Matrix: VDOL/lowThrust_2

Description: lowThrust optimal control problem (matrix 2 of 13)

VDOL/lowThrust_2 graph
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VDOL/lowThrust_2

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  • Matrix group: VDOL
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  • download as a MATLAB mat-file, file size: 230 KB. Use UFget(2706) or UFget('VDOL/lowThrust_2') in MATLAB.
  • download in Matrix Market format, file size: 176 KB.
  • download in Rutherford/Boeing format, file size: 156 KB.

    Matrix properties
    number of rows2,312
    number of columns2,312
    nonzeros26,261
    structural full rank?yes
    structural rank2,312
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    typereal
    structuresymmetric
    Cholesky candidate?no
    positive definite?no

    authorB. Senses, A. Rao
    editorT. Davis
    date2015
    kindoptimal control problem
    2D/3D problem?no

    Additional fieldssize and type
    bfull 2312-by-1
    rownamefull 2312-by-80
    mappingfull 2312-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/lowThrust                                                         
                                                                           
    Low-thrust orbit transfer optimal control problem is taken from        
    Ref.~\cite{betts2010practical}. The goal of the low-thrust orbit       
    transfer problem is to determine the state and the control that        
    minimize the fuel consumption during the orbit transfer of a           
    spacecraft that starts from a low-earth orbit and terminates at the    
    geostationary orbit via low-thrust propulsion systems.  The highly     
    nonlinear dynamics of the low-thrust orbit transfer problem is given   
    in modified equinoctial elements (state of the system) and the thrust  
    direction (control of the system).  Furthermore, the low-thrust        
    optimal control problem is a badly scaled problem because of the       
    small thrust-to-initial-mass ratio, that is typically on the order of  
    $O(10^{-4})$, and the long orbit transfer duration. Badly scaling of   
    the problem leads to a lot of delayed pivots. The specified accuracy   
    tolerance of $10^{-8}$ were satisfied after thirteen mesh iterations.  
    As the mesh refinement proceeds, the size of the KKT matrices          
    increases from 584 to 18476.                                           
                                                                           
    @book{betts2010practical,                                              
      title={Practical Methods for Optimal Control and Estimation Using    
         Nonlinear Programming},                                           
      author={Betts, John T},                                              
      volume={19},                                                         
      year={2010},                                                         
      publisher={SIAM Press},                                              
      address = {Philadelphia, Pennsylvania},                              
    }                                                                      
    

    Ordering statistics:result
    nnz(chol(P*(A+A'+s*I)*P')) with AMD38,870
    Cholesky flop count9.4e+05
    nnz(L+U), no partial pivoting, with AMD75,428
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD61,176
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD805,070

    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.