scholarly journals Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

2021 ◽  
Author(s):  
Dũng Đinh
Keyword(s):  
Convergence Rates ◽  
Polynomial Interpolation ◽  
Hermite Polynomials ◽  
Adaptive Methods ◽  
Sparse Grid ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Elliptic Pdes ◽  
Fully Discrete ◽  
Weighted Quadrature

By combining a certain  approximation property in the spatial domain, and weighted summability  of the Hermite polynomial expansion coefficients  in the parametric domain, we investigate  linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We explicitly construct  such methods and prove convergence rates of the approximations by them.  The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are  certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of  successive differences of  their parametric Lagrange interpolating polynomials. The  Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the  corresponding integration can be estimated via the error in Bochner space.  We also briefly  consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and  problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRO‐DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 15 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Raul Kangro ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Approximate Solutions ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Fully Discrete ◽  
Weakly Singular ◽  
Good Accordance ◽  
Theoretical Results ◽  
Simple Quadrature

In order to find approximate solutions of Volterra and Fredholm integro‐differential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non‐smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRAL EQUATIONS

2009 ◽  
Vol 14 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Raul Kangro ◽  
Inga Kangro
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Nonuniform Grids ◽  
Fully Discrete ◽  
Straightforward Application ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations

A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.

scholarly journals Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

2016 ◽  
Vol 51 (1) ◽  
pp. 341-363 ◽  
Author(s):  
Markus Bachmayr ◽  
Albert Cohen ◽  
Ronald DeVore ◽  
Giovanni Migliorati
Keyword(s):  
Convergence Rates ◽  
Hermite Polynomials ◽  
Sufficient Conditions ◽  
Elliptic Pdes ◽  
Linear Elliptic Equation ◽  
Solution Map ◽  
Infinite Dimensional ◽  
Sparse Polynomial ◽  
The Mean ◽  
Linear Pdes

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j ≥ 1yjψj where y = (yj) ∈ ℝNare i.i.d. standard scalar Gaussian variables and (ψj)j ≥ 1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(yj) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 <p ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥V)ν ∈ ℱ. Such results ensure convergence rates n− s with s = (1/p)−(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥ψj∥L∞)j ≥ 1 for some β>1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥V)ν ∈ ℱ. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

scholarly journals Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Balance Laws ◽  
Source Terms ◽  
Time Step ◽  
Systems Of Balance Laws ◽  
Reconstruction Operator ◽  
Non Linear ◽  
Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

scholarly journals MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

2021 ◽  
Author(s):  
Dong T.P. Nguyen ◽  
Dirk Nuyens
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Monte Carlo ◽  
Convergence Rates ◽  
Higher Order ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Quasi Monte Carlo ◽  
Higher Order Convergence ◽  
Element Method

We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE.   We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm.   Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

scholarly journals The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem.

2020 ◽  
Author(s):  
Andreas Neuenkirch ◽  
Michaela Szölgyenyi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Error Analysis ◽  
Strong Convergence ◽  
General Framework ◽  
Additive Noise ◽  
Convergence Rates ◽  
Convergence Order ◽  
Quadrature Problem ◽  
Weighted Quadrature

Abstract We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min \{3/4,(1+\kappa )/2\}-\epsilon$ for the equidistant EM scheme (for arbitrarily small $\epsilon&gt;0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of $(1+\kappa )/2-\epsilon$ for the corresponding EM scheme.

scholarly journals FULLY DISCRETE ADAPTIVE METHODS FOR A BLOW-UP PROBLEM

2004 ◽  
Vol 14 (10) ◽  
pp. 1425-1450 ◽  
Author(s):  
CRISTINA BRÄNDLE ◽  
PABLO GROISMAN ◽  
JULIO D. ROSSI
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Blow Up ◽  
Numerical Study ◽  
Adaptive Methods ◽  
Fully Discrete ◽  
Adaptive Procedures ◽  
Exact Asymptotic Behavior ◽  
Blow Up Rate ◽  
Continuous Problem

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem: [Formula: see text] with p,m>0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate and the blow-up set) of the continuous problem.

scholarly journals Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates

2017 ◽  
Vol 27 (14) ◽  
pp. 2781-2802 ◽  
Author(s):  
Annalisa Buffa ◽  
Carlotta Giannelli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Convergence Rates ◽  
Main Tool ◽  
Adaptive Methods ◽  
Interpolation Operator ◽  
B Splines ◽  
Optimal Convergence Rates ◽  
Error Indicators ◽  
Suitable Approximation

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces.

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