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<!doctype html>
<html>
<head>
<meta charset="utf-8">
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<img src='./img/logo.svg' style='height: 100%; padding: 0.3%; box-sizing: border-box;'/>
<div style='bottom: 2vh; position: absolute; width: 100%; text-align: center; font-family: "Source Sans Pro", Helvetica, sans-serif;'>
Jorn Hoofwijk - A multi-dimensional adaptive sampling algorithm and its application to Fermi surfaces
</div>
</div>
<div class="reveal">
<div class="slides">
<section>
<h4>A multi-dimensional adaptive sampling algorithm and its application to Fermi surfaces</h4>
<p>
Jorn Hoofwijk<br/>
<small>4396499</small>
</p>
<div>
<small style='text-align: left; margin-right: 30%'>
<em>Supervisors</em><br/>
Dr. Anton Akhmerov<br/>
Dr. ir. Dennis den Ouden-van der Horst
<!--
<br/><br/>
<em>Thanks to</em><br/>
Joe Weston<br/>
Bas Nijholt<br/> -->
</small>
<small style='text-align: right;'>
<em>Committee</em><br/>
Dr. Michael Wimmer<br/>
Dr. ir. Wolter Groenevelt
</small>
</div>
</section>
<section>
<h2>Fermi surface</h2>
<img src="img/L_fermi_surface/fermi_gold.jpg"/><br/>
Fermi surface of gold
<small style='position: absolute; bottom: 0; right: 0; font-size: 0.3em;'>Image from T. Li & W. Schroeder (2007)</small>
</section>
<section>
<h2>Outline</h2>
<ul class=''>
<li class='fragment'>Adaptive sampling algorithm</li>
<li class='fragment'>Generalisation to N dimensions</li>
<li class='fragment'>Band structures & Fermi surfaces</li>
</ul>
</section>
<section>
<h2>Intro</h2>
<!-- <div class='two-img'> -->
<img width=400 height=400 src="img/A_first_year_resonance/cantilever.svg" />
<img src="img/A_first_year_resonance/original.svg" />
<!-- </div> -->
</section>
<section>
<h2>Problem</h2>
<div class='two-img'>
<div class=''>
<img src='img/A_first_year_resonance/original.svg'/>
</div>
<div class=''>
<img src='img/A_first_year_resonance/homo_100.svg'/>
</div>
</div>
</section>
<section>
<h2>Solution</h2>
<div class='two-img'>
<div class=''>
<img src='img/A_first_year_resonance/homo_100.svg'/>
Homogeneous
</div>
<div class=''>
<img src='img/A_first_year_resonance/adaptive_100.svg'/>
Adaptive
</div>
</div>
</section>
<section>
<h2>Adaptive</h2>
<div style="position:relative; width:640px; height:480px; margin:0 auto;">
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_2.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_2_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_2_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_3.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_3_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_3_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_4.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_4_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_4_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_5.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_5_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_5_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_6.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_6_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_6_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_7.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_7_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_7_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_8.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_8_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_8_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_9.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_9_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_9_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_10.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_10_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_10_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_11.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_11_marked.svg" style="position:absolute;top:0;left:0;" /> -->
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_11_marked_2.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/B_Adaptive_iteration/img_12.svg" style="position:absolute;top:0;left:0;" />
</div>
</section>
<section>
<h2>Adaptive</h2>
<div class='two-img'>
<div class=''>
<img src='img/B_Adaptive_iteration/homogeneous.svg'/>
Homogeneous
</div>
<div class=''>
<img src='img/B_Adaptive_iteration/adaptive.svg'/>
Adaptive
</div>
</div>
</section>
<!-- <section>
<h2>Adaptive</h2>
<script type="math/tex; mode=display">
L_i = \sqrt{\left(\frac{\Delta x}{x_{\text{max}} - x_{\text{min}}}\right)^2 + \left(\frac{\Delta y}{y_{\text{max}} - y_{\text{min}}}\right)^2}
</script>
</section> -->
<section>
<div class='highlight-title'>A <b>multi-dimensional adaptive sampling</b> algorithm and its application to Fermi surfaces</div>
</section>
<section>
<h2>Adaptive 2D</h2>
<div class='two-img'>
<div class='' style='margin: 0 20px'>
<img src='img/C_adaptive_2d/homogeneous.png'/>
Homogeneous
</div>
<div class='' style='margin: 0 20px'>
<img src='img/C_adaptive_2d/learner2d.png'/>
Adaptive
</div>
</div>
</section>
<!-- <section>
<h2>Adaptive 2D</h2>
<div class='two-img'>
<div class='' style='margin: 0 20px'>
<img src='img/C_adaptive_2d/2_homogeneous.png'/>
Homogeneous
</div>
<div class='' style='margin: 0 20px'>
<img src='img/C_adaptive_2d/2_learner2d.png'/>
Adaptive
</div>
</div>
</section> -->
<section>
<h2>Loss in N dimenions</h2>
<img src='img/C_adaptive_2d/iso_homogeneous.png' width="50%"/>
</section>
<section>
<h2>Triangle area as loss</h2>
<img height="400" src='img/C_adaptive_2d/learnernd_no_curv.png'/>
</section>
<section>
<h2>Problems with length as loss</h2>
<div style='display: flex;'>
<div style='padding-top: 3em'>
<ul>
<li>Small features</li>
<li>Skipping peaks</li>
<!-- <li>Noise*</li> -->
<!-- <li>sin(1/x)</li> -->
</ul>
</div>
<div style="position:relative; width:640px; height:480px; margin:0 auto;">
<img class="fragment" width="640" height="480" src="img/D_problems_distance/bump_adaptive_32.svg" style="position:absolute;top:0;left:0;" />
<img class="fragment" width="640" height="480" src="img/D_problems_distance/sin_adaptive_200.svg" style="position:absolute;top:0;left:0;" />
<!-- <img class="fragment" width="640" height="480" src="img/D_problems_distance/noise_original.svg" style="position:absolute;top:0;left:0;" /> -->
<!-- <img class="fragment" width="640" height="480" src="img/D_problems_distance/noise_adaptive_100.svg" style="position:absolute;top:0;left:0;" /> -->
<!-- <img class="fragment" width="640" height="480" src="img/D_problems_distance/sin_over_xadaptive_250.svg" style="position:absolute;top:0;left:0;" /> -->
</div>
</div>
</section>
<section>
<h2>Line simplification</h2>
<p class='fragment'>
Visvalingam-Whyatt algorithm
</p>
<div class='fragment'>
<div style="position:relative; width:730px; height:280px; margin:0 auto;">
<img class=" " width="730" height="280" src="img/E_visvalingam_whyat/img_1.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_2.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_3.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_4.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_5.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_6.svg" style="position:absolute;top:0;left:0;background-color: white" />
<img class="fragment" width="730" height="280" src="img/E_visvalingam_whyat/img_2.svg" style="position:absolute;top:0;left:0;background-color: white" />
</div>
<small style='position: absolute; bottom: 0; right: 0; font-size: 0.3em;'>Modified images from M. Bostock (<a href='https://bost.ocks.org/mike/simplify/'>https://bost.ocks.org/mike/simplify/</a>)</small>
</div>
</section>
<section>
<h2>Loss function</h2>
<img height='300' src="img/F_curvature_loss/triangle_loss.svg"/>
<div class="">
<script type="math/tex; mode=display">
L_i = \frac12(A_{\text{left}} + A_{\text{right}}) = A_{\text{avg}}
</script>
</div>
</section>
<section>
<h2>Exploration term</h2>
<div style='display: flex'>
<div>
<script type="math/tex; mode=display">
L_i = A_{\text{avg}}
</script>
<img class='fragment' src="img/G_exploration_vs_exploit/curv_beta_0.0.svg" height=""/>
</div>
<div class='fragment' style='margin-left: 100px'>
<script type="math/tex; mode=display">
L_i = A_{\text{avg}} + { \beta \cdot \Delta x^3}
</script>
<img class='fragment' src="img/G_exploration_vs_exploit/curv_beta_0.05.svg" height="" />
</div>
</div>
<div class='fragment'>
<script type="math/tex; mode=display">
\beta = 0.05
</script>
</div>
</section>
<section>
<h2>Exploration term</h2>
<script type="math/tex; mode=display">
\text{Why}\ \Delta x^3?
</script>
<div style="position: relative; width: 600px; height: 400px;">
<img src='img/F_curvature_loss/big.svg' style='position: absolute; height: 100%;' class='fragment fade-in-then-out'/>
<img src='img/F_curvature_loss/small.svg' style='position: absolute; height: 100%;' class='fragment fade-in-then-out'/>
<img src='img/F_curvature_loss/frutselt.svg' style='position: absolute; height: 100%;' class='fragment fade-in-then-out'/>
</div>
</section>
<!-- <section>
<h2>Exploration term</h2>
<script type="math/tex; mode=display">
L_i = A_{\text{avg}} + { \beta \cdot \Delta x^3}
</script>
<img height=300 src='img/F_curvature_loss/relation_curvature_loss_2.png' />
<div class='fragment' style='font-size: 100%'>
<script type="math/tex; mode=display">
A = \frac12 f''(x) \cdot \color{red}{\Delta x^3} + \mathcal{O}(\Delta x^5)
</script>
</div>
</section> -->
<section>
<h2>Improvement on: small features</h2>
<div class='two-img'>
<div class=''>
<img src='img/D_problems_distance/bump_adaptive_32.svg'/>
Length as loss
</div>
<div class=''>
<img src='img/D_problems_distance/bump_curv_32.svg'/>
Curvature as loss
</div>
</div>
</section>
<section>
<h2>Improvement on: peaks</h2>
<div class='two-img'>
<div class=''>
<img src='img/D_problems_distance/sin_adaptive_200.svg'/>
Length as loss
</div>
<div class=''>
<img src='img/D_problems_distance/sin_curv_200.svg'/>
Curvature as loss
</div>
</div>
</section>
<!-- <section>
<h2>Improvement on: noise</h2>
<div class='two-img'>
<div class=''>
<img src='img/D_problems_distance/noise_adaptive_100.svg'/>
Length
</div>
<div class=''>
<img src='img/D_problems_distance/noise_curv_100.svg'/>
Curvature
</div>
</div>
</section> -->
<!-- <section>
<h2>Improvement on: sin(1/x)</h2>
<div class='two-img'>
<div class=''>
<img src='img/D_problems_distance/sin_over_xadaptive_250.svg'/>
Length
</div>
<div class=''>
<img src='img/D_problems_distance/sin_over_xcurv_250.svg'/>
Curvature
</div>
</div>
</section> -->
<section>
<h2>Comparison</h2>
<img src='img/H_interpolation_error/error_multi_interval_2.png'/>
<!-- <div class='fragment'>
<script type="math/tex; mode=display" >
\text{Err} = \int_a^b|f(x)-\tilde f(x)|\ \text{d}x
</script>
</div> -->
</section>
<!-- <section>
<h2>Error comparison</h2>
Error reduced 2x
<div class='two-img'>
<img height="450" src='img/H_interpolation_error/tanh10x.svg'/>
<img height="450" src='img/H_interpolation_error/err_tan10x.svg'/>
</div>
</section> -->
<section>
<h2>Error comparison</h2>
Error reduced 6x
<div class='two-img'>
<img height="450" src='img/H_interpolation_error/sin100x_times_gauss.svg'/>
<img height="450" src='img/H_interpolation_error/err_sin100x_times_gauss_correct.svg'/>
</div>
</section>
<section>
<h2>Error comparison</h2>
Error reduced 20x
<div class='two-img'>
<img height="450" src='img/H_interpolation_error/peaks.svg'/>
<img height="450" src='img/H_interpolation_error/err_peaks_correct.svg'/>
</div>
</section>
<!--
<section>
<h2>Loss function in 1 dimension</h2>
<img height='300' src="img/F_curvature_loss/triangle_loss.svg"/>
<div class="">
<script type="math/tex; mode=display">
L_i = \frac12(A_{\text{left}} + A_{\text{right}}) + \beta \cdot \Delta x^3
</script>
</div>
</section> -->
<section>
<h2>Loss function in N dimensions</h2>
<div class="two-img" >
<img class=" " style='width: 25%' src="img/J_extension_to_3d/start.png"/>
<img class="fragment" style='width: 25%' src="img/J_extension_to_3d/red_triangle_1.png"/>
<img class="fragment" style='width: 25%' src="img/J_extension_to_3d/red_triangle_2.png"/>
<img class="fragment" style='width: 25%' src="img/J_extension_to_3d/red_triangle_3.png"/>
</div>
<div class="fragment">
<script type="math/tex; mode=display">
L_s = \frac1{N+1}(\color{red}{V_1} + \color{green}{V_2} + ... + \color{blue}{V_{N+1}}) +{ \beta\cdot (\color{purple}{A_{\text{base}}}) ^ {(2+N)/N}}
</script>
</div>
</section>
<section>
<h2>Adaptive 2D</h2>
<div class='two-img' style='padding: 0 10%;'>
<div class=''>
<img src='img/C_adaptive_2d/learnernd_no_curv.png'/><br/>
Area as loss
</div>
<div class=''>
<img src='img/C_adaptive_2d/learnernd_curv.png'/><br/>
New loss
</div>
</div>
</section>
<section>
<h2>Error comparison</h2>
Error reduced 6x
<div class='two-img'>
<img height="400" style='width: initial' src='img/C_adaptive_2d/learnernd_curv.png'/>
<img height="450" src='img/H_interpolation_error/error_ring_correct.svg'/>
</div>
</section>
<!-- <section>
<h2>Error comparison</h2>
Error reduced 1.5x
<div class='two-img'>
<img height="450" src='img/H_interpolation_error/2d_sin_times_gauss.png'/>
<img height="450" src='img/H_interpolation_error/err_2d_sin_times_gauss.svg'/>
</div>
</section> -->
<section>
<div class='highlight-title'>A multi-dimensional adaptive sampling algorithm and its application to <b>Fermi surfaces</b></div>
</section>
<section>
<h2>Band structure</h2>
1D crystal
<div class='two-img'>
<img src='img/K_band_structure/crystal-1d.svg'/>
<img class='fragment' src='img/K_band_structure/band_structure_1d.svg'/>
</div>
</section>
<section>
<h2>Fermi level</h2>
<img height=300 src='img/K_band_structure/band_structure_1d_fermi_0.svg'/>
<img height=300 class=fragment src='img/K_band_structure/fermi_surface_1d.svg'/>
</section>
<section>
<h2>Fermi surface</h2>
<img src="img/L_fermi_surface/fermi_gold.jpg"/><br/>
Fermi surface of gold
<small style='position: absolute; bottom: 0; right: 0; font-size: 0.3em;'>Image from T. Li & W. Schroeder (2007)</small>
</section>
<section>
<h2>Modified adaptive sampling</h2>
<script type="math/tex; mode=display">
L_{\text{fermi}} =
\begin{cases}
\alpha\cdot L_{\text{curv}} \quad\text{simplex crosses Fermi level;}\quad\alpha>1\\
1\cdot L_{\text{curv}} \quad\text{otherwise}
\end{cases}
</script>
<img src="img/N_zoom_to_isosurf/isosurf.png" height="400" class='fragment'/>
<img src="img/N_zoom_to_isosurf/isosurf_zoom.png" height="400" class='fragment'/>
</section>
<section>
<h2>Fermi surface simple cubic</h2>
<img height=300 src="img/O_crystals_plus_fermi/cubic.png" style="margin-bottom: 100px"/>
<img height=500 src="img/O_crystals_plus_fermi/simple_cubic/iso_adaptive.png"/>
</section>
<section>
<h2>Fermi surface simple cubic</h2>
2.8x more triangles
<div class='mesh-img' style='text-align: center'>
<div>
<img src="img/O_crystals_plus_fermi/simple_cubic/mesh_homogeneous.png"/><br/>
Homogeneous
</div>
<div>
<img src="img/O_crystals_plus_fermi/simple_cubic/mesh_adaptive.png"/><br/>
Adaptive
</div>
</div>
</section>
<section>
<h2>Fermi surface FCC</h2>
<img height=300 src="img/O_crystals_plus_fermi/fcc.png" style="margin-bottom: 100px"/>
<img height=500 src="img/O_crystals_plus_fermi/fcc/iso_adaptive.png"/>
</section>
<section>
<h2>Fermi surface FCC</h2>
1.9x more triangles
<div class='mesh-img' style='text-align: center'>
<div>
<img src="img/O_crystals_plus_fermi/fcc/mesh_homogeneous.png"/><br/>
Homogeneous
</div>
<div>
<img src="img/O_crystals_plus_fermi/fcc/mesh_adaptive.png"/><br/>
Adaptive
</div>
</div>
</section>
<section>
<h2>Fermi surface hexagonal</h2>
<img height=300 src="img/O_crystals_plus_fermi/hex.png" style="margin-bottom: 100px"/>
<img height=500 src="img/O_crystals_plus_fermi/hexagon/iso_adaptive.png"/><br/>
</section>
<section>
<h2>Fermi surface Hexagonal</h2>
2.3x more triangles
<div class='mesh-img' style='text-align: center'>
<div>
<img src="img/O_crystals_plus_fermi/hexagon/mesh_homogeneous.png"/><br/>
Homogeneous
</div>
<div>
<img src="img/O_crystals_plus_fermi/hexagon/mesh_adaptive.png"/><br/>
Adaptive
</div>
</div>
</section>
<section>
<h3>Conclusion</h3>
<p class=fragment>Adaptive sampling suitable for finding Fermi surfaces</p>
<h4 class=fragment style='margin-top: 50px'>Future work</h4>
<ul>
<li class='fragment'>Higher order interpolation</li>
<li class='fragment'>Anisotropic triangulations</li>
</ul>
<img class='fragment fade-in-then-out' height=300 src="img/X_anisotropic/paper.png" style="position: absolute; left: -160px; bottom: 100px" />
<!-- <small style='position: absolute; bottom: 30px; right: 0; font-size: 0.3em;'>Image from Huang, Kamenski & Lang (2010)</small> -->
<div class='fragment' style='margin-top: 50px'>
<h2>Check out Adaptive</h2>
<a href='https://github.com/python-adaptive/adaptive'>github.com/python-adaptive/adaptive</a><br/>
</div>
</section>
<!-- <section>
<h2>Bonus slides</h2>
</section>
<section>
<h2>Error comparison</h2>
Error increased 1.14x
<div class='two-img'>
<img height="450" src='img/H_interpolation_error/2d_quadratic.png'/>
<img height="450" src='img/H_interpolation_error/err_2d_quadratic.svg'/>
</div>
</section>
<section>
<h2>Fermi surface BCC</h2>
1.3x more triangles
<div class='fermi-img' style='text-align: center'>
<img height=220 src="img/O_crystals_plus_fermi/bcc.png"/>
<div>
<img src="img/O_crystals_plus_fermi/bcc/iso_homogeneous.png"/><br/>
Homogeneous
</div>
<div>
<img src="img/O_crystals_plus_fermi/bcc/iso_adaptive.png"/><br/>
Adaptive
</div>
</div>
</section>
<section>
<h2>Exploration term</h2>
<div style="position:relative; width:500px; margin:0 auto;">
<div style='position:absolute;top:0; background-color: white;'>
<img src="img/G_exploration_vs_exploit/curv_beta_0.0.svg"/>
<script type="math/tex; mode=display">
\beta=0
</script>
</div>
<div class='fragment' style='position:absolute;top:0;left:0; background-color: white;'>
<img src="img/G_exploration_vs_exploit/curv_beta_10.svg"/>
<script type="math/tex; mode=display">
\beta=10
</script>
</div>
<div class='fragment' style='position:absolute;top:0;left:0; background-color: white;'>
<img src="img/G_exploration_vs_exploit/curv_beta_0.05.svg"/>
<script type="math/tex; mode=display">
\beta=0.05
</script>
</div>
</div>
</section>
<section>
<h2>Modified adaptive sampling</h2>
<div class='two-img' style='text-align: center'>
<div>
<img src="img/N_zoom_to_isosurf/adaptive_5.png"/><br/>
</div>
<div class='fragment'>
<img src="img/N_zoom_to_isosurf/adaptive_25.png"/><br/>
</div>
</div>
</section>
<section>
<h2>Bowyer Watson</h2>
<img height=500 src='img/Z_bowyer_watson/bowyer_watson.svg'/>
</section>
-->
</div>
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