Matrix: Dehghani/light_in_tissue

Description: Light transport in soft tissue. Hamid Dehghani, Univ. Exeter, UK

 (undirected graph drawing)

• Matrix group: Dehghani
• download as a MATLAB mat-file, file size: 2 MB. Use UFget(1873) or UFget('Dehghani/light_in_tissue') in MATLAB.

 Matrix properties number of rows 29,282 number of columns 29,282 nonzeros 406,084 structural full rank? yes structural rank 29,282 # of blocks from dmperm 1 # strongly connected comp. 1 explicit zero entries 0 nonzero pattern symmetry symmetric numeric value symmetry 0% type complex structure unsymmetric Cholesky candidate? no positive definite? no

 author H. Dehghani editor T. Davis date 2007 kind electromagnetics problem 2D/3D problem? yes

 Additional fields size and type b sparse 29282-by-1 Q sparse 14641-by-1 nodes full 14641-by-3 elements full 28800-by-3

Notes:

```% The problem is solving the fluence (PHI) of light in soft tissue using
% a simplified 3rd spherical harmonic expansion (SPN3) of the Radiative
% Transport Equation.  There are two coupled equations to solve:
% M1*phi1 = Q + (M2*phi2)                                   eq(1)
% (M4 - (M3*inv(M1)*M2))*phi2 = -2/3*Q + M3*inv(M1)*Q       eq(2)
% PHI = phi1 - (1/3).*phi2                                  eq(3)

Problem = UFget ('Dehghani/light_in_tissue') ;
A = Problem.A ;                   % get the problem
Q = Problem.aux.Q ;
k = size (A,1) / 2 ;
M1 = A (1:k,1:k) ;
M2 = A (1:k,k+1:end) ;
M3 = A (k+1:end, 1:k) ;
M4 = A (k+1:end, k+1:end) ;
elements = Problem.aux.elements ;
nodes = Problem.aux.nodes ;

Q2 = (-(2/3).*Q) + (M3*(M1\Q)) ;  % create rhs for equation 2
Q2 = [sparse(k,1) ; Q2] ;
phi2 = A\Q2 ;                     % solve for phi2
phi2 = phi2 (end/2+1:end,:) ;
Q1 = Q + M2*phi2 ;                % calculate rhs for equation 1
phi1 = M1\Q1;                     % solve for phi1
PHI = phi1 - (1/3).*phi2;
figure (1) ; clf                  % plot results
trisurf(elements, nodes(:,1), nodes(:,2), nodes(:,3), log(abs(PHI))) ;
view (2) ;
colorbar('horiz') ;
axis equal ;
axis off ;
colormap hot ;
```

 Ordering statistics: result nnz(chol(P*(A+A'+s*I)*P')) with AMD 1,390,043 Cholesky flop count 2.0e+08 nnz(L+U), no partial pivoting, with AMD 2,750,804 nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 2,925,391 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 5,672,099

 SVD-based statistics: norm(A) 2.66355 min(svd(A)) 0.000340525 cond(A) 7821.9 rank(A) 29,282 sprank(A)-rank(A) 0 null space dimension 0 full numerical rank? yes

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