Reduced-basis methods applied to locally non-affine and locally non-linear partial differential equations
Author(s)Solodukhov, Yuri Olegovich
Massachusetts Institute of Technology. Dept. of Mechanical Engineering.
Anthony T. Patera.
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In modern engineering and scientific applications there is a huge demand for solutions of parameter-based partial differential equations and associated outputs of interest expressed as functionals of these solutions. Areas that require solving partial differential equations include - but are not restricted to heat transfer, elasticity, and fluid dynamics. Since in most cases it is not feasible to obtain an analytic solution, many numerical approaches to obtain approximate numerical solutions - such as finite elements. finite differences, finite volumes have been developed. For applications like optimization. design, and inverse problems, where it is crucial to evaluate the field solution/output repeatedly, it might be overly computationally expensive to apply conventional numerical methods. To address this issue we present and compare two new reduced basis techniques for the rapid and reliable prediction of linear functional outputs of linear elliptic partial differential equations with locally non-affine parameter dependence: the partition of unity method (PUM) and the minimax coefficient approximation method (MCAM). We also describe the minimax coefficient approximation method (MCAM) in application to locally non-linear elliptic partial differential equations.(cont.) The essential components for both the PUM and the MCAM are (i) (provably) rapidly convergent global reduced basis approximations Galerkin projection onto a low-dimensional space spanned by the solutions of the governing partial differential equation at N selected points in the parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off- line/on-line computational procedures - methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound depends only on N (typically very small), the affine parametric complexity of the problem ad the number of points in the region where the non-affine/non-linear dependence is observed. The ratio of the error bound to the real error (which we call effectivity or the sharpness of our error estimate) typically does not exceed 100. The partition of unity approach relies on domain decomposition with respect to the separation of the affine part from the non-affine part and estimation of contributions to the error bound from these two parts. The minimax coefficient approximation approach is based on approximating the non-affine/non-linear dependence with an affine-like approximation and the subsequent treatment of the problem based on the ideas previously developed for affine problems. As a test for these new methods we consider several model problems involving steady heat transfer.(cont.) Numerical results are provided with respect to the accuracy and computational savings provided by the described reduced basis methods.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.Includes bibliographical references (p. 191-196).
DepartmentMassachusetts Institute of Technology. Dept. of Mechanical Engineering.
Massachusetts Institute of Technology