%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/VDOL/tumorAntiAngiogenesis_2
% name: VDOL/tumorAntiAngiogenesis_2
% [tumorAntiAngiogenesis optimal control problem (matrix 2 of 8)]
% id: 2749
% date: 2015
% author: B. Senses, A. Rao
% ed: T. Davis
% fields: name title A b id date author ed kind notes aux
% aux: rowname mapping
% kind: optimal control problem
%-------------------------------------------------------------------------------
% notes:
% Optimal control problem, Vehicle Dynamics & Optimization Lab, UF       
% Anil Rao and Begum Senses, University of Florida                       
% http://vdol.mae.ufl.edu                                                
%                                                                        
% This matrix arises from an optimal control problem described below.    
% Each optimal control problem gives rise to a sequence of matrices of   
% different sizes when they are being solved inside GPOPS, an optimal    
% control solver created by Anil Rao, Begum Senses, and others at in VDOL
% lab at the University of Florida.  This is one of the matrices in one  
% of these problems.  The matrix is symmetric indefinite.                
%                                                                        
% Rao, Senses, and Davis have created a graph coarsening strategy        
% that matches pairs of nodes.  The mapping is given for this matrix,    
% where map(i)=k means that node i in the original graph is mapped to    
% node k in the smaller graph.  map(i)=map(j)=k means that both nodes    
% i and j are mapped to the same node k, and thus nodes i and j have     
% been merged.                                                           
%                                                                        
% This matrix consists of a set of nodes (rows/columns) and the          
% names of these rows/cols are given                                     
%                                                                        
% Anil Rao, Begum Sense, and Tim Davis, 2015.                            
%                                                                        
% VDOL/tumorAntiogenesis                                                 
%                                                                        
% Tumor anti-angiogenesis optimal control problem is taken from          
% Ref.~\cite{ledzewicz2008analysis}. A tumor first uses the blood        
% vessels of its host but as the tumor grows oxygen that is carried by   
% the blood vessels of its host cannot defuse very far into the tumor.   
% Therefore, the tumor grows its own blood vessels by producing          
% vascular endothelial growth factor (VEGF). This process is called      
% angiogenesis. But blood vessels have a defense mechanism, called       
% endostatin, that tries to impede the development of new blood cells    
% by targeting VEGF. In addition, new pharmacological therapies that is  
% developed for tumor-type cancers also targets VEGF. The goal of the    
% tumor anti-angiogenesis problem is to determine the state and control  
% that minimizing the size of the tumor at the final time. The state of  
% the system is defined by the tumor volume, carrying capacity of a      
% vessel, and the total anti-angiogenic treatment administered and the   
% control of the system is the angiogenic dose rate.  The specified      
% accuracy tolerance of $10^{-7}$ were satisfied after eight mesh        
% iterations. As the mesh refinement proceeds, the size of the KKT       
% matrices increases from 205 to 490.                                    
%                                                                        
% @article{ledzewicz2008analysis,                                        
%   title={Analysis of Optimal Controls for a Mathematical Model of      
%      Tumour Anti-Angiogenesis},                                        
%   author={Ledzewicz, Urszula and Sch{\"a}ttler, Heinz},                
%   journal={Optimal Control Applications and Methods},                  
%   volume=29,                                                           
%   number=1,                                                            
%   pages={41--57},                                                      
%   year=2008,                                                           
%   publisher={Wiley Online Library}                                     
% }                                                                      
%-------------------------------------------------------------------------------
