Abstract:In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probabi… Show more
“…In this section, we briefly review the stochastic collocation method for computing the statistical moments of the solution to the micro problem (4), where Y ∈ Γ ⊂ R N is a vector of N independent random variables [5,29,22]. The stochastic collocation method consists of three main steps.…”
Section: Stochastic Collocation Methodsmentioning
confidence: 99%
“…Two typical examples of sparse grids include total degree and hyperbolic cross sparse grids. We refer to [5,29,22] for more details.…”
Section: Stochastic Collocation Methodsmentioning
confidence: 99%
“…Recently, it is shown in [22,21] that the solution of stochastic hyperbolic problems, unlike the solution of elliptic and parabolic problems, is not in general analytic with respect to the random variables. The wave solutions may posses high regularity for particular types of smooth data.…”
Section: Stochastic Regularity Of Quantities Of Interestmentioning
confidence: 99%
“…We note that both spatial mesh sizes ∆x 1 and ∆x 2 and time step ∆t are of order h and related by a proper CFL condition. Also Note that the constant in the term O(h r ) depends on Q(u h ) and is uniform with respect to Y , see [22].…”
Section: Discretization Error In the Deterministic Solvermentioning
We present a stochastic multilevel global-local algorithm for computing elastic waves propagating in fiber-reinforced composite materials. Here, the materials properties and the size and location of fibers may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems (macro, meso, micro) corresponding to the three natural scales, namely the sizes of laminate, ply, and fiber. The algorithm uses the homogenized global solution to construct a good local approximation that captures the microscale features of the real solution. We perform numerical experiments to show the applicability and efficiency of the method.
“…In this section, we briefly review the stochastic collocation method for computing the statistical moments of the solution to the micro problem (4), where Y ∈ Γ ⊂ R N is a vector of N independent random variables [5,29,22]. The stochastic collocation method consists of three main steps.…”
Section: Stochastic Collocation Methodsmentioning
confidence: 99%
“…Two typical examples of sparse grids include total degree and hyperbolic cross sparse grids. We refer to [5,29,22] for more details.…”
Section: Stochastic Collocation Methodsmentioning
confidence: 99%
“…Recently, it is shown in [22,21] that the solution of stochastic hyperbolic problems, unlike the solution of elliptic and parabolic problems, is not in general analytic with respect to the random variables. The wave solutions may posses high regularity for particular types of smooth data.…”
Section: Stochastic Regularity Of Quantities Of Interestmentioning
confidence: 99%
“…We note that both spatial mesh sizes ∆x 1 and ∆x 2 and time step ∆t are of order h and related by a proper CFL condition. Also Note that the constant in the term O(h r ) depends on Q(u h ) and is uniform with respect to Y , see [22].…”
Section: Discretization Error In the Deterministic Solvermentioning
We present a stochastic multilevel global-local algorithm for computing elastic waves propagating in fiber-reinforced composite materials. Here, the materials properties and the size and location of fibers may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems (macro, meso, micro) corresponding to the three natural scales, namely the sizes of laminate, ply, and fiber. The algorithm uses the homogenized global solution to construct a good local approximation that captures the microscale features of the real solution. We perform numerical experiments to show the applicability and efficiency of the method.
“…This representation is obtained from the eigenvalues and eigenfunctions of a homogeneous Fredholm integral equation of the second kind whose kernel is given by the covariance function of the random process. This approach has been widely used in the parametrization of parameters for elasticity problems, heat and mass transfer, fluid mechanic and acoustic [1,6,9].…”
Resumo:We consider the numerical approximation of homogeneous Fredholm integral equations of second kind. We employ the wavelet Galerkin method with 2D Haar wavelets as shape functions. We thoroughly describe the derivation of the shape functions and present a preliminary numerical experiment illustrating the computation of eigenvalues for a particular covariance kernel.Palavras-chave: Fredholm integral equations, Galerkin method, 2D Haar wavelets
The temperature developed in bondwires of integrated circuits (ICs) is a possible source of malfunction, and has to be taken into account during the design phase of an IC. Due to manufacturing tolerances, a bondwire's geometrical characteristics are uncertain parameters, and as such their impact has to be examined with the use of uncertainty quantification (UQ) methods. Considering a stochastic electrothermal problem featuring twelve (12) bondwire-related uncertainties, we want to quantify the impact of the uncertain inputs onto the temperature developed during the duty cycle of an IC. For this reason we apply the stochastic collocation (SC) method on sparse grids (SGs), which is considered the current state-of-the-art. We also implement an approach based on the recently introduced low-rank tensor decompositions, in particular the tensor train (TT) decomposition, which in theory promises to break the curse of dimensionality. A comparison of both methods is presented, with respect to accuracy and computational effort.keywords-electrothermal field simulation, IC packaging, low-rank tensor decompositions, sparse grids, stochastic collocation, tensor trains, uncertainty quantification.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.