Matrix: VDOL/tumorAntiAngiogenesis_8

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

VDOL/tumorAntiAngiogenesis_8 graph
(undirected graph drawing)


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  • download as a MATLAB mat-file, file size: 66 KB. Use UFget(2755) or UFget('VDOL/tumorAntiAngiogenesis_8') in MATLAB.
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    Matrix properties
    number of rows490
    number of columns490
    structural full rank?yes
    structural rank490
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    Cholesky candidate?no
    positive definite?no

    authorB. Senses, A. Rao
    editorT. Davis
    kindoptimal control problem
    2D/3D problem?no

    Additional fieldssize and type
    bfull 490-by-1
    rownamefull 490-by-80
    mappingfull 490-by-1


    Optimal control problem, Vehicle Dynamics & Optimization Lab, UF       
    Anil Rao and Begum Senses, University of Florida                                                                  
    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.                            
    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.                                    
      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},                  
      publisher={Wiley Online Library}                                     

    Ordering statistics:result
    nnz(chol(P*(A+A'+s*I)*P')) with AMD4,374
    Cholesky flop count4.8e+04
    nnz(L+U), no partial pivoting, with AMD8,258
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD26,376
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD119,014

    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.