Matrix: Bodendiek/CurlCurl_0

Description: Curl-Curl operator of 2nd order Maxwell's equations, A. Bodendiek

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Bodendiek/CurlCurl_0

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  • download as a MATLAB mat-file, file size: 273 KB. Use UFget(2569) or UFget('Bodendiek/CurlCurl_0') in MATLAB.
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    Matrix properties
    number of rows11,083
    number of columns11,083
    nonzeros113,343
    structural full rank?yes
    structural rank11,083
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    typereal
    structuresymmetric
    Cholesky candidate?yes
    positive definite?no

    authorA. Bodendiek
    editorT. Davis
    date2012
    kindmodel reduction problem
    2D/3D problem?yes

    Notes:

    Curl-Curl operator of 2nd order Maxwell's equations, A. Bodendiek    
                                                                         
    From Andre Bodendiek, Institut Computational Mathematics,            
    TU Braunschweig                                                      
                                                                         
    The following matrix collection consists of the curl-curl-operator   
    of a second-order Maxwell's equations with PEC boundary conditions,  
    i.e. E x n = 0, where E and n denote the electric field strength     
    and the unit outer normal of the computational domain. The           
    curl-curl-operator has been discretized using the Finite Element     
    Method with first-order Nedelec elements resulting in the weak       
    formulation                                                          
                                                                         
       1/mu0 ( curl E, curl v ),                                         
                                                                         
    where v resembles a test function of H(curl) and                     
    mu0 = 1.25 1e-9 H / mm denotes the magnetic permeability of vacuum,  
    see [Hipt02].                                                        
                                                                         
    In general, the underlying model problem of Maxwell's equations      
    results from a Coplanar Waveguide, which will be considered for      
    the analysis of parasitic effects in the development of new          
    semiconductors. Since the corresponding dynamical systems are often  
    high-dimensional, model order reduction techniques have become an    
    appealing approach for the efficient simulation and accurate analysis
    of the parasitic effects. However, different kinds of model order    
    techniques require the repeated solution of high-dimensional linear  
    systems of the original model problem, see [Bai02,An09]. Therefore,  
    the development of efficient solvers resembles an important task     
    in model order reduction.                                            
                                                                         
    Each matrix CurlCurl_ consists of a different number of degrees  
    of freedom, given in the following table:                            
                                                                         
     = 0:   11083                                                    
     = 1:  226451                                                    
     = 2:  806529                                                    
     = 3: 1219574                                                    
     = 4: 2380515                                                    
                                                                         
    References.                                                          
                                                                         
    @ARTICLE{Bai02,                                                      
      author = {Z. Bai},                                                 
      title = {Krylov subspace techniques for reduced-order modeling     
        of large-scale dynamical systems},                               
      journal = {Applied Numerical Mathematics},                         
      year = {2002},                                                     
      volume = {43},                                                     
      pages = {9--44},                                                   
      number = {1--2}                                                    
    }                                                                    
                                                                         
    @ARTICLE{Hipt02,                                                     
      author = {R. Hiptmair},                                            
      title = {Finite elements in computational electromagnetism},       
      journal = {Acta Numerica, Cambridge University Press},             
      year = {2002},                                                     
      pages = {237-339}                                                  
    }                                                                    
                                                                         
    @BOOK{An09,                                                          
      title = {Approximation of {L}arge-{S}cale {D}ynamical {S}ystems},  
      publisher = {Society for Industrial Mathematics},                  
      year = {2009},                                                     
      author = {Athanasios C. Antoulas}                                  
    }                                                                    
    

    Ordering statistics:result
    nnz(chol(P*(A+A'+s*I)*P')) with AMD1,160,115
    Cholesky flop count2.8e+08
    nnz(L+U), no partial pivoting, with AMD2,309,147
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD2,082,191
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD4,040,730

    SVD-based statistics:
    norm(A)4.27023e+10
    min(svd(A))8.51547e-10
    cond(A)5.01467e+19
    rank(A)9,884
    sprank(A)-rank(A)1,199
    null space dimension1,199
    full numerical rank?no
    singular value gap5.05125e+10

    singular values (MAT file):click here
    SVD method used:s = svd (full (A))
    status:ok

    Bodendiek/CurlCurl_0 svd

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

    Maintained by Tim Davis, last updated 12-Mar-2014.
    Matrix pictures by cspy, a MATLAB function in the CSparse package.
    Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.