extensions of modules for twisted current algebras abstract twisted current algebras are fixed point subalgebras of current algebras under a finite group action special cases include equivariant map algebras and twisted forms of current algebras their finite dimensional simple modules fall into two categories those which factor through an evaluation map and those which do not we show that there are no nontrivial extensions between finite dimensional simple evaluation and non evaluation modules we then compute extensions between any pair of finite dimensional simple modules for twisted current algebras and use this information to study the block decomposition for the category in the special case of twisted forms this decomposition can be described in terms of maps to the fundamental group of the underlying root system

preconditioning techniques for reduced basis methods for parameterized elliptic partial differential equations the reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values an offline step approximates the solution space and an online step utilizes this approximation the reduced basis to solve a smaller reduced problem which provides an accurate estimate of the solution traditionally the reduced problem is solved using direct methods however the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly in this scenario it may be more effective to use iterative methods to solve the reduced problem we construct preconditioners for reduced iterative methods which are derived from preconditioners for the full problem this approach permits reduced basis methods to be practical for larger bas

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

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

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

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

the structure of quotients of the onsager algebra by closed ideals we study the onsager algebra from the ideal theoretic point of view a complete classification of closed ideals and the structure of quotient algebras are obtained we also discuss the solvable algebra aspect of the onsager algebra through the use of formal lie algebras

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

quantized tensor structured finite elements for second order elliptic pdes in two dimensions we analyze the approximation of the solutions of second order elliptic problems which have point singularities and belong to a countably normed space of analytic functions with a first order h version finite element fe method based on uniform tensor product meshes the fe solutions are well known to converge with algebraic rate at most 1 2 in terms of the number of degrees of freedom and even slower in the presence of singularities we analyze the compression of the fe coefficient vectors represented in the so called quantized tensor train format we prove in a reference square that the resulting fe approximations converge exponentially in terms of the effective number n of degrees of freedom involved in the representation n mathcal o log 5 varepsilon 1 where varepsilon in 0 1 is the accuracy measured in the energy norm numerically we show for solutions from the same class that the entire process of solving the tensor structured galerkin first order fe discretization can achieve accuracy varepsilon in the energy norm with n mathcal o log kappa varepsilon 1 parameters where kappa 3

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

a reduced basis model with parametric coupling for fluid structure interaction problems we present a new model reduction technique for steady fluid structure interaction problems when the fluid domain deformation is suitably parametrized the coupling conditions between the fluid and the structure can be formulated in the low dimensional space of geometric parameters moreover we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid structure iterations in this way a reduced order model with reliable a posteriori error bounds is obtained the proposed method is validated with an example of steady stokes flow in an axisymmetric channel where the structure is described by a simple one dimensional generalized string model we demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased

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

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification abstract in this work we present a model order reduction mor technique for hyperbolic conservation laws with applications in uncertainty quantification uq the problem consists of a parametrized time dependent hyperbolic system of equations where the parameters affect the initial conditions and the fluxes in a non linear way the procedure utilized to reduce the order is a combination of a greedy algorithm in the parameter space a proper orthogonal decomposition pod in time and empirical interpolation method eim to deal with non linearities drohmann 2012 we provide under some hypothesis an error bound for the reduced solution with respect to the high order one the algorithm shows small errors and savings of the computational time up to 90 in the uq simulations which are performed to validate the algorithm

contraction and optimality properties of an adaptive legendre galerkin method the multi dimensional case we analyze the theoretical properties of an adaptive legendre galerkin method in the multidimensional case after the recent investigations for fourier galerkin methods in a periodic box and for legendre galerkin methods in the one dimensional setting the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral hp hp discretizations of elliptic boundary value problems the main contribution of the paper is a careful construction of a multidimensional riesz basis in h 1 h1 based on a quasi orthonormalization procedure this allows us to design an adaptive algorithm to prove its convergence by a contraction argument and to discuss its optimality properties in the sense of non linear approximation theory in certain sparsity classes of gevrey type

an hp certified reduced basis method for parametrized elliptic partial differential equations we present a new hp parameter multidomain certified reduced basis rb method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations we propose and provide theoretical justification for a new procedure for adaptive partition h refinement of the parameter domain into smaller parameter subdomains we pursue a hierarchical splitting of the parameter sub domains based on proximity to judiciously chosen parameter anchor points within each subdomain subsequently we construct individual standard rb approximation spaces p refinement over each subdomain greedy parameter sampling procedures and a posteriori error estimation play important roles in both the h type and p type stages of the new algorithm we present illustrative numerical results for a convection diffusion problem the new hp approach is considerably faster respectively more costly than the standard p type reduced basis method in the online respectively offline stage

galois cohomology and forms of algebras over laurent polynomial rings 3 loop torsors 8 3 1 the witt tits index of a loop torsor 9 3 2 reducibility 12 3 3 almost commutative subgroups 14 3 4 almost commuting families of elements of finite order 16 3 5 almost commuting pairs and their invariants 17 3 6 failure in the anisotropic case 24

a robust error estimator and a residual free error indicator for reduced basis methods abstract the reduced basis method rbm is a rigorous model reduction approach for solving parameterizedpartial differential equations it identifies a low dimensional subspace for approximation of the parametric solution manifold that is embedded in high dimensional space a reduced order model is subsequently constructed in this subspace rbm relies on residual based error indicators or a posteriori error bounds to guide construction of the reduced solution subspace to serve as a stopping criteria and to certify the resulting surrogate solutions unfortunately it is well known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision in this paper we develop two alternatives to the standard offline phase of reduced basis algorithms first we design a robust strategy for computation of residual error indicators that allows rbm algorithms to enrich the solution subspace with accuracy beyond root machine precision secondly we propose a new error indicator based on the lebesgue function in interpolation theory this error indicator does not require computation of residual norms and instead only requires the ability to compute the rbm solution this residual free indicator is rigorous in that it bounds the error committed by the rbm approximation but up to an uncomputable multiplicative constant because of this the residual free indicator is effective in choosing snapshots during the offline rbm phase but cannot currently be used to certify error that the approximation commits however it circumvents the need for a posteriori analysis of numerical methods and therefore can be effective on problems where such a rigorous estimate is hard to derive

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

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

adaptive stochastic galerkin fem with hierarchical tensor representations the solution of pde with stochastic data commonly leads to very high dimensional algebraic problems e g when multiplicative noise is present the stochastic galerkin fem considered in this paper then suffers from the curse of dimensionality this is directly related to the number of random variables required for an adequate representation of the random fields included in the pde with the presented new approach we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies namely a modern low rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations the adaptive adjustment includes the refinement of the fe mesh based on a residual estimator the problem adapted stochastic discretization in anisotropic legendre wiener chaos and the successive increase of the tensor rank computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned als scheme of the problem strikingly it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently a set of benchmark problems illustrates the performance of the adaptive algorithm with higher order fe moreover the influence of the tensor rank on the approximation quality is investigated

on the decay of the inverse of matrices that are sum of kronecker products abstract decay patterns of matrix inverses have recently attracted considerable interest due to their relevance in numerical analysis and in applications requiring matrix function approximations in this paper we analyze the decay pattern of the inverse of banded matrices of the form s m i n i n m where m is tridiagonal symmetric and positive definite i n is the identity matrix and stands for the kronecker product it is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal however the entries in s 1 show a non monotonic decay which is not caught by classical bounds by using an alternative expression for s 1 we derive computable upper bounds that closely capture the actual behavior of its entries we also show that similar estimates can be obtained when m has a larger bandwidth or when the sum of kronecker products involves two different matrices numerical experiments illustrating the new bounds are also reported

adaptive fourier galerkin methods we study the performance of adaptive fourier galerkin methods in a periodic box in r d with dimension d 1 these methods offer unlimited approximation power only restricted by solution and data regularity they are of intrinsic interest but are also a first step towards understanding adaptivity for the hp fem we examine two nonlinear approximation classes one classical corresponding to algebraic decay of fourier coefficients and another associated with exponential decay typical of spectral approximation we investigate the natural sparsity class for the operator range and find that the exponential class is not preserved thus in contrast with the algebraic class this entails a striking different behavior of the feasible residuals that lead to practical algorithms influencing the overall optimality the sparsity degradation for the exponential class is partially compensated with coarsening we present several feasible adaptive fourier algorithms prove their contraction properties and examine the cardinality of the activated sets the galerkin approximations at the end of each iteration are quasi optimal for both classes but inner loops or intermediate approximations are sub optimal for the exponential class

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

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

sparse grid reduced basis bayesian inversion we extend the reduced basis rb accelerated bayesian inversion methods for affine parametric linear operator equations which are considered in 16 17 to non affine nonlinear parametric operator equations we generalize the analysis of sparsity of parametric forward solution maps in 20 and of bayesian inversion in 48 49 to the fully discrete setting including petrov galerkin high fidelity hifi discretization of the forward maps we develop adaptive stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold the nonaffinity and nonlinearity with respect to w r t the distributed uncertain parameters and the unknown solution is collocated specifically by the so called empirical interpolation method eim for the corresponding bayesian inversion problems computational efficiency is enhanced in two ways first expectations w r t the posterior are computed by adaptive quadratures with dimension independent convergence rates proposed in 49 the present work generalizes 49 to account for the impact of the pg discretization in the forward maps on the convergence rates of the quantities of interest qoi for short second we propose to perform the bayesian estimation only w r t a parsimonious rb approximation of the posterior density based on the approximation results in 49 the infinite dimensional parametric deterministic forward map and operator admit n term rb and eim approximations which converge at rates which depend only on the sparsity of the parametric forward map in several numerical experiments the proposed algorithms exhibit dimension independent convergence rates which equal at least the currently known rate estimates for n term approximation we propose to accelerate bayesian estimation by first offline construction of reduced basis surrogates of the bayesian posterior density the parsimonious surrogates can then be employed for online data assimilation and for bayesian estimation they also open a perspective for optimal experimental design

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

analysis of the combined finite element and fourier interpolation we consider lagrange interpolation involving trigonometric polynomials of degree n in one space direction and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions we provide error estimates in non isotropic sobolev norms depending additively on the parametersh andn an application to the convergence analysis of an elliptic problem with some numerical results is given

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

fast computation in adaptive tree approximation adaptive methods of approximation arise in many settings including numerical methods for pdes and image processing they can usually be described by a tree which records the adaptive decisions this paper is concerned with the fast computation of near optimal trees based on n adaptive decisions the best tree based on n adaptive decisions could be found by examining all such possibilities however this is exponential in n and could be numerically prohibitive the main result of this paper is to show that it is possible to find near optimal trees using computations linear in n

reduced basis methods for nonlocal diffusion problems with random input data abstract the construction analysis and application of reduced basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models the need for reduced order modeling is much more acute in the nonlocal setting in this effort we develop reduced basis approximations for nonlocal diffusion equations with affine random coefficients efficiency estimates of the proposed greedy reduced basis methods are provided numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced basis method relative to sparse grid interpolation using the full finite element method it is shown that the proposed reduced basis approach can indeed provide substantial savings over standard sparse grid methods

a weighted reduced basis method for elliptic partial differential equations with random input data in this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations pdes with random input data the pdes are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure a priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations in this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by stokes equations we prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data the finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space respectively resulting in many large scale optimality systems to solve in order to reduce the unaffordable computational effort we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse grid techniques a weighted a posteriori error bound highlights the contribution stemming from each method numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient especially for solving high dimensional and large scale optimization problems

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

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

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

an offline online algorithm for a class of stochastic multiple obstacle scattering configurations in the half plane efficient computational models are required to understand quantities of interest qoi such as the intensity of sound radiated by non deterministic configurations comprising multiple obstacles in the half plane the stochastic nature of the configurations requires tens of thousands of realizations to compute the expected value and standard deviation of the random qoi the acoustic wave scattering configuration usually comprises multiple deterministic structures whose locations in the half plane are non deterministic thus understanding the nature of the deterministic structures is intrinsically an offline process whilst understanding the nature of the resulting qoi is an online process the online process includes interactions between the structures and the half plane boundary in accordance with the physical offline online process we develop a computational t matrix based offline online reduced order model for large realizations of stochastic acoustic wave scattering configurations in the half plane we present results with more than 60 000 monte carlo mc quasi mc qmc and generalized polynomial chaos gpc simulations which are evaluated using our efficient online scheme for the acoustic scattering half plane model our offline online deterministic and stochastic framework can be used in conjunction with any numerical method for the offline simulations of scattering from a single obstacle in free space

adaptive spectral galerkin methods with dynamic marking the convergence and optimality theory of adaptive galerkin methods is almost exclusively based on the dorfler marking this entails a fixed parameter and leads to a contraction constant bounded below away from zero for spectral galerkin methods this is a severe limitation which affects performance we present a dynamic marking strategy that allows for a superlinear relation between consecutive discretization errors and show exponential convergence with linear computational complexity whenever the solution belongs to a gevrey approximation class

sparse grid reduced basis bayesian inversion abstract we analyze reduced basis rb for short short acceleration of recently proposed sparse bayesian inversion algorithms for partial differential equations with uncertain distributed parameter for observation data subject to additive gaussian observation noise specifically bayesian inversion of affine parametric linear operator families on possibly high dimensional parameter spaces we consider high fidelity petrov galerkin pg discretizations of these countably parametric operator families we allow general families of inf sup stable pg finite element methods covering most conforming primal and mixed finite element discretizations of standard problems in mechanics rb acceleration of the high dimensional parametric forward response maps which need to be numerically solved numerous times in bayesian inversion is proposed and convergence rate bounds for the error in the bayesian estimate incurred by the use of rb are derived as consequence of recent theoretical results on dimension independent sparsity of parametric responses and preservation of sparsity for holomorphic parametric problems we establish new convergence rates of greedy rb compressions for both the parametric forward maps as well as for the countably parametric posterior densities which arise in bayesian inversion we show that the convergence rates for the rb compressions of the parametric forward maps as well as of the countably parametric sparse bayesian posterior densities are free from the curse of dimensionality and depend only on the sparsity of the uncertain input data in particular we establish the quadratic convergence of the rb compression for the posterior densities with respect to that for the parametric forward maps numerical experiments for linear elliptic affine parametric model problems in two space dimensions with hundreds of parameters are reported which confirm that the proposed adaptive sparse grid reduced basis algorithms indeed exploit sparsity of both the parametric forward maps as well as the bayesian posterior density

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization