phase transitions in the estimation of event rate a path integral analysis we try to capture the instantaneous rate of event occurrence as a function of time a jagged or smooth rate function can be arbitrarily chosen but it is nevertheless possible to select a plausible smoothness according to the principle of maximum likelihood upon optimization the smoothness may diverge indicating that the data are insufficient to uncover the time dependence of the underlying rate by evaluating the likelihood function through a marginalization path integral we found not only first order but also second order phase transitions leading to the divergence of the optimized smoothness

uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries abstract we present a stochastic finite volume ader sfv ader methodology for uncertainty quantification uq in the general framework of systems of hyperbolic balance laws with uncertainty in parameters the resulting method is weakly intrusive meaning that deterministic solvers need minor modifications to include random contributions and has no theoretical accuracy barrier an illustration of the second order version for the viscous burgers equation with uncertain viscosity coefficient is first given in detail under different deterministic initial conditions the attainment of the theoretically expected convergence rate is demonstrated empirically results and features of our scheme are discussed we then extend the sfv ader method to a non linear hyperbolic system with source terms that models one dimensional blood flow in arteries assuming uncertainties in a set of parameters of the problem results are then compared with measurements in a well defined 1 1 experimental replica of the network of largest arteries in the human systemic circulation for a 99 confidence level severe influence of parameter fluctuations is seen in pressure profiles while secondary effects on flow rate profiles become visible particularly in terminal branches results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws

inhomogeneous and modulated gamma processes summary two classes of point processes the inhomogeneous and modulated gamma processes are defined asymptotic test statistics of stationarity are derived for modulated gamma processes some examples are studied in detail and applied to two sets of data

estimating a state space model from point process observations a widely used signal processing paradigm is the state space model the state space model is defined by two equations an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit latent stimulus we develop an algorithm to estimate a state space model observed through point process measurements we represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus given the latent process neural spiking activity is characterized as a general point process defined by its conditional intensity function we develop an approximate expectation maximization em algorithm to estimate the unobservable state space process its parameters and the parameters of the point process the em algorithm combines a point process recursive nonlinear filter algorithm the fixed interval smoothing algorithm and the state space covariance algorithm to compute the complete data log likelihood efficiently we use a kolmogorov smirnov test based on the time rescaling theorem to evaluate agreement between the model and point process data we illustrate the model with two simulated data examples an ensemble of poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local bernoulli process

on the equivalence of dynamically orthogonal and bi orthogonal methods theory and numerical simulations abstract the karhunen loeve kl decomposition provides a low dimensional representation for random fields as it is optimal in the mean square sense although for many stochastic systems of practical interest described by stochastic partial differential equations spdes solutions possess this low dimensional character they also have a strongly time dependent form and to this end a fixed in time basis may not describe the solution in an efficient way motivated by this limitation of standard kl expansion sapsis and lermusiaux 2009 26 developed the dynamically orthogonal do field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty lives but also the stochastic characteristics of uncertainty recently cheng et al 2013 28 introduced an alternative approach the bi orthogonal bo method which performs the exact same tasks i e it evolves the spatial basis and the stochastic characteristics of uncertainty in the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence in the sense that one method is an exact reformulation of the other we show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the bo and the do solutions we also examine a pathology of the bo equations that occurs when two eigenvalues of the solution cross resulting in an instantaneous infinite speed internal rotation of the computed spatial basis we demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the bo approach on the other hand it is observed that the bo is more stable in nonlinear problems involving a relatively large number of modes several examples linear and nonlinear are presented to illustrate the do and bo methods as well as their equivalence

rate distortion functions for gamma type sources under absolute log distortion measure when the information source is a continuous distribution and the rate distortion function is strictly larger than the shannon lower bound the explicit evaluation of the rate distortion function is not straightforward we evaluate the rate distortion function for an independent identically distributed gamma source with respect to the absolute log distortion measure the logarithmic transformation reduces this rate distortion problem to that under the absolute distortion measure extending the explicit evaluation of the rate distortion function for the gaussian sources we obtain the parametric form of the rate distortion function we show that the optimal distribution of reconstruction consists of a continuous component enclosed by left and right discrete components and the left discrete component vanishes when the acceptable distortion is small we further extend the result for a wider class of source distributions

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

a dynamically bi orthogonal method for time dependent stochastic partial differential equations i derivation and algorithms we propose a dynamically bi orthogonal method dybo to solve time dependent stochastic partial differential equations spdes the objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi orthogonal basis it is well known that the karhunen loeve expansion kle minimizes the total mean squared error and gives the sparsest representation of stochastic solutions however the computation of the kl expansion could be quite expensive since we need to form a covariance matrix and solve a large scale eigenvalue problem the main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the kl expansion unlike other reduced model methods our method constructs the reduced basis on the fly without the need to form the covariance matrix or to compute its eigendecomposition in the first part of our paper we introduce the derivation of the dynamically bi orthogonal formulation for spdes discuss several theoretical issues such as the dynamic bi orthogonality preservation and some preliminary error analysis of the dybo method we also give some numerical implementation details of the dybo methods including the representation of stochastic basis and techniques to deal with eigenvalue crossing in the second part of our paper 11 we will present an adaptive strategy to dynamically remove or add modes perform a detailed complexity analysis and discuss various generalizations of this approach an extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the dybo method

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

cluster based generalized multiscale finite element method for elliptic pdes with random coefficients abstract we propose a generalized multiscale finite element method gmsfem based on clustering algorithm to study the elliptic pdes with random coefficients in the multi query setting our method consists of offline and online stages in the offline stage we construct a small number of reduced basis functions within each coarse grid block which can then be used to approximate the multiscale finite element basis functions in addition we coarsen the corresponding random space through a clustering algorithm in the online stage we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre computed multiscale basis the new gmsfem can be applied to multiscale spde starting with a relatively coarse grid without requiring the coarsest grid to resolve the smallest scale of the solution the new method offers considerable savings in solving multiscale spdes numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation

the ornstein uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex cortical neurons of behaving animals generate irregular spike sequences recently there has been a heated discussion about the origin of this irregularity softky and koch 1993 pointed out the inability of standard single neuron models to reproduce the irregularity of the observed spike sequences when the model parameters are chosen within a certain range that they consider to be plausible shadlen and newsome 1994 on the other hand demonstrated that a standard leaky integrate and fire model can reproduce the irregularity if the inhibition is balanced with the excitation motivated by this discussion we attempted to determine whether the ornstein uhlenbeck process which is naturally derived from the leaky integration assumption can in fact reproduce higher order statistics of biological data for this purpose we consider actual neuronal spike sequences recorded from the monkey prefrontal cortex to calculate the higher order statistics of the interspike intervals consistency of the data with the model is examined on the basis of the coefficient of variation and the skewness coefficient which are respectively a measure of the spiking irregularity and a measure of the asymmetry of the interval distribution it is found that the biological data are not consistent with the model if the model time constant assumes a value within a certain range believed to cover all reasonable values this fact suggests that the leaky integrate and fire model with the assumption of uncorrelated inputs is not adequate to account for the spiking in at least some cortical neurons

statistical mechanics of image restoration we develop the statistical mechanics formulation of the image restoration problem pioneered by geman and geman 1984 using bayesian methods we establish the posterior probability distribution for restored images for given data corrupted image and prior assumptions about source and corruption process in the simplest cases studied here the posterior is controlled by a cost function analagous to the configurational energy of an ising model with local fields whose sense is defined by the data through a combination of monte carlo simulation and mean field theory we address three key issues first we explore the sensitivity of the posterior distribution to the choice of prior parameters we find phase transitions separating regions in which the distribution is effective data dominated from regions in which it is ineffective prior dominated second we examine the question of how best to use the posterior distribution to prescribe a single optimal restored image we argue that the mean of the posterior is in general to be preferred over the mode both in principle and in practice finally borrowing from monte carlo techniques for free energy calculations we address the question of prior parameter estimation within the evidence framework of gull 1989 and mackay 1992 our results suggest that parameters identified by this framework provide effective priors leading to optimal restoration only to the extent that the forms of the priors are well matched to the processes they claim to represent

histogram bin width selection for time dependent poisson processes in constructing a time histogram of the event sequences derived from a nonstationary point process we wish to determine the bin width such that the mean squared error of the histogram from the underlying rate of occurrence is minimized we find that the optimal bin widths obtained for a doubly stochastic poisson process and a sinusoidally regulated poisson process exhibit different scaling relations with respect to the number of sequences time scale and amplitude of rate modulation but both diverge under similar parametric conditions this implies that under these conditions no determination of the time dependent rate can be made we also apply the kernel method to these point processes and find that the optimal kernels do not exhibit any critical phenomena unlike the time histogram method

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

a heterogeneous stochastic fem framework for elliptic pdes we introduce a new concept of sparsity for the stochastic elliptic operator div a x which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure this new concept of sparsity motivates a heterogeneous stochastic finite element method hsfem framework for linear elliptic equations which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space we also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques the resulting hsfem involves two stages and suits the multi query setting in the offline stage the local stochastic structure of the solution space is identified in the online stage the equation can be efficiently solved for multiple forcing functions an online error estimation and correction procedure through monte carlo sampling is given numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the hsfem in the online stage

occam factors and model independent bayesian learning of continuous distributions learning of a smooth but nonparametric probability density can be regularized using methods of quantum field theory we implement a field theoretic prior numerically test its efficacy and show that the data and the phase space factors arising from the integration over the model space determine the free parameter of the theory smoothness scale self consistently this persists even for distributions that are atypical in the prior and is a step towards a model independent theory for learning continuous distributions finally we point out that a wrong parametrization of a model family may sometimes be advantageous for small data sets

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

differences in spiking patterns among cortical neurons spike sequences recorded from four cortical areas of an awake behaving monkey were examined to explore characteristics that vary among neurons we found that a measure of the local variation of interspike intervals lv is nearly the same for every spike sequence for any given neuron while it varies significantly among neurons the distributions of lv values for neuron ensembles in three of the four areas were found to be distinctly bimodal two groups of neurons classified according to the spiking irregularity exhibit different responses to the same stimulus this suggests that neurons in each area can be classified into different groups possessing unique spiking statistics and corresponding functional properties

dual dynamically orthogonal approximation of incompressible navier stokes equations with random boundary conditions abstract in this paper we propose a method for the strong imposition of random dirichlet boundary conditions in the dynamical low rank dlr approximation of parabolic pdes and in particular incompressible navier stokes equations we show that the dlr variational principle can be set in the constrained manifold of all s rank random fields with a prescribed value on the boundary expressed in low rank format with rank smaller then s we characterize the tangent space to the constrained manifold by means of a dual dynamically orthogonal dual do formulation in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions consistent with the original problem the dual do formulation is also convenient to include the incompressibility constraint when dealing with incompressible navier stokes equations we show the performance of the proposed dual do approximation on two numerical test cases the classical benchmark of a laminar flow around a cylinder with random inflow velocity and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from mri data with random inflow conditions coming from doppler measurements

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

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

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 dynamical low rank approximation method abstract in this paper we extend the dynamical low rank approximation method to the space of finite signed measures under this framework we derive stochastic low rank dynamics for stochastic differential equations sdes coming from classical stochastic dynamics or unraveling of lindblad quantum master equations we justify the proposed method by error analysis and also numerical examples for applications in solving high dimensional sde stochastic burgers equation and high dimensional lindblad equation

computing nearly singular solutions using pseudo spectral methods in this paper we investigate the performance of pseudo spectral methods in computing nearly singular solutions of fluid dynamics equations we consider two different ways of removing the aliasing errors in a pseudo spectral method the first one is the traditional 2 3 dealiasing rule the second one is a high 36th order fourier smoothing which keeps a significant portion of the fourier modes beyond the 2 3 cut off point in the fourier spectrum for the 2 3 dealiasing method both the 1d burgers equation and the 3d incompressible euler equations are considered we demonstrate that the pseudo spectral method with the high order fourier smoothing gives a much better performance than the pseudo spectral method with the 2 3 dealiasing rule moreover we show that the high order fourier smoothing method captures about 12 15 more effective fourier modes in each dimension than the 2 3 dealiasing method for the 3d euler equations the gain in the effective fourier codes for the high order fourier smoothing method can be as large as 20 over the 2 3 dealiasing method another interesting observation is that the error produced by the high order fourier smoothing method is highly localized near the region where the solution is most singular while the 2 3 dealiasing method tends to produce oscillations in the entire domain the high order fourier smoothing method is also found be very stable dynamically no high frequency instability has been observed in the case of the 3d euler equations the energy is conserved up to at least six digits of accuracy throughout the computations 2007 elsevier inc all rights reserved

simple and accurate one sided inference based on a class of m estimators summary this paper presents a simple method of calculating one sided p values which yield accurate inferences for a simple null hypothesis or equivalently accurate one sided con fidence limits regarding a scalar parameter in the presence of nuisance parameters the method which extends the work of diciccio et al 2001 and lee young 2005 in the context of adjusted likelihood estimation is based on a test statistic analogous to the signed root of the loglikelihood but derived from the objective function of an m estimator from a certain class monte carlo simulation is used to avoid the need for onerous analytical calculations typical of competing procedures based on edgeworth or saddle point approximations the specific class of m estimators under consideration is character ised by the requirement that the associated test statistic has constant variance to second order this class contains among others the maximum likelihood estimator as well as variants of commonly used m estimators of location such as huber s 1964 proposal 2

estimating instantaneous irregularity of neuronal firing cortical neurons in vivo had been regarded as poisson spike generators that convey no information other than the rate of random firing recently using a metric for analyzing local variation of interspike intervals researchers have found that individual neurons express specific patterns in generating spikes which may symbolically be termed regular random or bursty rather invariantly in time in order to study the dynamics of firing patterns in greater detail we propose here a bayesian method for estimating firing irregularity and the firing rate simultaneously for a given spike sequence and we implement an algorithm that may render the empirical bayesian estimation practicable for data comprising a large number of spikes application of this method to electrophysiological data revealed a subtle correlation between the degree of firing irregularity and the firing rate for individual neurons irregularity of firing did not deviate greatly around the low degree of dependence on the firing rate and remained practically unchanged for individual neurons in the cortical areas v1 and mt whereas it fluctuated greatly in the lateral geniculate nucleus of the thalamus this indicates the presence and absence of autocontrolling mechanisms for maintaining patterns of firing in the cortex and thalamus respectively

empirical bayes interpretations of random point events given a sequence of apparently random point events such as neuronal spikes one may interpret them as being derived either irregularly in time from a constant rate or regularly from a fluctuating rate to determine which interpretation is more plausible in any given case we employ the empirical bayes method for a sequence of point events derived from a rate fluctuating gamma process the marginal likelihood function can possess two local maxima and the system exhibits a first order phase transition representing the switch of the most plausible interpretation from one to the other

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

an adaptive dynamically low dimensional approximation method for multiscale stochastic diffusion equations abstract in this paper we propose a dynamically low dimensional approximation method to solve a class of time dependent multiscale stochastic diffusion equations in cheng et al 2013 a dynamically bi orthogonal dybo method was developed to explore low dimensional structures of stochastic partial differential equations spdes and solve them efficiently however when the spdes have multiscale features in physical space the original dybo method becomes expensive to address this issue we construct multiscale basis functions within the framework of generalized multiscale finite element method gmsfem for dimension reduction in the physical space to further improve the accuracy we also perform online procedure to construct online adaptive basis functions in the stochastic space we use the generalized polynomial chaos gpc basis functions to represent the stochastic part of the solutions numerical results are presented to demonstrate the efficiency of the proposed method in solving time dependent pdes with multiscale and random features

optimizing time histograms for non poissonian spike trains the time histogram is a fundamental tool for representing the inhomogeneous density of event occurrences such as neuronal firings the shape of a histogram critically depends on the size of the bins that partition the time axis in most neurophysiological studies however researchers have arbitrarily selected the bin size when analyzing fluctuations in neuronal activity a rigorous method for selecting the appropriate bin size was recently derived so that the mean integrated squared error between the time histogram and the unknown underlying rate is minimized shimazaki shinomoto 2007 this derivation assumes that spikes are independently drawn from a given rate however in practice biological neurons express non poissonian features in their firing patterns such that the spike occurrence depends on the preceding spikes which inevitably deteriorate the optimization in this letter we revise the method for selecting the bin size by considering the possible non poissonian features improvement in the goodness of fit of the time histogram is assessed and confirmed by numerically simulated non poissonian spike trains derived from the given fluctuating rate for some experimental data the revised algorithm transforms the shape of the time histogram from the poissonian optimization method

real time reduced order modeling of stochastic partial differential equations via time dependent subspaces abstract we present a new methodology for the real time reduced order modeling of stochastic partial differential equations called the dynamically bi orthonormal dbo decomposition in this method the stochastic fields are approximated by a low rank decomposition to spatial and stochastic subspaces each of these subspaces is represented by a set of orthonormal time dependent modes we derive exact evolution equations of these time dependent modes and the evolution of the factorization of the reduced covariance matrix we show that dbo is equivalent to the dynamically orthogonal do 1 and bi orthogonal bo 2 decompositions via linear and invertible transformation matrices that connect dbo to do and bo however dbo shows several improvements compared to do and bo i dbo performs better than do and bo for cases with ill conditioned covariance matrix ii in contrast to bo the issue of eigenvalue crossing is not present in the dbo formulation iii in contrast to do the stochastic modes are orthonormal which leads to more accurate representation of the stochastic subspace we study the convergence properties of the method and compare it to the do and bo methods for demonstration we consider three cases i stochastic linear advection equation ii stochastic burgers equation and iii stochastic incompressible flow over a bump in a channel overall we observe improvements in the numerical accuracy of dbo compared against do and bo

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

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

field theories for learning probability distributions imagine being shown n samples of random variables drawn independently from the same distribution what can you say about the distribution in general of course the answer is nothing unless you have some prior notions about what to expect from a bayesian point of view one needs an a priori distribution on the space of possible probability distributions which defines a scalar field theory in one dimension free field theory with a normalization constraint provides a tractable formulation of the problem and we discuss generalizations to higher dimensions

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

dynamically orthogonal tensor methods for high dimensional nonlinear pdes abstract we develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high dimensional time dependent nonlinear partial differential equations pdes the key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets this process which can be conveniently visualized in terms of binary trees yields series expansions analogous to the classical tensor train and hierarchical tucker tensor formats by enforcing dynamic orthogonality conditions at each level of the binary tree we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition this allows us to effectively compute the solution to high dimensional time dependent nonlinear pdes on tensor manifolds of constant rank with no need for rank reduction methods we also propose new algorithms for dynamic addition and removal of modes within each subspace numerical examples are presented and discussed for high dimensional hyperbolic and parabolic pdes in bounded domains