computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

some prime factorization results for type ii1 factors we prove several unique prime factorization results for tensor products of type ii1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected lie groups of real rank 1 in particular we show that if mathcal r overline otimes mathcal l mathbb f r 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f r m is isomorphic to a subfactor in mathcal r overline otimes mathcal l mathbb f s 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f s n for some 2 ri sj then m n

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

polynomial chaos representation of databases on manifolds characterizing the polynomial chaos expansion pce of a vector valued random variable with probability distribution concentrated on a manifold is a relevant problem in data driven settings the probability distribution of such random vectors is multimodal in general leading to potentially very slow convergence of the pce in this paper we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold the proposed methodology constructs a pce of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold the method is robust and remains efficient for high dimension and large datasets the resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data driven queries

unique prime factorization for infinite tensor product factors abstract in this article we investigate a unique prime factorization property for infinite tensor product factors we provide several examples of type ii and iii factors which satisfy this property including all free product factors with diffuse free product components in the type iii setting this is the first classification result for infinite tensor product non amenable factors our proof is based on popa s intertwining techniques and a characterization of relative amenability on the continuous cores

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

weak exactness for c algebras and application to condition ao abstract kirchberg introduced weak exactness for von neumann algebras as an analogue of exactness for c algebras ozawa found useful characterizations of weak exactness and he proved that weak exactness for group von neumann algebras is equivalent to exactness of the groups we generalize this concept to inclusions of c algebras in von neumann algebras and study some fundamental properties and relationship to kirchberg s weak exactness we then give some permanence properties which are similar to those of exact groups in the last section we study a similar condition to ozawa s condition ao with our weak exactness and generalize ozawa s theorem for bi exact groups as a corollary we give new examples of prime factors

spectral power iterations for the random eigenvalue problem two computationally efficient algorithms are developed for solving the stochastic eigenvalue problem an algorithm based on the power iteration technique is proposed for the calculation of the dominant eigenpairs this algorithm is then extended to find other subdominant random eigenpairs the uncertainty in the operator is represented by a polynomial chaos expansion and a similar representation is considered for the random eigenvalues and eigenvectors the algorithms are distinguished due to their speed in converging to the true random eigenpairs and their ability to estimate a prescribed number of subdominant eigenpairs the algorithms are demonstrated on two examples with close agreement observed with the exact solution and a solution synthesized through monte carlo sampling

l 2 rigidity in von neumann algebras we introduce the notion of l2 rigidity for von neumann algebras a generalization of property t which can be viewed as an analogue for the vanishing of 1 cohomology into the left regular representation of a group we show that l2 rigidity passes to normalizers and is satisfied by nonamenable ii1 factors which are non prime have property or are weakly rigid as a consequence we obtain that if m is a free product of diffuse von neumann algebras or if m l where is a finitely generated group with 1 2 0 then any nonamenable regular subfactor of m is prime and does not have properties or t in particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of n ozawa

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

approximation properties and absence of cartan subalgebra for free araki woods factors abstract we show that all the free araki woods factors h r u t have the complete metric approximation property using ozawa popa s techniques we then prove that every nonamenable subfactor n h r u t which is the range of a normal conditional expectation has no cartan subalgebra we finally deduce that the type iii1 factors constructed by connes in the 70s can never be isomorphic to any free araki woods factor which answers a question of shlyakhtenko and vaes

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

subcell resolution in simplex stochastic collocation for spatial discontinuities subcell resolution has been used in the finite volume method fvm to obtain accurate approximations of discontinuities in the physical space stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions however the adaptive refinement in the probability space is ineffective in the non intrusive uncertainty quantification framework if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location the dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics which leads to first order error convergence and an underprediction of the maximum standard deviation to avoid these problems we introduce subcell resolution into the simplex stochastic collocation ssc method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space the presented ssc sr method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location the applications to a linear advection problem the inviscid burgers equation a shock tube problem and the transonic flow over the rae 2822 airfoil show that ssc sr resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples

unique cartan decomposition for ii1 factors arising from arbitrary actions of free groups we prove that for any free ergodic probability measure preserving action mathbb f n curvearrowright x mu of a free group on n generators mathbb f n 2 leq n leq infty the associated group measure space ii1 factor l infty x rtimes mathbb f n has l x as its unique cartan subalgebra up to unitary conjugacy we deduce that group measure space ii1 factors arising from actions of free groups with different number of generators are never isomorphic we actually prove unique cartan decomposition results for ii1 factors arising from arbitrary actions of a much larger family of groups including all free products of amenable groups and their direct products

unique prime factorization and bicentralizer problem for a class of type iii factors abstract we show that whenever m 1 and m 1 m m are nonamenable factors in a large class of von neumann algebras that we call c ao and which contains all free araki woods factors the tensor product factor m 1 m m retains the integer m and each factor m i up to stable isomorphism after permutation of the indices our approach unifies the unique prime factorization upf results from 33 25 and moreover provides new upf results in the case when m 1 m m are free araki woods factors in order to obtain the aforementioned upf results we show that connes s bicentralizer problem has a positive solution for all type i i i 1 factors in the class c ao

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

factoriality type classification and fullness for free product von neumann algebras abstract we give a complete answer to the questions of factoriality type classification and fullness for arbitrary free product von neumann algebras

full factors bicentralizer flow and approximately inner automorphisms we show that a factor m is full if and only if the c algebra generated by its left and right regular representations contains the compact operators we prove that the bicentralizer flow of a type mathrm iii 1 factor is always ergodic as a consequence for any type mathrm iii 1 factor m and any lambda in 0 1 there exists an irreducible afd type mathrm iii lambda subfactor with expectation in m moreover any type mathrm iii 1 factor m which satisfies m cong m mathbin overline otimes r lambda for some lambda in 0 1 has trivial bicentralizer finally we give a counter example to the characterization of approximately inner automorphisms conjectured by connes and we prove a weaker version of this conjecture in particular we obtain a new proof of kawahigashi sutherland takesaki s result that every automorphism of the afd type mathrm iii 1 factor is approximately inner

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved