Contents

function BT_TripleChain(version, istest)
%
% Computes a reduced order model (ROM) for the triple chain example of
% Truhar and Veselic [1] via Balanced truncation, e.g. [2].
%
% Usage:   BT_TripleChain(version, istest)
%
% Input:
%
% version  Decides the Balanced Truncation version to use.
%          Possible values:
%          'FO' for reduction of the first order form to first order form
%          'VV' velocity-velocity balancing of the second order form to
%               second order form.
%          'PP' position-position balancing of the second order form to
%               second order form.
%          'PV' position-velocity balancing of the second order form to
%               second order form.
%          'VP' velocity-position balancing of the second order form to
%               second order form.
%
% istest      flag to determine whether this demo runs as a CI test or
%             interactive demo
%             (optional, defaults to 0, i.e. interactive demo)
%
% References:
%
% [1] N. Truhar and K. Veselic, An efficient method for estimating the
%     optimal dampers’ viscosity for linear vibrating systems using
%     Lyapunov equation, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 18–39.
%
% [2] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Vol.
%     6 of Adv. Des. Control, SIAM Publications, Philadelphia, PA, 2005.
%     https://doi.org/10.1137/1.9780898718713.
%
%
% This program is free software; you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation; either version 2 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; if not, see <http://www.gnu.org/licenses/>.
%
% Copyright (C) Jens Saak, Martin Koehler, Peter Benner and others
%               2009-2020
%

narginchk(0,2)

if nargin==0, version = 'FO'; end
if nargin<2, istest=0; end

format long e;

set operation

oper = operatormanager('so_1');

Initialize problem data

n1=500;
alpha=.002;
beta=alpha;
v=5;

[eqn.M_,eqn.E_,eqn.K_]=triplechain_MSD(n1,alpha,beta,v);

s  = size(eqn.K_,1);
O = zeros(s,1);
Cv = O';
Cp = ones(1,size(eqn.K_,1));
B = ones(size(eqn.K_,1),1);
eqn.B = [O; B];
eqn.C = [Cp, Cv];

eqn.haveE=1;

ADI tolerances and maximum iteration number

opts.adi.maxiter = 300;
opts.adi.res_tol = 1e-10;
opts.adi.rel_diff_tol = 0;
opts.norm = 'fro';

opts.adi.info = 1;
opts.adi.accumulateK = 0;
opts.adi.accumulateDeltaK = 0;
opts.adi.compute_sol_fac = 1;
opts.norm = 'fro';
%Heuristic shift parameters via projection
opts.shifts.num_desired=5;

opts.shifts.info=0;
opts.shifts.method = 'projection';

Compute controlability Gramian factor

eqn.type = 'N';
tic;
outB = mess_lradi(eqn, opts, oper);
toc;

if istest
    if min(outB.res)>=1e-1
       error('MESS:TEST:accuracy','unexpectedly inaccurate result');
   end
else
    figure(1);
    semilogy(outB.res);
    title('0= AXM^T + MXA^T -BB^T');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end

disp('size outB.Z:');
disp(size(outB.Z));
ADI step:    1 normalized residual: 5.073506e-01 
ADI step:    2 normalized residual: 5.560323e-01 
ADI step:    4 normalized residual: 2.325197e-01 
ADI step:    6 normalized residual: 2.106015e-01 
ADI step:    7 normalized residual: 2.270676e-01 
ADI step:    8 normalized residual: 8.370974e-02 
ADI step:    9 normalized residual: 1.592188e-01 
ADI step:   10 normalized residual: 1.514040e-01 
ADI step:   12 normalized residual: 1.113157e-01 
ADI step:   14 normalized residual: 9.340643e-02 
ADI step:   15 normalized residual: 7.596297e-02 
ADI step:   17 normalized residual: 7.107314e-02 
ADI step:   19 normalized residual: 5.571037e-02 
ADI step:   20 normalized residual: 5.792667e-02 
ADI step:   21 normalized residual: 4.856437e-02 
ADI step:   22 normalized residual: 5.849958e-02 
ADI step:   24 normalized residual: 5.367959e-02 
ADI step:   26 normalized residual: 5.289025e-02 
ADI step:   28 normalized residual: 5.246617e-02 
ADI step:   29 normalized residual: 3.877912e-02 
ADI step:   31 normalized residual: 4.018146e-02 
ADI step:   33 normalized residual: 3.840383e-02 
ADI step:   34 normalized residual: 4.766065e-02 
ADI step:   36 normalized residual: 4.931627e-02 
ADI step:   37 normalized residual: 5.143612e-02 
ADI step:   39 normalized residual: 3.584359e-02 
ADI step:   41 normalized residual: 3.616331e-02 
ADI step:   42 normalized residual: 4.160551e-02 
ADI step:   44 normalized residual: 3.229327e-02 
ADI step:   46 normalized residual: 3.099704e-02 
ADI step:   47 normalized residual: 3.571161e-02 
ADI step:   49 normalized residual: 2.642895e-02 
ADI step:   51 normalized residual: 2.652633e-02 
ADI step:   52 normalized residual: 4.057050e-02 
ADI step:   54 normalized residual: 3.577600e-02 
ADI step:   56 normalized residual: 3.570631e-02 
ADI step:   58 normalized residual: 3.416888e-02 
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ADI step:   61 normalized residual: 3.721183e-02 
ADI step:   63 normalized residual: 3.735574e-02 
ADI step:   64 normalized residual: 2.840138e-02 
ADI step:   66 normalized residual: 2.859885e-02 
ADI step:   68 normalized residual: 3.284944e-02 
ADI step:   69 normalized residual: 3.733041e-02 
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ADI step:   72 normalized residual: 3.698424e-02 
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ADI step:   94 normalized residual: 3.169164e-02 
ADI step:   95 normalized residual: 3.138957e-02 
ADI step:   97 normalized residual: 2.562323e-02 
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ADI step:  100 normalized residual: 2.804258e-02 
ADI step:  102 normalized residual: 2.780144e-02 
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ADI step:  105 normalized residual: 2.745734e-02 
ADI step:  107 normalized residual: 2.742431e-02 
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ADI step:  112 normalized residual: 2.349271e-02 
ADI step:  114 normalized residual: 2.342001e-02 
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ADI step:  120 normalized residual: 2.203365e-02 
ADI step:  122 normalized residual: 2.168591e-02 
ADI step:  124 normalized residual: 2.221375e-02 
ADI step:  125 normalized residual: 1.851511e-02 
ADI step:  127 normalized residual: 1.817853e-02 
ADI step:  129 normalized residual: 1.969048e-02 
ADI step:  130 normalized residual: 1.861911e-02 
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ADI step:  134 normalized residual: 1.886756e-02 
ADI step:  135 normalized residual: 1.785993e-02 
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ADI step:  144 normalized residual: 1.830817e-02 
ADI step:  145 normalized residual: 1.768449e-02 
ADI step:  147 normalized residual: 1.759622e-02 
ADI step:  149 normalized residual: 1.603948e-02 
ADI step:  150 normalized residual: 1.602936e-02 
ADI step:  152 normalized residual: 1.591953e-02 
ADI step:  154 normalized residual: 1.572462e-02 
ADI step:  155 normalized residual: 1.574657e-02 
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ADI step:  159 normalized residual: 1.560752e-02 
ADI step:  160 normalized residual: 1.574328e-02 
ADI step:  162 normalized residual: 1.586069e-02 
ADI step:  164 normalized residual: 1.586593e-02 
ADI step:  165 normalized residual: 1.619715e-02 
ADI step:  167 normalized residual: 1.630662e-02 
ADI step:  169 normalized residual: 1.585621e-02 
ADI step:  170 normalized residual: 1.563056e-02 
ADI step:  172 normalized residual: 1.557036e-02 
ADI step:  174 normalized residual: 1.530038e-02 
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ADI step:  178 normalized residual: 1.519329e-02 
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ADI step:  218 normalized residual: 1.278199e-02 
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ADI step:  220 normalized residual: 1.196528e-02 
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ADI step:  252 normalized residual: 1.095684e-02 
ADI step:  254 normalized residual: 1.103310e-02 
ADI step:  255 normalized residual: 1.117404e-02 
ADI step:  257 normalized residual: 1.117405e-02 
ADI step:  259 normalized residual: 1.117017e-02 
ADI step:  260 normalized residual: 1.095007e-02 
ADI step:  262 normalized residual: 1.089079e-02 
ADI step:  264 normalized residual: 1.085239e-02 
ADI step:  265 normalized residual: 1.096006e-02 
ADI step:  267 normalized residual: 1.082644e-02 
ADI step:  269 normalized residual: 1.072482e-02 
ADI step:  270 normalized residual: 1.063044e-02 
ADI step:  272 normalized residual: 1.058604e-02 
ADI step:  274 normalized residual: 1.054422e-02 
ADI step:  275 normalized residual: 1.042107e-02 
ADI step:  277 normalized residual: 1.043413e-02 
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ADI step:  284 normalized residual: 1.032968e-02 
ADI step:  285 normalized residual: 1.029359e-02 
ADI step:  287 normalized residual: 1.019644e-02 
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ADI step:  290 normalized residual: 1.014151e-02 
ADI step:  292 normalized residual: 1.013369e-02 
ADI step:  294 normalized residual: 1.009579e-02 
ADI step:  295 normalized residual: 9.819899e-03 
ADI step:  297 normalized residual: 9.823660e-03 
ADI step:  299 normalized residual: 9.824014e-03 
ADI step:  301 normalized residual: 9.825429e-03 
Elapsed time is 0.531542 seconds.
size outB.Z:
        3002         301

Compute observability Gramian factor

eqn.type = 'T';
tic;
outC = mess_lradi(eqn, opts, oper);
toc;

if istest
    if min(outC.res)>=1e-1
       error('MESS:TEST:accuracy','unexpectedly inaccurate result');
   end
else
    figure(2);
    semilogy(outC.res);
    title('0= A^TXM + M^TXA -C^TC');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end

disp('size outC.Z:');
disp(size(outC.Z));


switch upper(version)
    case 'FO'

Compute first order ROM

        opts.srm.max_ord = 150;
        opts.srm.tol = eps;
        opts.srm.info = 1;

        [TL,TR] = mess_square_root_method(eqn,opts,oper,outB.Z,outC.Z);

        ROM.E = eye(size(TL,2));
        ROM.A = TL'*oper.mul_A(eqn, opts, 'N', TR, 'N');
        ROM.B = TL'*eqn.B;
        ROM.C = eqn.C*TR;
        ROM.D = [];
reduced system order: 150  (max possible/allowed: 301/150)

    case 'VV'
        U = outB.Z(1:s,:);
        V = outC.Z(1:s,:);
    case 'PP'
        U = outB.Z(s+1:end,:);
        V = outC.Z(s+1:end,:);
    case 'PV'
        U = outB.Z(s+1:end,:);
        V = outC.Z(1:s,:);
    case 'VP'
        U = outB.Z(1:s,:);
        V = outC.Z(s+1:end,:);
end
if not(strcmp(version,'FO'))
    max_ord = 75;
    tol = eps;
    info = 1;

    [TL,TR] = square_root_method_SO(eqn.M_, max_ord, tol, info, U, V);

    ROM.M = eye(size(TL,2));
    ROM.E = TL'*(eqn.E_*TR);
    ROM.K = TL'*(eqn.K_*TR);
    ROM.B = TL'*B;
    ROM.Cv = Cv*TR;
    ROM.Cp = Cp*TR;
end
Warning: Could not compute initial projection shifts. Going to retry with random
right hand side. 
ADI step:    1 normalized residual: 8.993042e+01 
ADI step:    2 normalized residual: 1.871791e+03 
ADI step:    4 normalized residual: 4.038661e+02 
ADI step:    6 normalized residual: 2.893253e+02 
ADI step:    8 normalized residual: 1.999807e+02 
ADI step:   10 normalized residual: 2.156404e+02 
ADI step:   12 normalized residual: 1.080239e+02 
ADI step:   14 normalized residual: 1.077604e+01 
ADI step:   16 normalized residual: 8.584757e+00 
ADI step:   18 normalized residual: 7.864729e+00 
ADI step:   20 normalized residual: 8.055562e+00 
ADI step:   22 normalized residual: 1.600772e+01 
ADI step:   24 normalized residual: 1.598200e+01 
ADI step:   26 normalized residual: 1.623130e+01 
ADI step:   28 normalized residual: 1.620898e+01 
ADI step:   30 normalized residual: 3.165546e+00 
ADI step:   32 normalized residual: 2.968809e+00 
ADI step:   34 normalized residual: 6.467635e+00 
ADI step:   36 normalized residual: 3.651244e+00 
ADI step:   38 normalized residual: 3.522482e+00 
ADI step:   40 normalized residual: 3.618827e+00 
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ADI step:   58 normalized residual: 8.749579e-01 
ADI step:   60 normalized residual: 9.494202e-01 
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ADI step:  269 normalized residual: 1.652104e-02 
ADI step:  270 normalized residual: 1.602112e-02 
ADI step:  272 normalized residual: 1.589620e-02 
ADI step:  273 normalized residual: 1.416399e-02 
ADI step:  275 normalized residual: 1.465861e-02 
ADI step:  277 normalized residual: 1.538592e-02 
ADI step:  278 normalized residual: 1.498473e-02 
ADI step:  280 normalized residual: 1.509517e-02 
ADI step:  282 normalized residual: 1.305592e-02 
ADI step:  283 normalized residual: 1.390804e-02 
ADI step:  285 normalized residual: 1.420802e-02 
ADI step:  287 normalized residual: 1.403487e-02 
ADI step:  288 normalized residual: 1.257510e-02 
ADI step:  290 normalized residual: 1.252698e-02 
ADI step:  292 normalized residual: 1.294582e-02 
ADI step:  294 normalized residual: 1.278812e-02 
ADI step:  296 normalized residual: 1.203515e-02 
ADI step:  297 normalized residual: 1.204933e-02 
ADI step:  299 normalized residual: 1.206596e-02 
ADI step:  301 normalized residual: 1.172421e-02 
Elapsed time is 0.516285 seconds.
size outC.Z:
        3002         301

plot results

opts.sigma.fmin = 1e-4;
opts.sigma.fmax = 1e0;
opts.sigma.nsample = 200;
if istest
    opts.sigma.info = 1;
else
    opts.sigma.info = 2;
end
out = mess_sigma_plot(eqn, opts, oper, ROM); err = out.err;
if istest
    if max(err) > 1000
        error('MESS:TEST:accuracy','unexpectedly inaccurate result %g',max(err));
    end
end
Computing TFMs of original and reduced order systems and MOR errors
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