random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

some counting questions for matrices with restricted entries we obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box in an arbitrary location and a given determinant we then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues finally we outline some open questions

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces stochastic collocation sc methods for uncertainty quantification uq in computational problems are usually limited to hypercube probability spaces due to the structured grid of their quadrature rules nonhypercube probability spaces with an irregular shape of the parameter domain do however occur in practical engineering problems for example production tolerances and other geometrical uncertainties can lead to correlated random inputs on nonhypercube domains in this paper a simplex stochastic collocation ssc method is introduced as a multielement uq method based on simplex elements that can efficiently discretize nonhypercube probability spaces it combines the delaunay triangulation of randomized sampling at adaptive element refinements with polynomial extrapolation to the boundaries of the probability domain the robustness of the extrapolation is quantified by the definition of the essentially extremum diminishing eed robustness principle numerical examples show that the resulting ssc eed method achieves superlinear convergence and a linear increase of the initial number of samples with increasing dimensionality these properties are demonstrated for uniform and nonuniform distributions and correlated and uncorrelated parameters in problems with 15 dimensions and discontinuous responses

reaction spreading on graphs we study reaction diffusion processes on graphs through an extension of the standard reaction diffusion equation starting from first principles we focus on reaction spreading i e on the time evolution of the reaction product m t at variance with pure diffusive processes characterized by the spectral dimension d s the important quantity for reaction spreading is found to be the connectivity dimension d l numerical data in agreement with analytical estimates based on the features of n independent random walkers on the graph show that m t t d l in the case of erdos renyi random graphs the reaction product is characterized by an exponential growth m t e t with proportional to ln k where k is the average degree of the graph

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

high dimensional integration of smooth functions over cubes we construct a new algorithm for the numerical integration of functions that are defined on a d dimensional cube it is based on the clenshaw curtis rule for d 1 and on smolyak s construction this way we make the best use of the smoothness properties of any nonperiodic function we prove error bounds showing that our algorithm is almost optimal up to logarithmic factors for different classes of functions with bounded mixed derivative numerical results show that the new method is very competitive in particular for smooth integrands and d ge 8

sparse pseudospectral approximation method multivariate global polynomial approximations such as polynomial chaos or stochastic collocation methods are now in widespread use for sensitivity analysis and uncertainty quantification the pseudospectral variety of these methods uses a numerical integration rule to approximate the fourier type coefficients of a truncated expansion in orthogonal polynomials for problems in more than two or three dimensions a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation however when using a sparse rule to approximately integrate these coefficients one often finds unacceptable errors in the coefficients associated with higher degree polynomials by reexamining smolyak s algorithm and exploiting the connections between interpolation and projection in tensor product spaces we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule the compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation

quasi monte carlo integration the standard monte carlo approach to evaluating multidimensional integrals using pseudo random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement as an alternative to the random methods it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo random sequences with more uniformly distributed sequences known as quasi random in this paper quasi random halton sobol and faure and pseudo random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand including variance variation smoothness and dimension the results show that variation which plays an important role in the theoretical upper bound given by the koksma hlawka inequality does not affect convergence while variance the determining factor in random monte carlo is shown to provide a rough upper bound but does not accurately predict performance in general quasi monte carlo methods are superior to random monte carlo but the advantage may be slight particularly in high dimensions or for integrands that are not smooth for discontinuous integrands we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi monte carlo is only slightly larger than 1 2 in high dimensions more 21 refs 6 figs 5 tabs less

stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method an adaptive stochastic spectral projection method is deployed for the uncertainty quantification in limit cycle oscillations of an elastically mounted two dimensional lifting surface in a supersonic flow field variabilities in the structural parameters are propagated in the aeroelastic system which accounts for nonlinear restoring force and moment by means of hardening cubic springs the physical nonlinearities promote sharp and sudden flutter onset for small change of the reduced velocity in a stochastic context this behavior translates to steep solution gradients developing in the parametric space a remedy is to expand the stochastic response of the airfoil on a piecewise generalized polynomial chaos basis accurate approximation andaffordable computational costs are obtained using sensitivity based adaptivity for various types of supersonic stochastic responses depending on the selected values of the mach number on the bifurcation map sensitivity analysis via sobol indices shows how the probability density function of the peak pitch amplitude responds to combined uncertainties e g the elastic axis location torsional stiffness and flap angle we believe that this work demonstrates the capability and flexibility of the approach for more reliable predictions of realistic aeroelastic systems subject to a moderate number of uncertainties

a gpc based approach to uncertain transonic aerodynamics abstract the present paper focus on the stochastic response of a two dimensional transonic airfoil to parametric uncertainties both the freestream mach number and the angle of attack are considered as random parameters and the generalized polynomial chaos gpc theory is coupled with standard deterministic numerical simulations through a spectral collocation projection methodology the results allow for a better understanding of the flow sensitivity to such uncertainties and underline the coupling process between the stochastic parameters two kinds of non linearities are critical with respect to the skin friction uncertainties on one hand the leeward shock movement characteristic of the supercritical profile and on the other hand the boundary layer separation on the aft part of the airfoil downstream the shock the sensitivity analysis thanks to the sobol decomposition shows that a strong non linear coupling exists between the uncertain parameters comparisons with the one dimensional cases demonstrate that the multi dimensional parametric study is required to get the correct shape and magnitude of the standard deviation distributions of the flow quantities such as pressure and skin friction

description of diffusive and propagative behavior on fractals the known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes after that we propose a description developed in the context of the intrinsic metric of fractals which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime we show that our approach has a stronger physical justification than previous works on this field the most important result we present is the introduction of a dependence on time and space for the conductivity in fractals which is deduced by scaling arguments and supported by computer simulations finally the diffusion equation is used to introduce the possibility of reaction diffusion processes on fractals and analyze their properties specifically an analytic expression for the speed of the corresponding travelling fronts which can be of great interest for application purposes is derived

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

stochastic design optimization application to reacting flows one concern in the optimization of realistic complex systems is to ensure that the optimal model response is reliable and robust with respect to the inherent uncertainties associated with the design variables constraints and the objective function traditional optimization techniques together with uncertainty analysis are computationally expensive and time consuming when it comes to identify what drives the response variability a stochastic optimization framework combining stochastic surrogate model representation and optimization algorithm is proposed a generalized polynomial chaos stochastic representation is used as the surrogate model this representation is obtained from a high order stochastic collocation method the surrogate model can then be put to use within an appropriate optimization algorithm and provides fast approximations at new design points this approach allows both sensitivity and optimization analysis the stochastic optimization method is applied to a multi layer reacting flow device the geometric configuration is assumed to be uncertain the structure design is optimized to maximize the energy transfer between the reacting flow and the device moving parts gradient descent and simulated annealing optimization techniques are successfully tested on the kinetic energy surrogate model of the device inner parts

front propagation in reactive systems with anomalous diffusion abstract we study front propagation of reactive fields in systems whose diffusive behavior is anomalous both superdiffusive and subdiffusive the features of the front propagation depend not only on the scaling exponent x t 2 t 2 but also on the detailed shape of the probability distribution of the diffusive process from the analysis of different systems we have three possible behavior of front propagation the usual fisher kolmogorov like scenario i e the field has a spatial exponential tail moving with constant speed v f and thickness the field has a spatial exponential tail but v f and change in time as a power law and finally the field has a spatial power law tail and v f increases exponentially in time a linear analysis of the front tail is in quantitative agreement with the numerical simulations it is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation moreover if the probability distribution of the transport process follows the scaling relation given by the flory argument the front propagation is standard fisher kolmogorov like even in presence of super or sub diffusion

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

exceptional splitting of reductions of abelian surfaces heuristics based on the sato tate conjecture and the lang trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction we prove this result for abelian surfaces with real multiplication as in previous work by charles and elkies this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite the proof relies on the arakelov intersection theory on hilbert modular surfaces

issues in computational fluid dynamics code verification and validation a broad range of mathematical modeling errors of fluid flow physics and numerical approximation errors are addressed in computational fluid dynamics cfd it is strongly believed that if cfd is to have a major impact on the design of engineering hardware and flight systems the level of confidence in complex simulations must substantially improve to better understand the present limitations of cfd simulations a wide variety of physical modeling discretization and solution errors are identified and discussed here discretization and solution errors refer to all errors caused by conversion of the original partial differential or integral conservation equations representing the physical process to algebraic equations and their solution on a computer the impact of boundary conditions on the solution of the partial differential equations and their discrete representation will also be discussed throughout the article clear distinctions are made between the analytical mathematical models of fluid dynamics and the numerical models lax s equivalence theorem and its frailties in practical cfd solutions are pointed out distinctions are also made between the existence and uniqueness of solutions to the partial differential equations as opposed to the discrete equations two techniques are briefly discussed for the detection and quantification of certain types ofmore discretization and grid resolution errors less

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

spreading dynamics in heterogeneous graphs beyond the assortativity coefficient abstract we study spreading dynamics of a reaction diffusion process in a special class of heterogeneous graphs with poissonian degree distribution and composed of both local and long range links the behavior of the spreading dynamics on such networks are investigated by relating them to the topological features of graphs we find that the degree of assortativity can give just some indication about the large scale behavior of the spreading dynamics while a detailed description of the process can be addressed by introducing new more appropriate topological quantities linked to the distance between nodes

propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach abstract this paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients these typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory the concepts of projection orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that in the stochastic case inner products refer to expectation operations specifically the karhunen loeve expansion is used to discretize these processes into a denumerable set of random variables thus providing a denumerable function space in which the problem is cast the polynomial chaos expansion is then used to represent the solution in this space and the coefficients in the expansion are evaluated as generalized fourier coefficients via a galerkin procedure in the hilbert space of random variables

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

dynamical features of reaction diffusion fronts in fractals the speed of front propagation in fractals is studied by using i the reduction of the reaction transport equation into a hamilton jacobi equation and ii the local equilibrium approach different equations proposed for describing transport in fractal media together with logistic reaction kinetics are considered finally we analyze the main features of wave fronts resulting from this dynamic process i e why they are accelerated and what is the exact form of this acceleration

a quantitative model independent method for global sensitivity analysis of model output a new method for sensitivity analysis sa of model output is introduced it is based on the fourier amplitude sensitivity test fast and allows the computation of the total contribution of each input factor to the output s variance the term total here means that the factor s main effect as well as all the interaction terms involving that factor are included although computationally different the very same measure of sensitivity is offered by the indices of sobol the main advantages of the extended fast are its robustness especially at low sample size and its computational efficiency the computational aspects of the extended fast are described these include 1 the definition of new sets of parametric equations for the search curve exploring the input space 2 the selection of frequencies for the parametric equations and 3 the procedure adopted to estimate the total contributions we also address the limitations of other global sa methods and suggest that the total effect indices are id