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

symmetrically trimmed least squares estimation for tobit models this paper proposes alternatives to maximum likelihood estimation of the censored and truncated regression models known to economists as tobit models the proposed estimators are based upon symmetric censoring or truncation of the upper tail of the distribution of the dependent variable unlike methods based on the assumption of identically distributed gaussian errors the estimators are semiparametric in the sense that they are consistent and asymptotically normal for a wide class of symmetric error distributions with heteroskedasticity of unknown form the paper gives the regularity conditions and proofs of these large sample properties demonstrates how to construct consistent estimators of the asymptotic covariance matrices and presents the results of a simulation study for the censored case extensions and limitations of the approach are also considered

asymptotic properties of the product limit estimate under random truncation on donne une description precise du comportement asymptotique de l estimateur limite produit incluant une forme explicite simple de la structure de covariance asymptotique

estimation in linear models with censored data summary three methods for linear regression with censored data are considered that of buckley james 1979 a proposed simpler nonparametric method and a normal model for censored data a new estimator for the variance of the error term in buckley james s 1979 procedure is proposed and simulations comparing the three methods are described many methods have been proposed for handling regression problems in which the dependent variable may be censored we focus on three methods for the linear model y xt3 e two methods are nonparametric so that the distribution of e is unspecified while the third assumes that e is normally distributed the nonparametric methods are analogous to the expectation maximization em algorithm introduced by dempster laird rubin 1977 which is broadly applicable for computing maximum likelihood estimates in 2 we discuss each of these methods and propose a new estimator of the variance of e for the buckley james method in 3 we give simulation results which compare the three methods

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

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

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

numerical methods for high dimensional probability density function equations in this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables these types of equations arise naturally in many different areas of mathematical physics e g in particle systems liouville and boltzmann equations stochastic dynamical systems fokker planck and dostupov pugachev equations random wave theory malakhov saichev equations and coarse grained stochastic systems mori zwanzig equations we propose three different classes of new algorithms addressing high dimensionality the first one is based on separated series expansions resulting in a sequence of low dimensional problems that can be solved recursively and in parallel by using alternating direction methods the second class of algorithms relies on truncation of interaction in low orders that resembles the bogoliubov born green kirkwood yvon bbgky framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations the third class of algorithms is based on high dimensional model representations e g the anova method and probabilistic collocation methods a common feature of all these approaches is that they are reducible to the problem of computing the solution to high dimensional equations via a sequence of low dimensional problems the effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations with up to 120 variables

low rank matrix completion by riemannian optimization the matrix completion problem consists of finding or approximating a low rank matrix based on a few samples of this matrix we propose a new algorithm for matrix completion that minimizes the least square distance on the sampling set over the riemannian manifold of fixed rank matrices the algorithm is an adaptation of classical nonlinear conjugate gradients developed within the framework of retraction based optimization on manifolds we describe all the necessary objects from differential geometry necessary to perform optimization over this low rank matrix manifold seen as a submanifold embedded in the space of matrices in particular we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions finally we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations the numerical experiments indicate that our approach

time integration of rank constrained tucker tensors dynamical low rank approximation in the tucker tensor format of given large time dependent tensors and of tensor differential equations is the subject of this paper in particular a discrete time integration method for rank constrained tucker tensors is presented and analyzed it extends the known projector splitting integrator for dynamical low rank approximation of matrices to tucker tensors and is shown to inherit the same favorable properties the integrator is based on iteratively applying the matrix projector splitting integrator to tensor unfoldings but with inexact solution in a substep it has the property that it reconstructs time dependent tucker tensors of the given rank exactly the integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings numerical examples with time dependent problems from quantum physics and tensor optimization methods illustrate our theoretical results

a pseudo r2 measure for limited and qualitative dependent variable models abstract a pseudo r 2 measure for limited and qualitative dependent variable models is proposed a consistent estimator of the variance of the disturbance term is also suggested the pseudo r 2 measure is shown to have the same asymptotic limit as the conventional r 2 the measure is adapted to the logit model in an empirical example of female car ownership modeling the pseudo r 2 is applicable to a wide set of limited and qualitative dependent variable models and offers a valuable tool for model evaluation

numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques we consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary as a paradigm for such problems we simply refer to stokes flow after a brief review of the problem and of its well posedness we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints we exploit a penalty method to account for them we perform the analysis of the method in terms of consistency boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization since the additional terms introduced to account for the defective boundary conditions are non local we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy in contrast to alternative discretization methods based on lagrange multipliers accounting for the constraints on the boundary the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation indeed it does not require neither multipliers nor sub iterations or additional adjoint problems with respect to the reference problem at hand

an r squared measure of goodness of fit for some common nonlinear regression models abstract for regression models other than the linear model r squared type goodness of fit summary statistics have been constructed for particular models using a variety of methods we propose an r squared measure of goodness of fit for the class of exponential family regression models which includes logit probit poisson geometric gamma and exponential this r squared is defined as the proportionate reduction in uncertainty measured by kullback leibler divergence due to the inclusion of regressors under further conditions concerning the conditional mean function it can also be interpreted as the fraction of uncertainty explained by the fitted model

parallel tensor methods for high dimensional linear pdes abstract high dimensional partial differential equations pdes arise in a number of fields of science and engineering where they are used to describe the evolution of joint probability functions their examples include the boltzmann and fokker planck equations we develop new parallel algorithms to solve such high dimensional pdes the algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition both implicit and explicit integration schemes are presented and discussed we demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the boltzmann equation

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

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

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

stability analysis of hierarchical tensor methods for time dependent pdes abstract we address the question of whether it is possible to stably integrate time dependent high dimensional linear pdes with hierarchical tensor methods and explicit time stepping schemes to this end we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank we also argue that the applicability of pde solvers with explicit time stepping may be limited by time step restriction dependent on the dimension of the problem numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic pdes

tensor methods for the boltzmann bgk equation abstract we present a tensor decomposition method to solve the boltzmann transport equation bte in the bhatnagar gross krook approximation the method represents the six dimensional bte as a set of six one dimensional problems which are solved with the alternating least squares algorithm and the discrete fourier transform at n collocation points we use this method to predict the equilibrium distribution steady state simulation and a non equilibrium distribution returning to the equilibrium transient simulation our numerical experiments demonstrate n log n scaling unlike many bte specific numerical techniques the numerical tensor decomposition method we propose is a general technique that can be applied to other high dimensional systems

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

inexact accurate partitioned algorithms for fluid structure interaction problems with finite elasticity in haemodynamics in this paper we consider the numerical solution of the three dimensional fluid structure interaction problem in haemodynamics in the case of real geometries physiological data and finite elasticity vessel deformations we study some new inexact schemes obtained from semi implicit approximations which treat exactly the physical interface conditions while performing just one or few iterations for the management of the interface position and of the fluid and structure non linearities we show that such schemes allow to improve the efficiency while preserving the accuracy of the related exact implicit scheme to do this we consider both a simple analytical test case and two real cases of clinical interest in haemodynamics we also provide an error analysis for a simple differential model problem when a bdf method is considered for the time discretization and only few newton iterations are performed at each temporal instant

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

parallel als algorithm for solving linear systems in the hierarchical tucker representation tensor network formats are an efficient tool for numerical computations in many dimensions yet even this tool often becomes too time and memory consuming for a single compute node when applied to problems of scientific interest intending to overcome such limitations we present and analyze a parallelization scheme for algorithms based on the hierarchical tucker representation which distributes the network vertices and their associated computations over a set of distributed memory processors we then propose a modified version of the alternating least squares algorithm for solving linear systems amenable to parallelization according to the aforementioned scheme and highlight technical considerations important for obtaining an efficient and stable implementation our numerical experiments support the theoretical assertion that the parallel scaling of this algorithm is constrained only by the dimensionality and the rank uniformity of the targeted problem

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

a fully symmetric nonlinear biorthogonal decomposition theory for random fields abstract we present a general approach for nonlinear biorthogonal decomposition of random fields the mathematical theory is developed based on a fully symmetric operator framework that unifies different types of expansions and allows for a simple formulation of necessary and sufficient conditions for their completeness the key idea of the method relies on an equivalence between nonlinear mappings of hilbert spaces and local inner products i e inner products that may be functionals of the random field being decomposed this extends previous work on the subject and allows for an effective formulation of field dependent and field independent representations the proposed new methodology can be applied in many areas of mathematical physics for stochastic low dimensional modelling of partial differential equations and dimensionality reduction of complex nonlinear phenomena an application to a transient stochastic heat conduction problem in a one dimensional infinite medium is presented and discussed

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

analysis and optimization of robin robin partitioned procedures in fluid structure interaction problems in the solution of fluid structure interaction fsi problems partitioned procedures are modular algorithms that involve separate fluid and structure solvers that interact in an iterative framework through the exchange of suitable transmission conditions at the fs interface in this work we study using fourier analysis the convergence of partitioned algorithms based on robin transmission conditions we derive for different models of the fluid and the structure a frequency dependent reduction factor at each iteration of the partitioned algorithm which is minimized by choosing optimal values of the coefficients in the robin transmission conditions two dimensional numerical results are also reported which highlight the effectiveness of the optimization procedure

least absolute deviations estimation for the censored regression model this paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression or censored tobit model the proposed estimator is a generalization of least absolute deviations estimation for the standard linear model and unlike estimation methods based on the assumption of normally distributed error terms the estimator is consistent and asymptotically normal for a wide class of error distributions and is also robust to heteroscedasticity the paper gives the regularity conditions and proofs of these large sample results and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances

a new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics we consider the incompressible navier stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed there are basically two strategies for solving these defective boundary problems the variational approach see j heywood r rannacher and s turek internat j numer methods fluids 22 1996 pp 325 352 and the augmented formulation see l formaggia j f gerbeau f nobile and a quarteroni siam j numer anal 40 2002 pp 376 401 and a veneziani and c vergara internat j numer methods fluids 47 2005 pp 803 816 however both of these approaches present some drawbacks for the flow rate problem the former resorts to nonstandard functional spaces which are quite difficult to discretize on the other hand for the mean pressure problem it yields exact solutions only in very special cases the latter is applicable only to the flow rate problem since for the mean pressure problem it provides unfeasible boundary conditions in this paper we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional this approach allows us to treat the two kinds of problems flow rate and mean pressure successfully within the same framework which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries moreover it is more versatile being prone to be extended to other kinds of defective conditions we analyze the problems obtained with this approach and propose some algorithms for their numerical solution several numerical results are presented supporting the effectiveness of our approach

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

new evolution equations for the joint response excitation probability density function of stochastic solutions to first order nonlinear pdes by using functional integral methods we determine new evolution equations satisfied by the joint response excitation probability density function pdf associated with the stochastic solution to first order nonlinear partial differential equations pdes the theory is presented for both fully nonlinear and for quasilinear scalar pdes subject to random boundary conditions random initial conditions or random forcing terms particular applications are discussed for the classical linear and nonlinear advection equations and for the advection reaction equation by using a fourier galerkin spectral method we obtain numerical solutions of the proposed response excitation pdf equations these numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi element probabilistic collocation methods it is found that the response excitation approach yields accurate predictions of the statistical properties of the system in addition it allows to directly ascertain the tails of probabilistic distributions thus facilitating the assessment of rare events and associated risks the computational cost of the response excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the pde is subject to high dimensional random boundary or initial conditions the question of high dimensionality for evolution equations involving multidimensional joint response excitation pdfs is also addressed

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

splitting methods based on algebraic factorization for fluid structure interaction we discuss in this paper the numerical approximation of fluid structure interaction fsi problems dealing with strong added mass effect we propose new semi implicit algorithms based on inexact block lu factorization of the linear system obtained after the space time discretization and linearization of the fsi problem as a result the fluid velocity is computed separately from the coupled pressure structure velocity system at each iteration reducing the computational cost we investigate explicit implicit decomposition through algebraic splitting techniques originally designed for the fsi problem this approach leads to two different families of methods which extend to fsi the algebraic pressure correction method and the yosida method two schemes that were previously adopted for pure fluid problems furthermore we have considered the inexact factorization of the fluid structure system as a preconditioner the numerical properties of these methods have been tested on a model problem representing a blood vessel system

exact solution to the inverse womersley problem for pulsatile flows in cylindrical vessels with application to magnetic particle targeting an exact solution to the inverse womersley problem was derived for the fully developed laminar pulsatile flow of a viscous newtonian fluid within a circular cylindrical vessel with rigid walls in particular given an arbitrary time periodic flow rate the axisymmetric velocity profile was obtained by means of two neat and computable maps relating the corresponding fourier coefficients the study of such an inverse problem is motivated by the fact that flow rate is the main physical quantity which can be actually measured in many practical situations the hypothesis of a fully developed flow was deliberately introduced in order to obtain an analytical solution otherwise hardly achievable despite the intrinsic simplifications associated with the adopted position which restrict the applicability of our results to 3d finite length complex domains and non newtonian fluids the obtained solution provides a benchmark and at the same time an approximation for the inverse problem of pulsatile flows it may serve as a debugging tool for more ambitious numerical approaches based on realistic data and can also be used as an improved source of boundary data as expected the main advantage of our analytical solutions compared to fully numerical approaches resides in computational efficiency this was quantitatively assessed through numerical tests moreover the proposed solution was applied in the context of magnetic particle targeting to highlight some peculiar effects on particle trajectories and capture efficiency due to pulsatility such a transport problem is increasingly drawing the attention of an interdisciplinary community ranging from physicians to biomedical engineers physicists and roboticists thanks to its potential for targeted therapy up to remote guidance of intravascular devices more in general the obtained benchmark solution holds potential for effectively exploitation in an interdisciplinary context

optimization on the hierarchical tucker manifold applications to tensor completion in this work we develop an optimization framework for problems whose solutions are well approximated by hierarchical tucker ht tensors an efficient structured tensor format based on recursive subspace factorizations by exploiting the smooth manifold structure of these tensors we construct standard optimization algorithms such as steepest descent and conjugate gradient for completing tensors from missing entries our algorithmic framework is fast and scalable to large problem sizes as we do not require svds on the ambient tensor space as required by other methods moreover we exploit the structure of the gramian matrices associated with the ht format to regularize our problem reducing overfitting for high subsampling ratios we also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries these samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios using these algorithms we successfully interpolate large scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow

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

adaptive leja sparse grid constructions for stochastic collocation and high dimensional approximation we propose an adaptive sparse grid stochastic collocation approach based upon leja interpolation sequences for approximation of parameterized functions with high dimensional parameters leja sequences are arbitrarily granular any number of nodes may be added to a current sequence producing a new sequence and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions when undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables this paper establishes that a certain weighted formulation of one dimensional leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the gauss quadrature nodes associated with the weight function this property is true even for unbounded domains we apply the leja sparse grid approach to several high dimensional problems and

on the asymptotic bias of the ordinary least squares estimator of the tobit model this paper presents a precise characterization of the bias of least squares in two limited dependent variable models the tobit model and the truncated regression model for the cases considered the method of moments can be used to correct the bias of ols for more general cases the results provide approximations which appear to be relatively robust 13 and o in this paper we present a precise characterization of that bias for the particular case in which xt as well as is normally distributed we also show that the bias of the ols slope estimator can be corrected by dividing each estimate by the sample proportion of nonlimit observations other structural parameters can be consistently estimated in a similar fashion we present some evidence on the effect of nonnormality with respect to the predictions obtained in the normal model the case in which the sample contains only nonlimit observation the truncated regression model is considered elsewhere olsen 7 we analyze the relationship between his results and ours and derive some predictions of the normal model with respect to the seriousness of truncation bias

an approximate method for solving incompressible navier stokes problems with flow rate conditions we consider the incompressible navier stokes problem with flow rate boundary conditions this problem has been investigated in j heywood r rannacher s turek artificial boundaries and flux and pressure conditions for the incompressible navier stokes equations int j numer methods fluids 22 1996 325 352 l formaggia j f gerbeau f nobile a quarteroni numerical treatment of defective boundary conditions for the navier stokes equation siam j numer anal 40 1 2002 376 401 a veneziani c vergara flow rate defective boundary conditions in haemodynamics simulations int j numer methods fluids 47 2005 803 816 in the latter two papers a feasible numerical formulation of this problem based on a lagrange multiplier approach has been investigated this approach has the drawback of high computational costs in this paper we propose an approximate formulation of this problem yielding a strong reduction of the computational costs the error analysis shows that the error introduced by this approximate formulation is confined in a small region of the boundary this is confirmed by the numerical simulations

a variational approach for estimating the compliance of the cardiovascular tissue an inverse fluid structure interaction problem estimation of the stiffness of a biological soft tissue is useful for the detection of pathologies such as tumors or atherosclerotic plaques elastography is a method based on the comparison between two images before and after a forced deformation of the tissue of interest an inverse elasticity problem is then solved for young s modulus estimation in the case of arteries no forced deformation is required since vessels naturally move under the action of blood young s modulus can therefore be estimated by solving a coupled inverse fluid structure interaction problem in this paper we focus on the mathematical properties of this problem and its numerical solution we give some well posedness analysis and some preliminary results based on a synthetic data set i e test cases where the exact young s modulus is known and the displacement dataset is numerically generated by solving a forward fluid structure interaction problem we address the problem of the presence of the noise in the measured displacement and of the proper sampling frequency for obtaining reliable estimates

on the decay rate of the singular values of bivariate functions in this work we establish a new truncation error estimate of the singular value decomposition svd for a class of sobolev smooth bivariate functions kappa in l 2 omega h s d s

a projection semi implicit scheme for the coupling of an elastic structure with an incompressible fluid we address the numerical simulation of fluid structure systems involving an incompressible viscous fluid this issue is particularly difficult to face when the fluid added mass acting on the structure is strong as it happens in hemodynamics for example indeed several works have shown that in such situations implicit coupling seems to be necessary in order to avoid numerical instabilities although significant improvements have been achieved during the last years solving implicit coupling often exhibits a prohibitive computational cost in this work we introduce a semi implicit coupling scheme which remains stable for a reasonable range of the discretization parameters the first idea consists in treating implicitly the added mass effect whereas the other contributions geometrical non linearities viscous and convective effects are treated explicitly the second idea relies on the fact that this kind of explicit implicit splitting can be naturally performed using a chorin temam projection scheme in the fluid we prove conditional stability of the scheme for a fully discrete formulation several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches

spatiotemporal and statistical symmetries the notion of symmetries either statistical or deterministic can be useful for the characterization of complex systems and their bifurcations in this paper we investigate the connection between the microscopic spatiotemporal symmetries of a space time functionu x t on the one hand and the macroscopic symmetries of statistical quantities such as the spatial resp temporal two point correlations and the spatial resp temporal average on the other hand we show how under certain conditions these symmetries are related to the symmetries of the orbits described byu x t in the characteristic phase spaces we also determine the largest group of spatiotemporal symmetries in the sense introduced in our earlier work satisfied by a given space time functionu x t and indicate how to extract the subgroups of point symmetries namely those directly implemented on the space and time variables conversely we determine all the functions invariant by a given space time symmetry group finally we illustrate all the previous points with specific examples

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

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

on local convergence of alternating schemes for optimization of convex problems in the tensor train format alternating linear schemes als with the alternating least squares algorithm a notable special case provide one of the simplest and most popular choices for the treatment of optimization tasks by tensor methods an according adaptation of als for the recent tt tensor train format i v oseledets siam j sci comput 33 2011 pp 2295 2317 known in quantum computations as matrix product states has recently been investigated in s holtz t rohwedder and r schneider siam j sci comput 34 2012 pp a683 a713 with the present work the positive practical experience with tt als is backed up with an according local linear convergence theory for the optimization of convex functionals j the main assumption entering the proof is that the redundancy introduced by the tt parametrization tau matches the null space of the hessian of the induced functional j j circ tau and we give conditions under which this assumption can be expected to hold in particular this is the case if

measures of dependence for censored survival data in the linear regression model with normal errors the squared product moment correlation provides the standard measure of dependence between the explanatory variable and the response variable using the concept of information gain a measure of dependence can also be defined for more general regression models used in survival analysis such as the weibull regression model or cox s proportional hazards model further this measure of dependence can be conveniently estimated even when the response variable is subject to censoring and a simple approximation is available

a computable evolution equation for the joint response excitation probability density function of stochastic dynamical systems by using functional integral methods we determine a computable evolution equation for the joint response excitation probability density function of a stochastic dynamical system driven by coloured noise this equation can be represented in terms of a superimposition of differential constraints i e partial differential equations involving unusual limit partial derivatives the first one of which was originally proposed by sapsis athanassoulis a connection with the classical response approach is established in the general case of random noise with arbitrary correlation time yielding a fully consistent new theory for non markovian systems we also address the question of computability of the joint response excitation probability density function as a solution to a boundary value problem involving only one differential constraint by means of a simple analytical example it is shown that in general such a problem is undetermined in the sense that it admits an infinite number of solutions this issue can be overcome by completing the system with additional relations yielding a closure problem which is similar to the one arising in the standard response theory numerical verification of the equations for the joint response excitation density is obtained for a tumour cell growth model under immune response

robin robin preconditioned krylov methods for fluid structure interaction problems in this work we propose a robin robin preconditioner combined with krylov iterations for the solution of the interface system arising in fluid structure interaction fsi problems it can be seen as a partitioned fsi procedure and in this respect it generalizes the ideas introduced in s badia f nobile c vergara j comput phys 227 2008 7027 7051 we analyze the convergence of gmres iterations with the robin robin preconditioner on a model problem and compare its efficiency with some existing algorithms the method is shown to be very efficient for many challenging fluid structure interaction problems such as those characterized by a large added mass effect or by enclosed fluids in particular the possibility to solve balloon type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches c 2009 elsevier b v all rights reserved