Higher-order sensitivity analysis of finite element method by automatic differentiation

1995 ◽  
Vol 16 (4) ◽  
pp. 223-234 ◽  
Author(s):  
I. Ozaki ◽  
F. Kimura ◽  
M. Berz
Author(s):  
Dong T.P. Nguyen ◽  
Dirk Nuyens

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 $\{\phi_j\}_{j \ge 1}$.


2016 ◽  
Vol 64 (1) ◽  
pp. 7-13 ◽  
Author(s):  
Onic Islam Shuvo ◽  
Md Naimul Islam

One of the major problems with Electrical Impedance Tomography (EIT) is the lack of spatial sensitivity within the measured volume. In this paper, sensitivity distribution of the tetrapolar impedance measurement system was visualized considering a cylindrical phantom consisting of homogeneous and inhomogeneous medium. Previously, sensitivity distribution was analysed analytically only for the homogeneous medium considering simple geometries and the distribution was found to be complex1,2. However, for the inhomogeneous volume conductors sensitivity analysis needs to be done using finite element method (FEM). In this paper, the results of sensitivity analysis based on finite element method using COMSOL Multiphysics simulation software are presented. A cylindrical non-uniform, inhomogeneous phantom, which mimics the human upper arm, was chosen to do the experiments by varying different parameters of interest. A successful method for controlling the region of interest was found where the sensitivity was maximum. Refining the finite element mesh size and introducing multifrequency input current (up to 1 MHz) this simulation method can be further improved.Dhaka Univ. J. Sci. 64(1): 7-13, 2016 (January)


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