**Matrix: VDOL/tumorAntiAngiogenesis_1**

Description: tumorAntiAngiogenesis optimal control problem (matrix 1 of 8)

(undirected graph drawing) |

Matrix properties | |

number of rows | 205 |

number of columns | 205 |

nonzeros | 1,783 |

structural full rank? | yes |

structural rank | 205 |

# of blocks from dmperm | 1 |

# strongly connected comp. | 1 |

explicit zero entries | 0 |

nonzero pattern symmetry | symmetric |

numeric value symmetry | symmetric |

type | real |

structure | symmetric |

Cholesky candidate? | no |

positive definite? | no |

author | B. Senses, A. Rao |

editor | T. Davis |

date | 2015 |

kind | optimal control problem |

2D/3D problem? | no |

Additional fields | size and type |

b | full 205-by-1 |

rowname | full 205-by-79 |

mapping | full 205-by-1 |

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} }

Ordering statistics: | result |

nnz(chol(P*(A+A'+s*I)*P')) with AMD | 1,546 |

Cholesky flop count | 1.3e+04 |

nnz(L+U), no partial pivoting, with AMD | 2,887 |

nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD | 4,058 |

nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD | 20,612 |

For a description of the statistics displayed above, click here.

*Maintained by Tim Davis, last updated 04-Jun-2015.Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.
*