build based on c1ce90c

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-31T16:04:49","documenter_version":"1.2.1"}}
+{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-02-14T11:35:09","documenter_version":"1.2.1"}}

dev/guide/index.html

@@ -87,4 +87,4 @@
     S_\ell \vec v_\ell &\approx K^\mathrm{T} \vec u_\ell.
 \end{aligned}
 \right\}
-\end{equation}\]</p><p>Together with the property <span>$\vec u_\ell^\mathrm{T} \vec u_{\ell&#39;} \approx \delta_{\ell\ell&#39;} \approx \vec v_\ell^\mathrm{T} \vec v_{\ell&#39;}$</span> we have successfully translated the original SVE problem into an SVD, because</p><p class="math-container">\[    K = \sum_\ell S_\ell \vec u_\ell \vec v_\ell^\mathrm{T}.\]</p></li><li><p>The next step is calling the <code>matrices</code> function which computes the matrix <span>$K$</span> derived in the previous step.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The function is named in the plural because in the centrosymmetric case it actually returns two matrices <span>$K_+$</span> and <span>$K_-$</span>, one for the even and one for the odd kernel. These matrices&#39; SVDs are later concatenated, so for simplicity, we will refer to <span>$K$</span> from here on out.</p></div></div><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Special care is taken here to avoid FP-arithmetic cancellation around <span>$x = -1$</span> and <span>$x = +1$</span>.</p></div></div><p><img src="../assets/img/kernel_red_matrices.png" alt="Kernel matrices"/> Note that in the plot, the matrices are rotated 90 degrees to the left to make the connection with the (subregion <span>$[0, 1] × [0, 1]$</span> of the) previous figure more obvious. Thus we can see how the choice of sampling points has magnified and brought to the matrices&#39; centers the regions of interest. Furthermore, elements with absolute values smaller than <span>$10\%$</span> of the maximum have been omitted to emphasize the structure; this should however not be taken to mean that there is any sparsity to speak of we could exploit in the next step.</p></li><li><p>Take the truncated singular value decompostion (TSVD) of <span>$K$</span>, or rather, of <span>$K_+$</span> and <span>$K_-$</span>. We use here a custom TSVD routine written by Markus Wallerberger which combines a homemade rank-revealing QR decomposition with <code>GenericLinearAlgebra.svd!</code>. This is necessary because there is currently no TSVD for arbitrary types available.</p></li><li><p>Via the function <code>truncate</code>, we throw away superfluous terms in our expansion. More specifically, we choose <span>$L$</span> in \eqref{SVE} such that <span>$S_\ell / S_0 &gt; \varepsilon$</span> for all <span>$\ell \leq L$</span>. Here <span>$\varepsilon$</span> is our selected precision, in our case it&#39;s equal to the double precision machine epsilon, <span>$2^{-52} \approx 2.22 \times 10^{-16}$</span>.</p></li></ol></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../public/">Public »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 31 January 2024 16:04">Wednesday 31 January 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{equation}\]</p><p>Together with the property <span>$\vec u_\ell^\mathrm{T} \vec u_{\ell&#39;} \approx \delta_{\ell\ell&#39;} \approx \vec v_\ell^\mathrm{T} \vec v_{\ell&#39;}$</span> we have successfully translated the original SVE problem into an SVD, because</p><p class="math-container">\[    K = \sum_\ell S_\ell \vec u_\ell \vec v_\ell^\mathrm{T}.\]</p></li><li><p>The next step is calling the <code>matrices</code> function which computes the matrix <span>$K$</span> derived in the previous step.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The function is named in the plural because in the centrosymmetric case it actually returns two matrices <span>$K_+$</span> and <span>$K_-$</span>, one for the even and one for the odd kernel. These matrices&#39; SVDs are later concatenated, so for simplicity, we will refer to <span>$K$</span> from here on out.</p></div></div><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Special care is taken here to avoid FP-arithmetic cancellation around <span>$x = -1$</span> and <span>$x = +1$</span>.</p></div></div><p><img src="../assets/img/kernel_red_matrices.png" alt="Kernel matrices"/> Note that in the plot, the matrices are rotated 90 degrees to the left to make the connection with the (subregion <span>$[0, 1] × [0, 1]$</span> of the) previous figure more obvious. Thus we can see how the choice of sampling points has magnified and brought to the matrices&#39; centers the regions of interest. Furthermore, elements with absolute values smaller than <span>$10\%$</span> of the maximum have been omitted to emphasize the structure; this should however not be taken to mean that there is any sparsity to speak of we could exploit in the next step.</p></li><li><p>Take the truncated singular value decompostion (TSVD) of <span>$K$</span>, or rather, of <span>$K_+$</span> and <span>$K_-$</span>. We use here a custom TSVD routine written by Markus Wallerberger which combines a homemade rank-revealing QR decomposition with <code>GenericLinearAlgebra.svd!</code>. This is necessary because there is currently no TSVD for arbitrary types available.</p></li><li><p>Via the function <code>truncate</code>, we throw away superfluous terms in our expansion. More specifically, we choose <span>$L$</span> in \eqref{SVE} such that <span>$S_\ell / S_0 &gt; \varepsilon$</span> for all <span>$\ell \leq L$</span>. Here <span>$\varepsilon$</span> is our selected precision, in our case it&#39;s equal to the double precision machine epsilon, <span>$2^{-52} \approx 2.22 \times 10^{-16}$</span>.</p></li></ol></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../public/">Public »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 14 February 2024 11:35">Wednesday 14 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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