# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Implementation of smooth interpolation for optimization

(2015)
Department of Automatic Control
Abstract
Models of physical systems are often expressed as a system of mathematical expressions derived from first principles. However some of the relationships present in a model are more conveniently expressed as a table, i.e. for certain values of independent variables the values of dependent variables are known. This imposes a limitation on the optimization software to be used, since optimization software is often dependent on all relationships being expressed as mathematical functions. Many useful optimization methods also require that the functions be twice continuously differentiable.

In this thesis software for interpolating table data relationships between two variables as a twice continuously differentiable mathematical function has... (More)
Models of physical systems are often expressed as a system of mathematical expressions derived from first principles. However some of the relationships present in a model are more conveniently expressed as a table, i.e. for certain values of independent variables the values of dependent variables are known. This imposes a limitation on the optimization software to be used, since optimization software is often dependent on all relationships being expressed as mathematical functions. Many useful optimization methods also require that the functions be twice continuously differentiable.

In this thesis software for interpolating table data relationships between two variables as a twice continuously differentiable mathematical function has been developed. This software has also been prototypically made callable from the automatic differentiation tool CasADi. CasADi is used in the optimization tool chain in JModelica.org, an open source platform for optimization and simulation. By implementing support for table based relations a larger range of problems may be solved using CasADi.

The software developed for interpolating tables uses cubic b-splines and de Boor evaluation. Using it one may evaluate the interpolant and its derivatives up to the third order. The resultant function is demonstrated to be twice continuously differentiable and to interpolate the value within machine epsilon range of the correct one at the data points, provided the table data points are equidistantly distributed. The oscillations that occur when interpolating non-equidistant table data points are also examined. (Less)
author
supervisor
organization
year
type
H3 - Professional qualifications (4 Years - )
subject
other publication id
ISRN LUTFD2/TFRT--59964--SE
language
English
id
8349327
2015-12-18 13:17:48
date last changed
2015-12-18 13:17:48
```@misc{8349327,
abstract     = {Models of physical systems are often expressed as a system of mathematical expressions derived from first principles. However some of the relationships present in a model are more conveniently expressed as a table, i.e. for certain values of independent variables the values of dependent variables are known. This imposes a limitation on the optimization software to be used, since optimization software is often dependent on all relationships being expressed as mathematical functions. Many useful optimization methods also require that the functions be twice continuously differentiable.

In this thesis software for interpolating table data relationships between two variables as a twice continuously differentiable mathematical function has been developed. This software has also been prototypically made callable from the automatic differentiation tool CasADi. CasADi is used in the optimization tool chain in JModelica.org, an open source platform for optimization and simulation. By implementing support for table based relations a larger range of problems may be solved using CasADi.

The software developed for interpolating tables uses cubic b-splines and de Boor evaluation. Using it one may evaluate the interpolant and its derivatives up to the third order. The resultant function is demonstrated to be twice continuously differentiable and to interpolate the value within machine epsilon range of the correct one at the data points, provided the table data points are equidistantly distributed. The oscillations that occur when interpolating non-equidistant table data points are also examined.},