https://modelreduction.org/morwiki/index.php?action=history&feed=atom&title=Projection_based_MOR
Projection based MOR - Revision history
2026-04-11T23:27:09Z
Revision history for this page on the wiki
MediaWiki 1.43.6
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=3709&oldid=prev
Lund: remove link
2023-05-11T12:24:01Z
<p>remove link</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:24, 11 May 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l45">Line 45:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Reduced basis methods and <del style="font-weight: bold; text-decoration: none;">[[List_of_abbreviations#</del>POD<del style="font-weight: bold; text-decoration: none;">|POD]] </del>methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Reduced basis methods and POD methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Lund
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=3704&oldid=prev
Lund: remove link to list of abbreviations
2023-05-11T12:20:53Z
<p>remove link to list of abbreviations</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:20, 11 May 2023</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>as an example. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>as an example. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction (<del style="font-weight: bold; text-decoration: none;">[[List_of_abbreviations#</del>MOR<del style="font-weight: bold; text-decoration: none;">|MOR]]</del>) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction (MOR) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>. </div></td></tr>
</table>
Lund
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=2714&oldid=prev
Himpe at 09:16, 6 December 2018
2018-12-06T09:16:44Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:16, 6 December 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12">Line 12:</td>
<td colspan="2" class="diff-lineno">Line 12:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using <math>S_1</math> as the test subspace. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[<ins style="font-weight: bold; text-decoration: none;">wikipedia:Petrov–Galerkin_method|</ins>Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[<ins style="font-weight: bold; text-decoration: none;">wikipedia:Galerkin_method|</ins>Galerkin projection]] using <math>S_1</math> as the test subspace. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\hat x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{x} (t)=V z(t)</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\hat x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{x} (t)=V z(t)</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td></tr>
</table>
Himpe
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1613&oldid=prev
Feng at 08:54, 11 December 2013
2013-12-11T08:54:42Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:54, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td>
<td colspan="2" class="diff-lineno">Line 17:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Once <math>z(t)</math> is computed, an approximate solution <math>\hat x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Once <math>z(t)</math> is computed, an approximate solution <math>\hat x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The vector <math>z(t)</math> can be computed from the reduced model <del style="font-weight: bold; text-decoration: none;">which is </del>derived by the following two steps.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The vector <math>z(t)</math> can be computed from the reduced model<ins style="font-weight: bold; text-decoration: none;">, </ins>derived by the following two steps.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get</div></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1612&oldid=prev
Feng at 08:53, 11 December 2013
2013-12-11T08:53:09Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:53, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16">Line 16:</td>
<td colspan="2" class="diff-lineno">Line 16:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Once <math>z(t)</math> is computed, an approximate solution <math>\<del style="font-weight: bold; text-decoration: none;">tilde </del>x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Once <math>z(t)</math> is computed, an approximate solution <math>\<ins style="font-weight: bold; text-decoration: none;">hat </ins>x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The vector <math>z(t)</math> can be computed from the reduced model which is derived by the following two steps.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The vector <math>z(t)</math> can be computed from the reduced model which is derived by the following two steps.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1611&oldid=prev
Feng at 08:52, 11 December 2013
2013-12-11T08:52:23Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:52, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction ([[List_of_abbreviations#MOR|MOR]]) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction ([[List_of_abbreviations#MOR|MOR]]) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\<del style="font-weight: bold; text-decoration: none;">tilde </del>x(t)</math> in <math>S_1</math>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\<ins style="font-weight: bold; text-decoration: none;">hat </ins>x(t)</math> in <math>S_1</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using <math>S_1</math> as the test subspace. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using <math>S_1</math> as the test subspace. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\<del style="font-weight: bold; text-decoration: none;">tilde </del>x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\<del style="font-weight: bold; text-decoration: none;">tilde</del>{x} (t)=V z(t)</math>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\<ins style="font-weight: bold; text-decoration: none;">hat </ins>x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\<ins style="font-weight: bold; text-decoration: none;">hat</ins>{x} (t)=V z(t)</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Here <math>z</math> is a vector of length <math>q \ll n</math>.</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that the approximation <math>\<del style="font-weight: bold; text-decoration: none;">tilde </del>x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that the approximation <math>\<ins style="font-weight: bold; text-decoration: none;">hat </ins>x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2). </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1). </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1). </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model. </div></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1610&oldid=prev
Feng at 08:50, 11 December 2013
2013-12-11T08:50:18Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:50, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l36">Line 36:</td>
<td colspan="2" class="diff-lineno">Line 36:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that the approximation <math>\<del style="font-weight: bold; text-decoration: none;">hat </del>x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that the approximation <math>\<ins style="font-weight: bold; text-decoration: none;">tilde </ins>x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2). </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1). </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1). </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model. </div></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1609&oldid=prev
Feng at 08:49, 11 December 2013
2013-12-11T08:49:06Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:49, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22">Line 22:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math>E \frac{d{Vz}}{dt} <del style="font-weight: bold; text-decoration: none;">\approx </del>A Vz+Bu(t),\quad y(t) <del style="font-weight: bold; text-decoration: none;">\approx </del>CV z.</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math>E \frac{d{Vz}}{dt}<ins style="font-weight: bold; text-decoration: none;">=</ins>A Vz+Bu(t)<ins style="font-weight: bold; text-decoration: none;">+e</ins>,\quad <ins style="font-weight: bold; text-decoration: none;">\hat </ins>y(t)<ins style="font-weight: bold; text-decoration: none;">=</ins>CV z.</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1608&oldid=prev
Feng at 08:45, 11 December 2013
2013-12-11T08:45:29Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:45, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12">Line 12:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\tilde x(t)</math> in <math>S_1</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated by its projection <math>\tilde x(t)</math> in <math>S_1</math>. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using <del style="font-weight: bold; text-decoration: none;">the </del><math>S_1</math> as the test subspace. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using <math>S_1</math> as the test subspace. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\tilde{x} (t)=V z(t)</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\tilde{x} (t)=V z(t)</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.</div></td></tr>
</table>
Feng
https://modelreduction.org/morwiki/index.php?title=Projection_based_MOR&diff=1607&oldid=prev
Feng at 08:44, 11 December 2013
2013-12-11T08:44:55Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:44, 11 December 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction ([[List_of_abbreviations#MOR|MOR]]) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All the existing model order reduction ([[List_of_abbreviations#MOR|MOR]]) methods are based on projection<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated its projection <math>\tilde x(t)</math> in <math>S_1</math>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Afterwards, <math>x(t)</math> is approximated <ins style="font-weight: bold; text-decoration: none;">by </ins>its projection <math>\tilde x(t)</math> in <math>S_1</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using the <math>S_1</math> as the test subspace. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reduced model is produced by [[Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[Galerkin projection]] using the <math>S_1</math> as the test subspace. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\tilde{x} (t)=V z(t)</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\tilde x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\tilde{x} (t)=V z(t)</math>. </div></td></tr>
</table>
Feng