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 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 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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> ADI step: 1 normalized residual: 1.039363e+03 ADI step: 2 normalized residual: 7.977314e+03 ADI step: 4 normalized residual: 1.911899e+02 ADI step: 6 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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 Step 40 / 400 Step 80 / 400 Step 120 / 400 Step 160 / 400 Step 200 / 400 Step 240 / 400 Step 280 / 400 Step 320 / 400 Step 360 / 400 Step 400 / 400

