Thesis Abstract and Dissertation Abstracts
Global Sensitivity Analysis Of Ordinary Differential Equation
Abstract Category: Other Categories
Course / Degree: Ph.D.,
Institution / University: Magadh University, India
Published in: 2011
Introduction
Mathematical modeling is a key tool for the anlysis of a wide range of real-world phenomena ranging from physics and engineering to chemistry, biology and economics. the recently growing influence of modeling in the nalsis of biological processes poses challenging mathematical problems. among the different modeling approaches, ordinary differential equations are particularly important and have led to significant advances . ordinary differential equations model the temporal evolution of the relevant variables by describing their deterministic dynamics. the study of dynamical systems with ordinary dirrerential equations ia mature field and therefore, there is a reich litereature devoted go their anlysis and solution.The systematic study of how uncertainty and variability affect the model outputs is called sensitivity analysis and is a crucial step of any practical ,odeling approach. sensitivity anlaysis of ordinary differential equations can be addressed from different mathematical perspectives, which give access to different numerical methods. the advantages and disadvantages of those motivated the development of a novel approach. Depending on the provlem under study, the uncertainty and variability of an ordinary differential equations model may affect initial values, the paraeters, or both. these will be referred to as the model input. in many cases uncertainty can be regarded as small variations, or perturbations, around reference input values, while variability generally refers to larger variations. effects of small variations are often studies using a local approach. local sensitivity anlaysis is based on linearized solutions of the ordinary differential equation around a reference input values. linearization facilitates the anlysis of the problem considerably. it involves the computation of partial dereivtives of the ordinary differential equation with respect to the uncertain onput variables so called sisitivity indices, which describe the variance of the output undertainty. the two terms local and linear sensitivity anlysis are often used interchangeably.
ABSTRACT
Ordinary differential equations play an important role in the modelling of many real-world problems. to guarantee reliable results, model design and anlysis must account for uncertainty and in the model input. the propagation of uncertainty and variability through the model dynamics and their effect on the output is studied by sensitivity anlysis. global sensitivity analysis is concerned with variations in the model input that possible span a large domain. two major problems that complicate the anlysis are high dimensionality and quality control, i.e. contrlling the approximation error of the estimated output uncertainty. current numberical approaches to global sensitivity anlysis mainly fcus on scalability to high dimensional models. howerver, to what extent the estimated output uncertainty approximates the true output uncertainty generally remains unclear.In this thesis we suggest an error controlled approach to global sensitivity anlysis of ordinary differential equations. the approach exploits an equivalent formulation of the problem as a partial differential wquation, which describes the evolution of the state uncertainty in terms of a probability density function. we combine recent advances from numerical anlysis and approximation theroy to solve this paritla differential equation. the method automatically controls the approximation error by adapting both temporal and spatial discretization of the numerical solution. error control is realized using a Rothe method that provides a framework for estimating temporal and spatial errors such that the discretization can be adapted accordingly. we use a novel technique called approximate approximations for the spatial discretization, it is the first time that these are used in the context of an adaptive Rothe scheme.
We analyze the convergence of the method and investigate the performance of approximate approximations in the adaptive scheme. the method is show to converge, and the theroetical results directly indicate how to design an efficient implementation. Numerical examples illustrate the theretical results and show that the method yields highly accurate estimates of the true output uncertainty. furthermore, approximate approximations have favorable properties in terms of readily available error estimates and high approximation order at frasible computational costs. Recent advances in the theory of approximate approximations, ased on a meshfree discretization of the state space, promise that the applicability of the adaptive density propagation framework developed herein can be extended to higher dimensional problems.
Summary
We established and implemented a framework for adaptive density propagation with approximate approximations and studies its asymptotic properties. the method was shown to converge. numerical examples is one and two space dimensions illustrated the tehrotical results and showed that the method is applicable to nonlinear problesm as well as problems that give rist to solutions with steep gradients or bimodal sturcture. our analysis further revealed dependencies between temporal and spatial discretization, imposing strong constraints on the spatial accuracy. an efficient solution of thes constraints necessitates a high applroximation order of the spatial discretization scheme. compared to classical discretization methods such as finite element or finite volume methods, approximate approxmiations offere three substantial advantages.Error estimates are readily avilable. the approximation order can be increased at feasible computational costs, which allows for an efficient solution of the spatio temporal accuracy constraints. although in this work we considered approximate approximations with bases functions positioned on a uniform grids as well as unstructured, scattered grids.
Long Essay Keywords/Search Tags:
deterministic, sensitivity analysis, local sensitivity analysis, model input
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Submission Details: Long Essay Abstract submitted by Mohammed Khader Ali Khan from India on 13-Nov-2013 08:20.
Abstract has been viewed 2027 times (since 7 Mar 2010).
Mohammed Khader Ali Khan Contact Details: Email: mrkhansir@gmail.com Phone: 009885271835
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