the diameter of the first nodal line of a convex domains the main goal of this paper is to prove that the first nodal line for the dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain more precisely we locate the nodal line to within a distance comparable to the inradius near the zero of an ordinary differential equation which is associated to the domain in a natural way we also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction two examples a rectangle and a circular sector illustrate the two extreme possibilities for the location of the nodal line for the rectangle r x y o 1 we have the second eigenfunction

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

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

compressive sampling of polynomial chaos expansions sampling orthogonal polynomial bases via monte carlo is of interest for uncertainty quantification of models with random inputs using polynomial chaos pc expansions it is known that bounding a probabilistic parameter referred to as coherence yields a bound on the number of samples necessary to identify coefficients in a sparse pc expansion via solution to an 1 minimization problem utilizing results for orthogonal polynomials we bound the coherence parameter for polynomials of hermite and legendre type under their respective natural sampling distribution in both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation for more general orthonormal bases we propose the coherence optimal sampling a markov chain monte carlo sampling which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support we demonstrate these different sampling strategies numerically in both high order and high dimensional manufactured pc expansions in addition the quality of each sampling method is compared in the identification of solutions to two differential equations one with a high dimensional random input and the other with a high order pc expansion in both cases the coherence optimal sampling scheme leads to similar or considerably improved accuracy

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

non intrusive low rank separated approximation of high dimensional stochastic models this work proposes a sampling based non intrusive approach within the context of low rank separated representations to tackle the issue of curse of dimensionality associated with the solution of models e g pdes odes with high dimensional random inputs under some conditions discussed in details the number of random realizations of the solution required for a successful approximation grows linearly with respect to the number of random inputs the construction of the separated representation is achieved via a regularized alternating least squares regression together with an error indicator to estimate model parameters the computational complexity of such a construction is quadratic in the number of random inputs the performance of the method is investigated through its application to three numerical examples including two ode problems with high dimensional random inputs

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

a fast and scalable method for a optimal design of experiments for infinite dimensional bayesian nonlinear inverse problems we address the problem of optimal experimental design oed for bayesian nonlinear inverse problems governed by partial differential equations pdes the inverse problem seeks to infer an infinite dimensional parameter from experimental data observed at a set of sensor locations and from the governing pdes the goal of the oed problem is to find an optimal placement of sensors so as to minimize the uncertainty in the inferred parameter field specifically we seek an optimal subset of sensors from among a fixed set of candidate sensor locations we formulate the oed objective function by generalizing the classical a optimal experimental design criterion using the expected value of the trace of the posterior covariance this expected value is computed through sample averaging over the set of likely experimental data to cope with the infinite dimensional character of the parameter field we construct a gaussian approximation to the posterior at the maximum a posteriori probability map point and use the

a low rank control variate for multilevel monte carlo simulation of high dimensional uncertain systems abstract multilevel monte carlo mlmc is a recently proposed variation of monte carlo mc simulation that achieves variance reduction by simulating the governing equations on a series of spatial or temporal grids with increasing resolution instead of directly employing the fine grid solutions mlmc estimates the expectation of the quantity of interest from the coarsest grid solutions as well as differences between each two consecutive grid solutions when the differences corresponding to finer grids become smaller hence less variable fewer mc realizations of finer grid solutions are needed to compute the difference expectations thus leading to a reduction in the overall work this paper presents an extension of mlmc referred to as multilevel control variates mlcv where a low rank approximation to the solution on each grid obtained primarily based on coarser grid solutions is used as a control variate for estimating the expectations involved in mlmc cost estimates as well as numerical examples are presented to demonstrate the advantage of this new mlcv approach over the standard mlmc when the solution of interest admits a low rank approximation and the cost of simulating finer grids grows fast

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

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

a robust and efficient stepwise regression method for building sparse polynomial chaos expansions abstract polynomial chaos pc expansions are widely used in various engineering fields for quantifying uncertainties arising from uncertain parameters the computational cost of classical pc solution schemes is unaffordable as the number of deterministic simulations to be calculated grows dramatically with the number of stochastic dimension this considerably restricts the practical use of pc at the industrial level a common approach to address such problems is to make use of sparse pc expansions this paper presents a non intrusive regression based method for building sparse pc expansions the most important pc contributions are detected sequentially through an automatic search procedure the variable selection criterion is based on efficient tools relevant to probabilistic method two benchmark analytical functions are used to validate the proposed algorithm the computational efficiency of the method is then illustrated by a more realistic cfd application consisting of the non deterministic flow around a transonic airfoil subject to geometrical uncertainties to assess the performance of the developed methodology a detailed comparison is made with the well established lar based selection technique the results show that the developed sparse regression technique is able to identify the most significant pc contributions describing the problem moreover the most important stochastic features are captured at a reduced computational cost compared to the lar method the results also demonstrate the superior robustness of the method by repeating the analyses using random experimental designs

locating the first nodal set in higher dimensions we extend the two dimensional results of jerison 2000 on the location of the nodal set of the first neumann eigenfunction of a convex domain to higher dimensions if a convex domain in n is contained in a long and thin cylinder 0 n x b 0 with nonempty intersections with x 1 0 and x 1 n then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation by a bound proportional to whose coefficients are expressed in terms of the volume of the cross sections of the domain also the first nodal set is located within a distance comparable to e near the zero of the corresponding ordinary differential equation

locating the first nodal linein the neumann problem the location of the nodal line of the first nonconstant neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius this is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain a variant of these estimates is given for domains that are thin strips and satisfy a lipschitz condition

a near optimal sampling strategy for sparse recovery of polynomial chaos expansions abstract compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited originally these expensive simulation samples would be obtained at random locations in the parameter space it was later shown that the choice of sample locations could significantly impact the accuracy of resulting surrogates this motivated new sampling strategies or design of experiment approaches such as coherence optimal sampling which aim at improving the coherence property in this paper we propose a sampling strategy that can identify near optimal sample locations that lead to improvement in local coherence property and also enhancement of cross correlation properties of measurement matrices we provide theoretical motivations for the proposed sampling strategy along with several numerical examples that show that our near optimal sampling strategy produces substantially more accurate results compared to other sampling strategies

simulation based optimal bayesian experimental design for nonlinear systems the optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction particularly in situations where experiments are time consuming and expensive to conduct we propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation based models in particular we focus on finding sets of experiments that provide the most information about targeted sets of parameters our framework employs a bayesian statistical setting which provides a foundation for inference from noisy indirect and incomplete data and a natural mechanism for incorporating heterogeneous sources of information an objective function is constructed from information theoretic measures reflecting expected information gain from proposed combinations of experiments polynomial chaos approximations and a two stage monte carlo sampling method are used to evaluate the expected information gain stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high dimensional settings these algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics

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

sign pattern matrices that admit m n p or inverse m matrices in this paper we identify the sign pattern matrices that occur among the n matrices the p matrices and the m matrices we also address to the class of inverse m matrices and the related admissibility of sign pattern matrices problem

divide and conquer an incremental sparsity promoting compressive sampling approach for polynomial chaos expansions this paper introduces an efficient sparse recovery approach for polynomial chaos pc expansions which promotes the sparsity by breaking the dimensionality of the problem the proposed algorithm incrementally explores sub dimensional expansions for a sparser recovery and shows success when removal of uninfluential parameters that results in a lower coherence for measurement matrix allows for a higher order and or sparser expansion to be recovered the incremental algorithm effectively searches for the sparsest pc approximation and not only can it decrease the prediction error but it can also reduce the dimensionality of pce model four numerical examples are provided to demonstrate the validity of the proposed approach the results from these examples show that the incremental algorithm substantially outperforms conventional compressive sampling approaches for pce in terms of both solution sparsity and prediction error

the n matrix completion problem under digraphs assumptions fundacao para a ciencia e a tecnologia fct programa operacional ciencia tecnologia inovacao pocti spanish dgi grant number bfm2001 0081 c03 02

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources

gaussian process regression and conditional polynomial chaos for parameter estimation abstract we present a new approach for constructing a data driven surrogate model and using it for parameter estimation in partial differential equation pde models we first use parameter observations and gaussian process regression to condition the karhunen loeve kl expansion of the unknown space dependent parameters and then build the conditional generalized polynomial chaos gpc surrogate model of the pde states next we estimate the unknown parameters by computing coefficients in the kl expansion by minimizing the square difference between the gpc predictions and measurements of the states our approach addresses two major challenges in the parameter estimation first it reduces dimensionality of the parameter space and replaces expensive direct solutions of pdes with the conditional gpc surrogates second the estimated parameter field exactly matches the parameter measurements in addition we show that the conditional gpc surrogate can be used to estimate the state variance that in turn can be used to guide data acquisition we demonstrate that our approach improves the accuracy of parameter estimation with application to one and two dimensional darcy equations that have unknown space dependent hydraulic conductivity we also discuss the effect of hydraulic conductivity and head locations on the accuracy of the hydraulic conductivity estimations

the symmetric n matrix completion problem abstract an n n matrix is called an n matrix if all its principal minors are negative in this paper we are interested in the symmetric n matrix completion problem that is when a partial symmetric n matrix has a symmetric n matrix completion here we prove that a partial symmetric n matrix has a symmetric n matrix completion if the graph of its specified entries is chordal furthermore if this graph is not chordal then examples exist without symmetric n matrix completions necessary and sufficient conditions for the existence of a symmetric n matrix completion of a partial symmetric n matrix whose associated graph is a cycle are given

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

n matrix completion problem abstract an n n matrix is called an n matrix if all principal minors are negative in this paper we are interested in n matrix completion problems that is when a partial n matrix has an n matrix completion in general a combinatorially or non combinatorially symmetric partial n matrix does not have an n matrix completion here we prove that a combinatorially symmetric partial n matrix has an n matrix completion if the graph of its specified entries is a 1 chordal graph we also prove that there exists an n matrix completion for a partial n matrix whose associated graph is an undirected cycle

selection of polynomial chaos bases via bayesian model uncertainty methods with applications to sparse approximation of pdes with stochastic inputs generalized polynomial chaos gpc expansions allow us to represent the solution of a stochastic system using a series of polynomial chaos basis functions the number of gpc terms increases dramatically as the dimension of the random input variables increases when the number of the gpc terms is larger than that of the available samples a scenario that often occurs when the corresponding deterministic solver is computationally expensive evaluation of the gpc expansion can be inaccurate due to over fitting we propose a fully bayesian approach that allows for global recovery of the stochastic solutions in both spatial and random domains by coupling bayesian model uncertainty and regularization regression methods it allows the evaluation of the pc coefficients on a grid of spatial points via 1 the bayesian model average bma or 2 the median probability model and their construction as spatial functions on the spatial domain via spline interpolation the former accounts for the model uncertainty and provides bayes optimal predictions while the latter provides a sparse representation of the stochastic solutions by evaluating the expansion on a subset of dominating gpc bases moreover the proposed methods quantify the importance of the gpc bases in the probabilistic sense through inclusion probabilities we design a markov chain monte carlo mcmc sampler that evaluates all the unknown quantities without the need of ad hoc techniques the proposed methods are suitable for but not restricted to problems whose stochastic solutions are sparse in the stochastic space with respect to the gpc bases while the deterministic solver involved is expensive we demonstrate the accuracy and performance of the proposed methods and make comparisons with other approaches on solving elliptic spdes with 1 14 and 40 random dimensions

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

basis adaptive sample efficient polynomial chaos base pc abstract for a large class of orthogonal basis functions there has been a recent identification of expansion methods for computing accurate stable approximations of a quantity of interest this paper presents within the context of uncertainty quantification a practical implementation using basis adaptation and coherence motivated sampling which under assumptions has satisfying guarantees this implementation is referred to as basis adaptive sample efficient polynomial chaos base pc a key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner leading to consistently stable approximation for a practical class of smooth functionals this fully adaptive non intrusive method requires no a priori information of the solution and has satisfying theoretical guarantees of recovery a key contribution to stability is the use of a presented correction sampling for coherence optimal sampling in order to improve stability and accuracy within the adaptive basis scheme theoretically the method may dramatically reduce the impact of dimensionality in function approximation and numerically the method is demonstrated to perform well on problems with dimension up to 1000

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

on characterizing n matrices using linear complementarity we show that a square matrix a with at least one positive entry and all principal minors negative can be characterized in terms of the number of solutions the linear complementarity problem q a with the matrix a has for different vectors q such a matrix can also be characterized in terms of a sign nonreversal property these results complement the known results for a square matrix a 0 with all its principal minors negative

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

a rapid and efficient isogeometric design space exploration framework with application to structural mechanics abstract in this paper we present an isogeometric analysis framework for design space exploration while the methodology is presented in the setting of structural mechanics it is applicable to any system of parametric partial differential equations the design space exploration framework elucidates design parameter sensitivities used to inform initial and early stage design moreover this framework enables the visualization of a full system response including the displacement and stress fields throughout the domain by providing an approximation to the system solution vector this is accomplished through a collocation like approach where various geometries throughout the design space under consideration are sampled the sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method a surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion examples of this framework are presented with applications to the scordelis lo roof a flat l bracket and an nrel 5 mw wind turbine blade

weighted discrete least squares polynomial approximation using randomized quadratures we discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation the problem stems from uncertainty quantification uq where the independent variables of the functions are random variables with specified probability measure we propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product gaussian quadrature points we analyze the stability properties of this method and prove that the method is asymptotically stable provided that the number of points scales linearly up to a logarithmic factor with the cardinality of the polynomial space specific results in both bounded and unbounded domains are obtained along with a convergence result for chebyshev measure numerical examples are provided to verify the theoretical results

n0 completions on partial matrices an nxn matrix is called an n 0 matrix if all its principal minors are non positive in this paper we are interested in n 0 matrix completion problems that is when a partial n 0 matrix has an n 0 matrix completion in general a combinatorially or non combinatorially symmetric partial n 0 matrix does not have an n 0 matrix completion here we prove that a combinatorially symmetric partial n 0 matrix with no null main diagonal entries has an n 0 matrix completion if the graph of its specified entries is a 1 chordal graph or a cycle we also analyze the mentioned problem when the partial matrix has some null main diagonal entries

the size of the first eigenfunction of a convex planar domain the goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains in particular we will locate the place where u achieves its maximum to within a distance comparable to the inradius uniformly for arbitrarily large diameter in addition we will estimate the location of other level sets of u by showing that u is well approximated by the first eigenfunction of a naturally associated ordinary differential schrodinger operator we intend to show in a separate paper that the estimates here are best possible in order of magnitude the present paper depends on the ideas and results of our earlier work j and gj where detailed estimates for the zero set of the second eigenfunction or first nodal line are obtained the paper j also contains some estimates for the first eigenfunction and lowest eigenvalue but the techniques of gj and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here the maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency the first nodal line is the stationary curve of the drum at the second resonant frequency the maximum is harder to find experimentally than the nodal line because it is a single point its location has less influence on the eigenvalue or dirichlet integral so it is also harder to locate mathematically another way to describe the difficulty is as follows to find the maximum of the first eigenfunction we will need to estimate its first directional derivative derivatives of the first eigenfunction are roughly speaking analogous to the second eigenfunction because they are solutions to an eigenfunction equation moreover convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components but the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary

on polynomial chaos expansion via gradient enhanced l 1 minimization gradient enhanced uncertainty quantification uq has received recent attention in which the derivatives of a quantity of interest qoi with respect to the uncertain parameters are utilized to improve the surrogate approximation polynomial chaos expansions pces are often employed in uq and when the qoi can be represented by a sparse pce 1 minimization can identify the pce coefficients with a relatively small number of samples in this work we investigate a gradient enhanced 1 minimization where derivative information is computed to accelerate the identification of the pce coefficients for this approach stability and convergence analysis are lacking and thus we address these here with a probabilistic result in particular with an appropriate normalization we show the inclusion of derivative information will almost surely lead to improved conditions e g related to the null space and coherence of the measurement matrix for a successful solution recovery further we demonstrate our analysis empirically via three numerical examples a manufactured pce an elliptic partial differential equation with random inputs and a plane poiseuille flow with random boundaries these examples all suggest that including derivative information admits solution recovery at reduced computational cost an approach for sparse polynomial chaos pc expansions including derivative information is presented method relies on 1 minimization to solve for the pc coefficients formal analysis is presented to guarantee the stability and convergence of gradient enhanced 1 minimization three examples presented to illustrate cost accuracy improvements achieved by gradient enhanced 1 minimization

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

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

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate