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AuthorGuermond, Jean-Luc (6)Jin, Bangti (6)Liang, Faming (5)Efendiev, Yalchin R. (3)Jun, Mikyoung (3)View MoreJournalJournal of Computational Physics (7)Computer Methods in Applied Mechanics and Engineering (4)Journal of Computational and Applied Mathematics (4)Computational Statistics & Data Analysis (3)International Journal of Engineering Science (3)View MoreKAUST Grant Number

KUS-C1-016-04 (50)

Publisher
Elsevier BV (50)

SubjectFinite elements (6)Finite element method (3)Sturm-Liouville problem (3)Adaptive mesh refinement (2)Arterial wall (2)View MoreTypeArticle (50)Year (Issue Date)2016 (1)2015 (3)2014 (3)2013 (8)2012 (13)View MoreItem AvailabilityMetadata Only (50)

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A simple finite element method for boundary value problems with a Riemann–Liouville derivative

Jin, Bangti; Lazarov, Raytcho; Lu, Xiliang; Zhou, Zhi (Journal of Computational and Applied Mathematics, Elsevier BV, 2016-02) [Article]

© 2015 Elsevier B.V. All rights reserved. We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα-<sup>1</sup> in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and <sup>L2</sup>(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

Muhamadiev, Èrgash; Nazarov, Murtazo (Journal of Mathematical Analysis and Applications, Elsevier BV, 2015-03) [Article]

© 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p<sup>-2</sup>, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln<sup>2</sup> p which is an increasing function. Moreover, we prove that this estimate is sharp.

A class of Matérn-like covariance functions for smooth processes on a sphere

Jeong, Jaehong; Jun, Mikyoung (Spatial Statistics, Elsevier BV, 2015-02) [Article]

© 2014 Elsevier Ltd. There have been noticeable advancements in developing parametric covariance models for spatial and spatio-temporal data with various applications to environmental problems. However, literature on covariance models for processes defined on the surface of a sphere with great circle distance as a distance metric is still sparse, due to its mathematical difficulties. It is known that the popular Matérn covariance function, with smoothness parameter greater than 0.5, is not valid for processes on the surface of a sphere with great circle distance. We introduce an approach to produce Matérn-like covariance functions for smooth processes on the surface of a sphere that are valid with great circle distance. The resulting model is isotropic and positive definite on the surface of a sphere with great circle distance, with a natural extension for nonstationarity case. We present extensive numerical comparisons of our model, with a Matérn covariance model using great circle distance as well as chordal distance. We apply our new covariance model class to sea level pressure data, known to be smooth compared to other climate variables, from the CMIP5 climate model outputs.

The Galerkin finite element method for a multi-term time-fractional diffusion equation

Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi (Journal of Computational Physics, Elsevier BV, 2015-01) [Article]

© 2014 The Authors. We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.

Matérn-based nonstationary cross-covariance models for global processes

Jun, Mikyoung (Journal of Multivariate Analysis, Elsevier BV, 2014-07) [Article]

Many spatial processes in environmental applications, such as climate variables and climate model errors on a global scale, exhibit complex nonstationary dependence structure, in not only their marginal covariance but also their cross-covariance. Flexible cross-covariance models for processes on a global scale are critical for an accurate description of each spatial process as well as the cross-dependences between them and also for improved predictions. We propose various ways to produce cross-covariance models, based on the Matérn covariance model class, that are suitable for describing prominent nonstationary characteristics of the global processes. In particular, we seek nonstationary versions of Matérn covariance models whose smoothness parameters vary over space, coupled with a differential operators approach for modeling large-scale nonstationarity. We compare their performance to the performance of some existing models in terms of the aic and spatial predictions in two applications: joint modeling of surface temperature and precipitation, and joint modeling of errors in climate model ensembles. © 2014 Elsevier Inc.

A maximum-principle preserving finite element method for scalar conservation equations

Guermond, Jean-Luc; Nazarov, Murtazo (Computer Methods in Applied Mechanics and Engineering, Elsevier BV, 2014-04) [Article]

This paper introduces a first-order viscosity method for the explicit approximation of scalar conservation equations with Lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension. Provided the lumped mass matrix is positive definite, the method is shown to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions. © 2014 Elsevier B.V.

Use of SAMC for Bayesian analysis of statistical models with intractable normalizing constants

Jin, Ick Hoon; Liang, Faming (Computational Statistics & Data Analysis, Elsevier BV, 2014-03) [Article]

Statistical inference for the models with intractable normalizing constants has attracted much attention. During the past two decades, various approximation- or simulation-based methods have been proposed for the problem, such as the Monte Carlo maximum likelihood method and the auxiliary variable Markov chain Monte Carlo methods. The Bayesian stochastic approximation Monte Carlo algorithm specifically addresses this problem: It works by sampling from a sequence of approximate distributions with their average converging to the target posterior distribution, where the approximate distributions can be achieved using the stochastic approximation Monte Carlo algorithm. A strong law of large numbers is established for the Bayesian stochastic approximation Monte Carlo estimator under mild conditions. Compared to the Monte Carlo maximum likelihood method, the Bayesian stochastic approximation Monte Carlo algorithm is more robust to the initial guess of model parameters. Compared to the auxiliary variable MCMC methods, the Bayesian stochastic approximation Monte Carlo algorithm avoids the requirement for perfect samples, and thus can be applied to many models for which perfect sampling is not available or very expensive. The Bayesian stochastic approximation Monte Carlo algorithm also provides a general framework for approximate Bayesian analysis. © 2012 Elsevier B.V. All rights reserved.

Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

Bonito, Andrea; Guermond, Jean-Luc; Luddens, Francky (Journal of Mathematical Analysis and Applications, Elsevier BV, 2013-12) [Article]

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients. © 2013 Elsevier Ltd.

Reconstruction of the residual stresses in a hyperelastic body using ultrasound techniques

Joshi, Sunnie; Walton, Jay R. (International Journal of Engineering Science, Elsevier BV, 2013-09) [Article]

This paper focuses on a novel approach for characterizing the residual stress field in soft tissue using ultrasound interrogation. A nonlinear inverse spectral technique is developed that makes fundamental use of the finite strain nonlinear response of the material to a quasi-static loading. The soft tissue is modeled as a nonlinear, prestressed and residually stressed, isotropic, slightly compressible elastic body with a rectangular geometry. A boundary value problem is formulated for the residually stressed and prestressed soft tissue, the boundary of which is subjected to a quasi-static pressure, and then an idealized model for the ultrasound interrogation is constructed by superimposing small amplitude time harmonic infinitesimal vibrations on static finite deformation via an asymptotic construction. The model is studied, through a semi-inverse approach, for a specific class of deformations that leads to a system of second order differential equations with homogeneous boundary conditions of Sturm-Liouville type. By making use of the classical theory of inverse Sturm-Liouville problems, and root finding and optimization techniques, several inverse spectral algorithms are developed to approximate the residual stress distribution in the body, given the first few eigenfrequencies of several induced static pressures. © 2013 Elsevier Ltd. All rights reserved.

Analysis of a Cartesian PML approximation to acoustic scattering problems in and

Bramble, James H.; Pasciak, Joseph E. (Journal of Computational and Applied Mathematics, Elsevier BV, 2013-08) [Article]

We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift ("stretching") and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain, the complement of the bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem whose solution coincides with the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). This paper provides new stability estimates for the Cartesian PML approximations both on the infinite and the truncated domain. We first investigate the stability of the infinite PML approximation as a function of the PML strength σ0. This is done for PML methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions. We next introduce a truncation parameter M which determines the size of the PML layer. Our analysis shows that the truncated PML problem is stable provided that the product of Mσ0 is sufficiently large, in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer. This justifies the simple computational strategy of selecting a fixed PML layer and increasing σ0 to obtain the desired accuracy. The results of numerical experiments varying M and σ0 are given which illustrate the theoretically predicted behavior. © 2013 Elsevier B.V. All rights reserved.

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