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 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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 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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 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

