# Differentiation of numerical simulations with embedded nonlinear systems and integrals

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

The operation of a system or process can be studied by simulation. A simulation of a system starts with designing and developing a model of the selected (actual or theoretical) system by considering its key characteristics and behaviors. Computer simulation is to execute the corresponding model on a digital computer and analyze the results of the execution. The purpose of simulation is to examine and analyze the effect of different conditions or changes on a system. In case that a computer simulation model needs to be optimized, an optimization algorithm should be applied on it. Due to dependence of a lot of numerical algorithms for optimization on derivative information of the underlying problem, efficient evaluation of the derivatives is very important. In the case that the derivative information of the simulation system is required (for example for optimization), the derivatives of its embedded systems should be evaluated. Nonlinear systems are the systems in which the output is not directly roportional to the input and this is the case in many mathematical and physical systems. The Navier-Stokes equations in fluid dynamics are examples of nonlinear differential equations. In physics, integration is used very often, for example, for computing the work, where work is the integral of force over a distance, or as another example, for computing electric flux, which is the integral of the electric field over a surface. According to the above mentioned systems, there exists many simulation systems with embedded nonlinear systems and/or embedded (one- or multidimensional) integrals. For optimization of the corresponding simulation systems with a derivative-based method, the derivatives of the embedded integral(s) and embedded nonlinear equation(s) should also be evaluated. Therefore, different approaches to compute the derivatives of integrals and nonlinear equations are compared in terms of computational complexity, memory requirement and convergence and at the end, one will be able to choose the optimal differentiation method in case of having nonlinear system and/or (one- or multidimensional) integral in the system. On the other hand, the derivatives are not only needed for optimization, but also for example for approximating a function applying Taylor series. Considering the fact that a function can be approximated by using a finite number of terms of its Taylor series, the derivatives are also used for this approximation and if a certain amount of precision is required, higher-order derivatives should be computed, therefore, it is also crucial to have an efficient evaluation of the higher-order derivatives. The goal is to choose the better differentiation alternative in case of evaluating the first- and higher-order derivatives of a system with embedded nonlinear system and/or (one- or multidimensional) integral in terms of run time and memory requirement.