modeling is applied to three benchmark examples and then to a full scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device the performances of the lars kriging pc are compared to an ordinary kriging model and to a classical sparse polynomial chaos expansion the lars kriging pc appears to have better performances than the two other approaches a significant accuracy improvement is observed compared to the ordinary kriging or to the sparse polynomial chaos depending on the studied case this approach seems to be an optimal solution between the two other classical approaches a global sensitivity analysis is finally performed on the lars kriging pc model of the fetus exposure problem

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

stochastic field representation using bi fidelity combination of proper orthogonal decomposition and kriging abstract in the current paper efficient uncertainty quantification uq of high dimensional stochastic fields is performed via a bi fidelity surrogate model the method is based on combination of proper orthogonal decomposition pod and kriging method in the developed method the trend function of the kriging model is estimated via a low cost pod whilst the stochastic gaussian contribution is tuned using a limited number of high fidelity computations the proposed method is applied to three test cases including a highly nonlinear test function thermally driven flow in a cavity with stochastic wall temperatures and turbulent heat transfer in a ribbed channel with stochastic heat flux boundary condition implementing both pod and pod kriging methods leads to significant reduction of computational cost in comparison to the classical full polynomial chaos expansion pce with least cost saving of more than 60 furthermore in all cases combined pod kriging method performs either superior or at least similar to the pod method more specifically in surrogate models with lower number of low and high fidelity samples greater gains in accuracy are obtained consequently for a specific level of accuracy pod kriging method requires lower computational cost

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

improved bound for complexity of matrix multiplication in 1968 strassen 16 described a method of multiplying two 2 2 matrices using only seven multiplications by applying it recursively it was shown that two n n matrices could be multiplied using o n operations in all where log2 7 2 81 this discovery led to a period of activity devoted to the improvement of this bound one can define an exponent of complexity which is in effect the smallest number such that two n n matrices can be multiplied using o n operations using an method of the type described by strassen a precise definition is given in 2 then strassen s result says that log2 7 so it is natural to wish to determine the true value of in the decade or so after 1978 several successive reductions in the upper bound for were achieved 2 3 6 8 10 12 14 17 leading to the bound 2 375477 obtained by coppersmith and winograd 7 it is an open question whether 2 we present a small improvement of the coppersmith winograd bound namely 2 3736897 which was first obtained in stothers 15 the proof uses an extension of the ideas of 7 and relies heavily on a combinatorial construction from 7 which depends on probabilistic arguments and a theorem of salem and spencer on sets of integers containing no three terms in arithmetical progression we have tried to organize the proof so that it is reasonably concise but also self contained to make it more accessible in the belief that at least parts of the argument may be of wider interest outside the algebraic complexity community to this end we derive the results we need from algebraic complexity theory in 2 which also contains general background material in 3 we present the essence of the combinatorial construction as a lemma which we subsequently apply several times we first apply it in 4 to recover the coppersmith winograd bound although our presentation is rather different the proof is essentially the same as that of 7 except for the proof of lemma 4 1 for which we give a somewhat simpler argument using 4 as preparation and motivation we obtain our new bound in 5 the argument is similar but more complicated and technical

more algorithms for all pairs shortest paths in weighted graphs in the first part of the paper we reexamine the all pairs shortest path apsp problem and present a new algorithm with running time o n 3 log 3 log n log 2n which improves all known algorithms for general real weighted dense graphs in the second part of the paper we use fast matrix multiplication to obtain truly subcubic apsp algorithms for a large class of geometrically weighted graphs where the weight of an edge is a function of the coordinates of its vertices for example for graphs embedded in euclidean space of a constant dimension d we obtain a time bound near o n 3 3 omega 2d 4 where omega 2 376 in two dimensions this is o n 2 922 our framework greatly extends the previously considered case of small integer weighted graphs and incidentally also yields the first truly subcubic result near o n 3 3 omega 4 o n 2 844 time for apsp in real vertex weighted graphs as well as an improved result near o n 3 omega 2 o n 2 688 time for the all pairs lightest shortest path problem for small integer weighted graphs

structural optimization using kriging approximation an optimization method using kriging approximation is applied to a structural optimization problem the method involves two main processes the first is a space estimation process that uses the kriging method and the second is an optimization process the use of the kriging method makes it easier to perform the approximation optimization as an example of the estimation performed as part of structural optimization a response surface for layout optimization of beam reinforcement is estimated to evaluate the applicability of the kriging method kriging estimation is compared with neural network approximation as a numerical example the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated the obtained results clearly show the applicability of the method

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

approximating the cut norm via grothendieck s inequality the cut norm a c of a real matrix a a ij i in r j in s is the maximum over all i subset r j subset s of the quantity sum i in i j in j a ij this concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems here we show that the problem of approximating the cut norm of a given real matrix is max snp hard and we provide an efficient approximation algorithm this algorithm finds for a given matrix a a ij i in r j in s two subsets i subset r and j subset s such that sum i in i j in j a ij geq rho a c where rho 0 is an absolute constant satisfying rho 0 56 the algorithm combines semidefinite programming with a rounding technique based on grothendieck s inequality we present three known proofs of grothendieck s inequality with the necessary modifications which emphasize their algorithmic aspects these proofs contain rounding techniques which go beyond the random hyperplane rounding of goemans and williamson j acm 42 1995 pp 1115 1145 allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case

on the k shortest simple paths problem in weighted directed graphs we present the first approximation algorithm for finding the k shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of mn it is deterministic and has a running time of o k m sqrt n n 3 2 log n where m is the number of edges in the graph and n is the number of vertices let s t in v the length of the i th simple path from s to t computed by our algorithm is at most frac 3 2 times the length of the i th shortest simple path from s to t the best algorithms for computing the exact k shortest simple paths connecting a pair of vertices in a weighted directed graph are due to yen management sci 17 1970 1971 pp 712 716 and lawler management sci 18 1971 1972 pp 401 405 the running time of their algorithms using modern data structures is o k mn n 2 log n both algorithms are from the early 70s although this problem and other variants of the k shortest path problem has drawn a lot of attention during the last three and a half decades the o k mn n 2 log n bound is still unbeaten

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

stability of bsdes with random terminal time and homogenization of semilinear elliptic pdes in this paper we extend the probabilistic method for homogenization of semi linear parabolic pdes developed by buckdahn hu and peng to the case of elliptic pdes first we give a stability result for bsdes with random terminal time which are related to elliptic pdes as shown in peng stochastics stochastics rep 37 1991 61 74 in the one dimensional case we also partially relax the monotonicity assumption on the coefficient then we use these stability results for bsdes with random terminal time to study homogenization of systems of semilinear elliptic pdes

finding the hidden path time bounds for all pairs shortest paths the all pairs shortest paths problem in weighted graphs is investigated an algorithm the hidden paths algorithm that finds these paths in time o m n n 2 log n where m is the number of edges participating in shortest paths is presented the algorithm is a practical substitute for dijkstra s algorithm it is argued that m is likely to be small in practice since m o n log n with high probability for many probability distributions on edge weights an omega mn lower bound on the running time of any path comparison based algorithm for the all pairs shortest paths problem is also proved path comparison based algorithms form a natural class containing the hidden paths algorithm as well as the algorithms of e w dijkstra numer math 1 1959 pp 269 271 and r w floyd comm acm 5 1962 p 345 lastly generalized forms of the shortest paths problem are considered and it is shown that many of the standard shortest paths algorithms are effective in this more genera

ergodic bsdes and optimal ergodic control in banach spaces in this paper we introduce a new kind of backward stochastic differential equations called ergodic bsdes which arise naturally in the study of optimal ergodic control we study the existence uniqueness and regularity of solution to ergodic bsdes then we apply these results to the optimal ergodic control of a banach valued stochastic state equation we also establish the link between the ergodic bsdes and the associated hamilton jacobi bellman equation applications are given to the optimal ergodic control of stochastic partial differential equations

ergodic bsdes driven by markov chains we consider ergodic backward stochastic differential equations bsdes in a setting where noise is generated by a countable state uniformly ergodic markov chain we show that for lipschitz drivers such that a comparison theorem holds these equations admit unique solutions to obtain this result we show by coupling and splitting techniques that uniform ergodicity estimates of markov chains are robust to perturbations of the rate matrix and that these perturbations correspond in a natural way to ergodic bsdes we then consider applications of this theory to markov decision problems with a risk averse average reward criterion

subcubic equivalences between path matrix and triangle problems we say an algorithm on n n matrices with integer entries in l m mr or n node graphs with edge weights from l m m is truly subcubic if it runs in o n3 poly log m time for some 0 we define a notion of subcubic reducibility and show that many important problems on graphs and matrices solvable in o n3 time are equivalent under subcubic reductions namely the following weighted problems either all have truly subcubic algorithms or none of them do the all pairs shortest paths problem on weighted digraphs apsp detecting if a weighted graph has a triangle of negative total edge weight listing up to n2 99 negative triangles in an edge weighted graph finding a minimum weight cycle in a graph of non negative edge weights the replacement paths problem on weighted digraphs finding the second shortest simple path between two nodes in a weighted digraph checking whether a given matrix defines a metric verifying the correctness of a matrix product over the min semiring finding a maximum subarray in a given matrix therefore if apsp cannot be solved in n3 e time for any e 0 then many other problems also need essentially cubic time in fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization verifying a matrix product over the same structure and corresponding triangle detection problems over the structure these equivalences simplify prior work on subcubic algorithms for all pairs path problems since it now suffices to give appropriate subcubic triangle detection algorithms other consequences of our work are new combinatorial approaches to boolean matrix multiplication over the or and semiring abbreviated as bmm we show that practical advances in triangle detection would imply practical bmm algorithms among other results building on our techniques we give two improved bmm algorithms a derandomization of the combinatorial bmm algorithm of bansal and williams focs 09 and an improved quantum algorithm for bmm

drift and monotonicity conditions for continuous time controlled markov chains with an average criterion we give conditions for the existence of average optimal policies for continuous time controlled markov chains with a denumerable state space and borel action sets the transition rates are allowed to be unbounded and the reward cost rates may have neither upper nor lower bounds in the spirit of the drift and monotonicity conditions for continuous time markov processes we propose a new set of conditions on the controlled process primitive data under which the existence of optimal deterministic stationary policies in the class of randomized markov policies is proved using the extended generator approach instead of kolmogorov s forward equation used in the previous literature and under which the convergence of a policy iteration method is also shown moreover we use a controlled queueing system to show that all of our conditions are satisfied whereas those in the previous literature fail to hold

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

counting triangles in massive graphs with mapreduce graphs and networks are used to model interactions in a variety of contexts there is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure some of the most useful metrics are triangle based and give a measure of the connectedness of mutual friends this is often summarized in terms of clustering coefficients which measure the likelihood that two neighbors of a node are themselves connected computing these measures exactly for large scale networks is prohibitively expensive in both memory and time however a recent wedge sampling algorithm has proved successful in efficiently and accurately estimating clustering coefficients in this paper we describe how to implement this approach in mapreduce to deal with massive graphs we show results on publicly available networks the largest of which is 132m nodes and 4 7b edges as well as artificially generated networks using the graph500 benchmark the largest of which has 240m nodes and 8 5b edges

a domain decomposition strategy for reduced order models application to the incompressible navier stokes equations abstract in this work a domain decomposition strategy for non linear hyper reduced order models is presented the basic idea consists of restricting the reduced order basis functions to the nodes belonging to each of the subdomains into which the physical domain is partitioned an extension of the proposed domain decomposition strategy to a hybrid full order reduced order model is then described the general domain decomposition approach is particularized for the reduced order finite element approximation of the incompressible navier stokes equations with hyper reduction when solving the reduced incompressible navier stokes equations instabilities in the form of large gradients of the recovered reduced order unknown at the subdomain interfaces may appear which is the motivation for the design of additional stability terms giving rise to penalty matrices numerical examples illustrate the behavior of the proposed method for the simulation of the reduced order systems showing the capability of the approach to adapt to configurations which are not present in the original snapshot set

finding the largest triangle in a graph in expected quadratic time abstract finding the largest triangle in an n nodes edge weighted graph belongs to a set of problems all equivalent under subcubic reductions namely a truly subcubic algorithm for any one of them would imply that they are all subcubic a recent strong conjecture states that none of them can be solved in less than n3 time but this negative result does not rule out the possibility of algorithms with average rather than worst case subcubic running time indeed in this work we describe the first truly subcubic average complexity procedure for this problem for graphs whose edge lengths are uniformly distributed in 0 1 our procedure finds the largest triangle in average quadratic time which is the best possible complexity of any algorithm for this problem we also give empirical evidence that the quadratic average complexity holds for many other random distributions of the edge lengths a notable exception is when the lengths are distances between random points in euclidean space for which the algorithm takes average cubic time

filtration consistent nonlinear expectations and related g expectations from a general definition of nonlinear expectations viewed as operators preserving monotonicity and constants we derive under rather general assumptions the notions of conditional nonlinear expectation and nonlinear martingale we prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation and in particular admits continuous paths in other words it is a g martingale

finding the k shortest paths we give algorithms for finding the k shortest paths not required to be simple connecting a pair of vertices in a digraph our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges in time o m n log n k we can also find the k shortest paths from a given source s to each vertex in the graph in total time o m n log n kn we describe applications to dynamic programming problems including the knapsack problem sequence alignment maximum inscribed polygons and genealogical relationship discovery

dynamic ordered sets with exponential search trees we introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully dynamic linear space data structures this leads to an optimal bound of o slog n log log n for searching and updating a dynamic set x of n integer keys in linear space searching x for an integer y means finding the maximum key in x which is smaller than or equal to y this problem is equivalent to the standard text book problem of maintaining an ordered set the best previous deterministic linear space bound was o log n log log n due to fredman and willard from stoc 1990 no better deterministic search bound was known using polynomial space we also get the following worst case linear space trade offs between the number n the word length w and the maximal key u our results are generalized to finger searching and string searching providing optimal results for both in terms of n

designing programs that check their work a program correctness checker is an algorithm for checking the output of a computation that is given a program and an instance on which the program is run the checker certifies whether the output of the program on that instance is correct this paper defines the concept of a program checker it designs program checkers for a few specific and carefully chosen problems in the class fp of functions computable in polynomial time problems in fp for which checkers are presented in this paper include sorting matrix rank and gcd it also applies methods of modern cryptography especially the idea of a probabilistic interactive proof to the design of program checkers for group theoretic computations two structural theorems are proven here one is a characterization of problems that can be checked the other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable all problems in the class are checkable