Contents

function BT_TripleChain(variant, 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:
%
% variant  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 file is part of the M-M.E.S.S. project
% (http://www.mpi-magdeburg.mpg.de/projects/mess).
% Copyright © 2009-2022 Jens Saak, Martin Koehler, Peter Benner and others.
% All rights reserved.
% License: BSD 2-Clause License (see COPYING)
%
narginchk(0,2)

if nargin==0, variant = '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 controllability Gramian factor

eqn.type = 'N';
t_mess_lradi = tic;
outB = mess_lradi(eqn, opts, oper);
t_elapsed1 = toc(t_mess_lradi);
fprintf(1,'mess_lradi took %6.2f seconds \n',t_elapsed1);
if istest
    if min(outB.res)>=1e-1
       error('MESS:TEST:accuracy','unexpectedly inaccurate result');
   end
else
    figure(1);
    semilogy(outB.res,'LineWidth',3);
    title('0= A X E^T + E X A^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 
ADI step:   59 normalized residual: 3.416453e-02 
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 
ADI step:   71 normalized residual: 3.606839e-02 
ADI step:   72 normalized residual: 3.698424e-02 
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ADI step:   94 normalized residual: 3.169166e-02 
ADI step:   95 normalized residual: 3.138964e-02 
ADI step:   97 normalized residual: 2.562329e-02 
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ADI step:  100 normalized residual: 2.804265e-02 
ADI step:  102 normalized residual: 2.780155e-02 
ADI step:  104 normalized residual: 2.672199e-02 
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ADI step:  109 normalized residual: 2.552206e-02 
ADI step:  110 normalized residual: 2.514486e-02 
ADI step:  112 normalized residual: 2.349278e-02 
ADI step:  114 normalized residual: 2.342010e-02 
ADI step:  116 normalized residual: 2.345786e-02 
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ADI step:  119 normalized residual: 2.265592e-02 
ADI step:  120 normalized residual: 2.203523e-02 
ADI step:  122 normalized residual: 2.168775e-02 
ADI step:  124 normalized residual: 2.221522e-02 
ADI step:  125 normalized residual: 1.852156e-02 
ADI step:  127 normalized residual: 1.818338e-02 
ADI step:  129 normalized residual: 1.968910e-02 
ADI step:  130 normalized residual: 1.864035e-02 
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ADI step:  147 normalized residual: 1.716120e-02 
ADI step:  149 normalized residual: 1.591096e-02 
ADI step:  150 normalized residual: 1.604931e-02 
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ADI step:  155 normalized residual: 1.659504e-02 
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ADI step:  159 normalized residual: 1.641254e-02 
ADI step:  160 normalized residual: 1.704029e-02 
ADI step:  162 normalized residual: 1.698300e-02 
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ADI step:  169 normalized residual: 1.551576e-02 
ADI step:  170 normalized residual: 1.578635e-02 
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ADI step:  184 normalized residual: 1.592599e-02 
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ADI step:  224 normalized residual: 1.408914e-02 
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ADI step:  227 normalized residual: 1.378953e-02 
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ADI step:  254 normalized residual: 1.134657e-02 
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ADI step:  257 normalized residual: 1.091387e-02 
ADI step:  259 normalized residual: 1.087427e-02 
ADI step:  260 normalized residual: 1.083111e-02 
ADI step:  262 normalized residual: 1.084872e-02 
ADI step:  264 normalized residual: 1.077242e-02 
ADI step:  265 normalized residual: 1.079865e-02 
ADI step:  267 normalized residual: 1.076889e-02 
ADI step:  269 normalized residual: 1.054557e-02 
ADI step:  270 normalized residual: 1.046478e-02 
ADI step:  272 normalized residual: 1.037342e-02 
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ADI step:  275 normalized residual: 1.032450e-02 
ADI step:  277 normalized residual: 1.032852e-02 
ADI step:  279 normalized residual: 1.032709e-02 
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ADI step:  282 normalized residual: 1.048461e-02 
ADI step:  284 normalized residual: 1.039864e-02 
ADI step:  285 normalized residual: 1.012241e-02 
ADI step:  287 normalized residual: 1.011572e-02 
ADI step:  289 normalized residual: 9.953143e-03 
ADI step:  290 normalized residual: 1.019119e-02 
ADI step:  292 normalized residual: 1.020078e-02 
ADI step:  294 normalized residual: 1.014819e-02 
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ADI step:  298 normalized residual: 1.013405e-02 
ADI step:  299 normalized residual: 1.044240e-02 
ADI step:  300 normalized residual: 1.015609e-02 
Warning: LR-ADI reached maximum iteration number.results may be inaccurate! 
mess_lradi took   0.80 seconds 
size outB.Z:
        3002         300

Compute observability Gramian factor

eqn.type = 'T';
t_mess_lradi =tic;
outC = mess_lradi(eqn, opts, oper);
t_elapsed2 = toc(t_mess_lradi);
fprintf(1,'mess_lradi took %6.2f seconds \n' ,t_elapsed2);


if istest
    if min(outC.res)>=1e-1
       error('MESS:TEST:accuracy','unexpectedly inaccurate result');
   end
else
    figure(2);
    semilogy(outC.res,'LineWidth',3);
    title('0 = A^T X E + E^T X A - C^T C');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end

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


switch upper(variant)
    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: 300/150)

    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(variant,'FO'))
    max_ord = 75;
    tol = eps;
    inform = 1;

    [TL,TR] = square_root_method_SO(eqn.M_, max_ord, tol, inform, 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: 7.157459e+01 
ADI step:    2 normalized residual: 1.553258e+03 
ADI step:    4 normalized residual: 4.276002e+02 
ADI step:    6 normalized residual: 2.972375e+02 
ADI step:    8 normalized residual: 2.015019e+02 
ADI step:   10 normalized residual: 2.179720e+02 
ADI step:   12 normalized residual: 1.078765e+02 
ADI step:   14 normalized residual: 1.058741e+01 
ADI step:   16 normalized residual: 1.659045e+01 
ADI step:   18 normalized residual: 1.760458e+01 
ADI step:   20 normalized residual: 1.888842e+01 
ADI step:   22 normalized residual: 9.218597e+00 
ADI step:   24 normalized residual: 7.581765e+00 
ADI step:   26 normalized residual: 5.307815e+00 
ADI step:   28 normalized residual: 4.429565e+00 
ADI step:   30 normalized residual: 3.905992e+00 
ADI step:   32 normalized residual: 3.958810e+00 
ADI step:   33 normalized residual: 1.444905e+01 
ADI step:   35 normalized residual: 4.995354e+00 
ADI step:   37 normalized residual: 3.885198e+00 
ADI step:   38 normalized residual: 3.085614e+00 
ADI step:   40 normalized residual: 3.665910e+00 
ADI step:   42 normalized residual: 3.412363e+00 
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ADI step:   46 normalized residual: 1.805953e+00 
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ADI step:   49 normalized residual: 1.861243e+00 
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ADI step:   52 normalized residual: 1.941017e+00 
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ADI step:   56 normalized residual: 1.446102e+00 
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ADI step:   60 normalized residual: 2.016358e+00 
ADI step:   62 normalized residual: 1.969557e+00 
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ADI step:  251 normalized residual: 2.761973e-02 
ADI step:  252 normalized residual: 2.514674e-02 
ADI step:  254 normalized residual: 2.504548e-02 
ADI step:  255 normalized residual: 2.926822e-02 
ADI step:  257 normalized residual: 3.002182e-02 
ADI step:  259 normalized residual: 2.815447e-02 
ADI step:  260 normalized residual: 2.528701e-02 
ADI step:  262 normalized residual: 2.596840e-02 
ADI step:  264 normalized residual: 2.454883e-02 
ADI step:  266 normalized residual: 1.981306e-02 
ADI step:  267 normalized residual: 2.586688e-02 
ADI step:  268 normalized residual: 2.398569e-02 
ADI step:  270 normalized residual: 2.421163e-02 
ADI step:  272 normalized residual: 2.244289e-02 
ADI step:  273 normalized residual: 2.339550e-02 
ADI step:  275 normalized residual: 2.323619e-02 
ADI step:  277 normalized residual: 2.061973e-02 
ADI step:  278 normalized residual: 2.072000e-02 
ADI step:  280 normalized residual: 2.011700e-02 
ADI step:  282 normalized residual: 1.849902e-02 
ADI step:  283 normalized residual: 1.846989e-02 
ADI step:  285 normalized residual: 1.901308e-02 
ADI step:  287 normalized residual: 1.941725e-02 
ADI step:  288 normalized residual: 1.965287e-02 
ADI step:  290 normalized residual: 1.795240e-02 
ADI step:  292 normalized residual: 1.772159e-02 
ADI step:  293 normalized residual: 1.867071e-02 
ADI step:  295 normalized residual: 1.866235e-02 
ADI step:  297 normalized residual: 1.858484e-02 
ADI step:  298 normalized residual: 1.983339e-02 
ADI step:  300 normalized residual: 1.955150e-02 
Warning: LR-ADI reached maximum iteration number.results may be inaccurate! 
mess_lradi took   0.79 seconds 
size outC.Z:
        3002         300

plot results

opts.sigma.fmin = 1e-4;
opts.sigma.fmax = 1e0;
opts.sigma.nsample = 400;
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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