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 (c) 2009-2025 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 = false;
end

format long e;

set operation

opts = struct();
[oper, opts] = operatormanager(opts, '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 = true;

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 = false;
opts.adi.accumulateDeltaK = false;
opts.adi.compute_sol_fac = true;
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);
mess_fprintf(opts, 'mess_lradi took %6.2f seconds \n', t_elapsed1);
if istest
    if min(outB.res) >= 1e-1
        mess_err(opts, '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
[mZ, nZ] = size(outB.Z);
mess_fprintf(opts, 'size outB.Z: %d x %d\n\n', mZ, nZ);
ADI step:    1 normalized residual: 5.073506e-01   
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ADI step:  300 normalized residual: 1.023827e-02   
Warning: LR-ADI reached maximum iteration number. Results may be inaccurate! 
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/package/package.m',13)">package (line 13)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/publish_demos.m',18)">publish_demos (line 18)</a>
↳ In <a href="matlab:opentoline('/matlab/R2020b/toolbox/matlab/codetools/publish.p',0)">publish (line 0)</a>
↳ In <a href="matlab:opentoline('',21)">evalmxdom (line 21)</a>
↳ In <a href="matlab:opentoline('',109)">instrumentAndRun (line 109)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/DEMOS/TripleChain/BT_TripleChain.m',102)">BT_TripleChain (line 102)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_lradi.m',865)">mess_lradi (line 865)</a>
mess_lradi took   0.82 seconds   
size outB.Z: 3002 x 300  
  

Compute observability Gramian factor

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

if istest
    if min(outC.res) >= 1e-1
        mess_err(opts, '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

[mZ, nZ] = size(outC.Z);
mess_fprintf(opts, 'size outC.Z: %d x %d\n\n', mZ, nZ);

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. 
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/package/package.m',13)">package (line 13)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/publish_demos.m',18)">publish_demos (line 18)</a>
↳ In <a href="matlab:opentoline('/matlab/R2020b/toolbox/matlab/codetools/publish.p',0)">publish (line 0)</a>
↳ In <a href="matlab:opentoline('',21)">evalmxdom (line 21)</a>
↳ In <a href="matlab:opentoline('',109)">instrumentAndRun (line 109)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/DEMOS/TripleChain/BT_TripleChain.m',124)">BT_TripleChain (line 124)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_lradi.m',621)">mess_lradi (line 621)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/shifts/mess_para.m',379)">mess_para (line 379)</a>
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ADI step:  164 normalized residual: 6.175020e-02   
ADI step:  166 normalized residual: 5.999222e-02   
ADI step:  167 normalized residual: 6.714471e-02   
ADI step:  169 normalized residual: 6.637702e-02   
ADI step:  171 normalized residual: 6.695174e-02   
ADI step:  172 normalized residual: 6.576657e-02   
ADI step:  174 normalized residual: 6.369356e-02   
ADI step:  176 normalized residual: 6.051356e-02   
ADI step:  177 normalized residual: 5.433492e-02   
ADI step:  179 normalized residual: 5.405664e-02   
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ADI step:  183 normalized residual: 5.369673e-02   
ADI step:  184 normalized residual: 4.650131e-02   
ADI step:  185 normalized residual: 4.600174e-02   
ADI step:  186 normalized residual: 4.180418e-02   
ADI step:  187 normalized residual: 5.902269e-02   
ADI step:  189 normalized residual: 5.955937e-02   
ADI step:  190 normalized residual: 6.078923e-02   
ADI step:  192 normalized residual: 3.514186e-02   
ADI step:  194 normalized residual: 3.499859e-02   
ADI step:  195 normalized residual: 5.462922e-02   
ADI step:  196 normalized residual: 4.539451e-02   
ADI step:  198 normalized residual: 3.982283e-02   
ADI step:  199 normalized residual: 4.255023e-02   
ADI step:  200 normalized residual: 4.110830e-02   
ADI step:  202 normalized residual: 3.939838e-02   
ADI step:  204 normalized residual: 3.632452e-02   
ADI step:  206 normalized residual: 3.559814e-02   
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ADI step:  212 normalized residual: 3.738489e-02   
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ADI step:  222 normalized residual: 3.747796e-02   
ADI step:  224 normalized residual: 3.418642e-02   
ADI step:  225 normalized residual: 2.567229e-02   
ADI step:  227 normalized residual: 2.556362e-02   
ADI step:  229 normalized residual: 2.525263e-02   
ADI step:  230 normalized residual: 2.469037e-02   
ADI step:  232 normalized residual: 2.504278e-02   
ADI step:  234 normalized residual: 2.509127e-02   
ADI step:  235 normalized residual: 2.402671e-02   
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ADI step:  245 normalized residual: 2.854640e-02   
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ADI step:  250 normalized residual: 1.918979e-02   
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ADI step:  254 normalized residual: 1.902006e-02   
ADI step:  256 normalized residual: 1.900664e-02   
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ADI step:  259 normalized residual: 2.366267e-02   
ADI step:  261 normalized residual: 2.387701e-02   
ADI step:  262 normalized residual: 2.125206e-02   
ADI step:  264 normalized residual: 2.122155e-02   
ADI step:  265 normalized residual: 2.576238e-02   
ADI step:  267 normalized residual: 2.519779e-02   
ADI step:  269 normalized residual: 2.463721e-02   
ADI step:  271 normalized residual: 1.888866e-02   
ADI step:  272 normalized residual: 1.725708e-02   
ADI step:  274 normalized residual: 1.727830e-02   
ADI step:  275 normalized residual: 2.176824e-02   
ADI step:  277 normalized residual: 2.130534e-02   
ADI step:  279 normalized residual: 1.998423e-02   
ADI step:  281 normalized residual: 1.842274e-02   
ADI step:  282 normalized residual: 1.622261e-02   
ADI step:  284 normalized residual: 1.554208e-02   
ADI step:  285 normalized residual: 1.445818e-02   
ADI step:  287 normalized residual: 1.405300e-02   
ADI step:  289 normalized residual: 1.364326e-02   
ADI step:  290 normalized residual: 1.862752e-02   
ADI step:  292 normalized residual: 1.605889e-02   
ADI step:  294 normalized residual: 1.587883e-02   
ADI step:  295 normalized residual: 1.791207e-02   
ADI step:  297 normalized residual: 1.363707e-02   
ADI step:  299 normalized residual: 1.353945e-02   
ADI step:  300 normalized residual: 1.643088e-02   
Warning: LR-ADI reached maximum iteration number. Results may be inaccurate! 
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/package/package.m',13)">package (line 13)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/publish_demos.m',18)">publish_demos (line 18)</a>
↳ In <a href="matlab:opentoline('/matlab/R2020b/toolbox/matlab/codetools/publish.p',0)">publish (line 0)</a>
↳ In <a href="matlab:opentoline('',21)">evalmxdom (line 21)</a>
↳ In <a href="matlab:opentoline('',109)">instrumentAndRun (line 109)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/DEMOS/TripleChain/BT_TripleChain.m',124)">BT_TripleChain (line 124)</a>
↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_lradi.m',865)">mess_lradi (line 865)</a>
mess_lradi took   0.76 seconds   
size outC.Z: 3002 x 300  
  

plot results

opts.tf_plot.fmin = 1e-4;
opts.tf_plot.fmax = 1e0;
opts.tf_plot.nsample = 400;
if istest
    opts.tf_plot.info = 1;
else
    opts.tf_plot.info = 2;
end

opts.tf_plot.type = 'sigma';

out = mess_tf_plot(eqn, opts, oper, ROM);
err = out.err;
if istest
    if max(err) > 1000
        mess_err(opts, '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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