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tidy up research
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@Hadrien-Montanelli
Hadrien-Montanelli committed Jan 17, 2024
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Expand Up @@ -102,7 +102,7 @@ <h1>Publications</h1>

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<p>Garnier, Haddar & M., <i><b>The linear sampling method for small random scatterers</b></i>, in prep.</p>
<p>Garnier, Haddar & M., <i><b>The linear sampling method with data generated by small random scatterers</b></i>, in prep.</p>

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29 changes: 9 additions & 20 deletions research.html
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Expand Up @@ -119,7 +119,7 @@ <h4>Background</h4>

<h4>Results</h4>

<p>In 2023, I presented an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of my sampling method is based on the Helmholtz&ndash;Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which I introduced and analyzed. The resulting method gives comparable results to the standard linear sampling method with deterministic sources. In 2024, I extended it to the case of small random scatterers&mdash;a seemingly simple yet powerful model of a random medium that allowed me to apply the linear sampling method in a novel manner.</p>
<p>In 2023, I presented an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of my sampling method is based on the Helmholtz&ndash;Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which I introduced and analyzed. The resulting method gives comparable results to the standard linear sampling method with deterministic sources.</p> <!--In 2024, I extended it to the case of small random scatterers&mdash;a seemingly simple yet powerful model of a random medium that allowed me to apply the linear sampling method in a novel manner.</p>-->

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Expand Down Expand Up @@ -156,12 +156,12 @@ <h4>Software</h4>

<h4>Papers</h4>

<p>2024 &nbsp; <i><b>The linear sampling method for small random scatterers</b></i></p>
<!--<p>2024 &nbsp; <i><b>The linear sampling method with data generated by small random scatterers</b></i></p>-->
<p>2023 &nbsp; <i><b>The linear sampling method for random sources</b></i>&nbsp;&nbsp;<a href="https://epubs.siam.org/doi/full/10.1137/22M1531336"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/2210.15560.pdf"><mark>arXiv</mark></a></p>

<h4>Blog posts</h4>

<p>2024 &nbsp; <a href="2023-12-22.html">The linear sampling method for small random scatterers</a></p>
<!--<p>2024 &nbsp; <a href="2023-12-22.html">The linear sampling method with data generated by small random scatterers</a></p>-->
<p>2022 &nbsp; <a href="2022-11-04.html">The linear sampling method for random sources</a></p>

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Expand Down Expand Up @@ -254,9 +254,7 @@ <h4>Software</h4>
<h4>Papers</h4>

<p>2021 &nbsp; <i><b>Deep ReLU networks overcome the curse of dimensionality for generalized bandlimited functions</b></i>&nbsp;&nbsp;<a href="https://global-sci.org/intro/article_detail/jcm/19912.html"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1903.00735.pdf"><mark>arXiv</mark></a></p>

<p>2020 &nbsp; <i><b>Error bounds for deep ReLU networks using the Kolmogorov&ndash;Arnold superposition theorem</b></i>&nbsp;&nbsp;<a href="https://linkinghub.elsevier.com/retrieve/pii/S0893608019304058"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1906.11945.pdf"><mark>arXiv</mark></a></p>

<p>2019 &nbsp; <i><b>New error bounds for deep ReLU networks using sparse grids</b></i>&nbsp;&nbsp;<a href="https://epubs.siam.org/doi/abs/10.1137/18M1189336"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1712.08688.pdf"><mark>arXiv</mark></a></p>

<h4>Blog posts</h4>
Expand All @@ -275,7 +273,7 @@ <h4>Background</h4>

<p>These packages are particularly relevant to the astrophysics, computational fluid dynamics, and biology communities, where simple geometries such as the sphere are prevalent. For example, nonlinear advection equations on the sphere, such as the shallow water equations, are of significant importance in atmospheric numerical modeling, while reaction-diffusion equations in a spherical shell are widely used as a model for convection patterns within the Earth's mantle, as well as for the modeling of morphogenesis in embryos. On top of these standard local differential equations, their nonlocal integral analogs are becoming more and more popular to model a wide range of phenomena in these communities.</p>

<h4>Results (Ph.D. at Oxford)</h4>
<h4>Results</h4>

<p>My Ph.D. introduced new numerical methods for the simulation of periodic physical phenomena.</p>

Expand Down Expand Up @@ -322,9 +320,7 @@ <h4>Results (Ph.D. at Oxford)</h4>
<img src="/blog/pdesphere.jpg" class="img-responsive">
</center>

<h4>Results (Postdoc at Columbia)</h4>

<p>With discretizations of operators in hand, I explored potential applications in biology. I used the fast algorithms I developed to investigate pattern formation and explain symmetry breaking on the sphere.</p>
<p>With discretizations of operators in hand, I explored potential applications in biology during a postdoc at Columbia. I used the fast algorithms I developed to investigate pattern formation and explain symmetry breaking on the sphere.</p>

<center>
<img src="/blog/sphericalcaps3.jpg" class="img-responsive">
Expand All @@ -341,7 +337,7 @@ <h4>Results (Postdoc at Columbia)</h4>
<img src="/blog/nonlocalpdesphere.jpg" class="img-responsive">
</center>

<p>Several years later, in 2024, I introduced a nonlocal vector calculus on the sphere based on weakly singular integral operators. When employing scalar and vector spherical harmonics as bases, I showed that these nonlocal operators exhibit diagonal behavior. Through analysis, I also established strong convergence to the operators of local vector calculus as the interaction range approaches zero. This work is consistent with my previous work on nonlocal diffusion operators. This extends to the sphere, a prototype of a manifold, the nonlocal calculus for Euclidean domains.</p>
<!--<p>Several years later, in 2024, I introduced a nonlocal vector calculus on the sphere based on weakly singular integral operators. When employing scalar and vector spherical harmonics as bases, I showed that these nonlocal operators exhibit diagonal behavior. Through analysis, I also established strong convergence to the operators of local vector calculus as the interaction range approaches zero. This work is consistent with my previous work on nonlocal diffusion operators. This extends to the sphere, a prototype of a manifold, the nonlocal calculus for Euclidean domains.</p>-->

<h4>Collaborators</h4>

Expand All @@ -355,25 +351,18 @@ <h4>Software</h4>

<h4>Papers</h4>

<p>2024 &nbsp; <i><b>Nonlocal vector calculus on the sphere using vector spherical harmonics</b></i></p>

<!--<p>2024 &nbsp; <i><b>Nonlocal vector calculus on the sphere using vector spherical harmonics</b></i></p>-->
<p>2020 &nbsp; <i><b>Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators</b></i>&nbsp;&nbsp;<a href="https://www.sciencedirect.com/science/article/abs/pii/S037847542030210X"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1604.08900.pdf"><mark>arXiv</mark></a></p>

<p>2018 &nbsp; <i><b>Spherical caps in cell polarization</b></i>&nbsp;&nbsp;<a href="https://www.cell.com/biophysj/fulltext/S0006-3495(18)30672-6"><mark>doi</mark></a></p>

<p>2018 &nbsp; <i><b>A spectral method for nonlocal diffusion operators on the sphere</b></i>&nbsp;&nbsp;<a href="https://www.sciencedirect.com/science/article/pii/S0021999118304029"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1801.04902.pdf"><mark>arXiv</mark></a></p>

<p>2018 &nbsp; <i><b>A spectral method for nonlocal diffusion operators on the sphere</b></i>&nbsp;&nbsp;<a href="https://www.sciencedirect.com/science/article/pii/S0021999118304029"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1801.04902.pdf"><mark>arXiv</mark></a></p>
<p>2018 &nbsp; <i><b>Fourth-order time-stepping for stiff PDEs on the sphere</b></i>&nbsp;&nbsp;<a href="https://epubs.siam.org/doi/abs/10.1137/17M1112728"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1701.06030.pdf"><mark>arXiv</mark></a></p>

<p>2016 &nbsp; <i><b>Computing hyperbolic choreographies</b></i>&nbsp;&nbsp;<a href="https://link.springer.com/article/10.1134/S1560354716050038"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1606.01850.pdf"><mark>arXiv</mark></a></p>

<p>2016 &nbsp; <i><b>Computing planar and spherical choreographies</b></i>&nbsp;&nbsp;<a href="https://epubs.siam.org/doi/abs/10.1137/15M1024652"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1505.04848.pdf"><mark>arXiv</mark></a></p>

<p>2015 &nbsp; <i><b>Extension of Chebfun to periodic functions</b></i>&nbsp;&nbsp;<a href="https://epubs.siam.org/doi/10.1137/141001007"><mark>doi</mark></a> <a href="https://arxiv.org/pdf/1511.00166.pdf"><mark>arXiv</mark></a></p>

<h4>Blog posts</h4>

<p>2024 &nbsp; <a href="2023-12-15.html">Nonlocal vector calculus on the sphere</a></p>
<!--<p>2024 &nbsp; <a href="2023-12-15.html">Nonlocal vector calculus on the sphere</a></p>-->
<p>2020 &nbsp; <a href="2020-05-19.html">Exponential integrators for stiff PDEs</a></p>
<p>2018 &nbsp; <a href="2018-12-05.html">Computer-assisted proofs for PDEs</a></p>
<p>2018 &nbsp; <a href="2018-02-27.html">Spherical caps in cell polarization</a></p>
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Expand Up @@ -102,10 +102,10 @@ <h3>IP Paris</h3>

<h4>2023&ndash;2024</h4>

<p>Optimization & control (École Polytechnique)</p>
<p>Optimization & Control (École Polytechnique)</p>
<p>Statistics (ENSTA Paris)</p>
<p>Probability (ENSTA Paris)</p>
<p>Modelling & simulations (École Polytechnique)</p>
<p>Modelling & Simulations (École Polytechnique)</p>
<p>Dynamical systems (ENSTA Paris)</p>

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