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@Hadrien-Montanelli
Hadrien-Montanelli committed Feb 2, 2024
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2 changes: 1 addition & 1 deletion 2022-11-04.html
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Expand Up @@ -239,7 +239,7 @@ <h2>Numerical experiments</h2>
<hr>
<h4>Blog posts about inverse scattering</h4>

<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for data generated by small random scatterers</a></p>
<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for 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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14 changes: 9 additions & 5 deletions 2024-02-04.html
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<title>The linear sampling method for data generated by small random scatterers</title>
<title>The linear sampling method for small random scatterers</title>
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<h1>The linear sampling method for data generated by small random scatterers</h1>
<h1>The linear sampling method for small random scatterers</h1>
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Expand Down Expand Up @@ -129,6 +129,8 @@ <h2>Introduction</h2>

<h2>The modified Helmholtz&ndash;Kirchoff identity</h2>

<p>In progress.</p>

$$
\begin{align}
u^s(\boldsymbol{x},\boldsymbol{x}') - \overline{u^s(\boldsymbol{x},\boldsymbol{x}')} \approx 2ik\sigma_\epsilon\int_{\Sigma_\epsilon}\overline{\tilde{v}_\epsilon(\boldsymbol{x},\boldsymbol{y}_\epsilon,\boldsymbol{z}_\epsilon)} \tilde{v}_\epsilon(\boldsymbol{x}',\boldsymbol{y}_\epsilon,\boldsymbol{z}_\epsilon)dS(\boldsymbol{y}_\epsilon) - [\phi(\boldsymbol{x},\boldsymbol{x}') - \overline{\phi(\boldsymbol{x},\boldsymbol{x}')}]
Expand All @@ -143,20 +145,22 @@ <h2>The modified Helmholtz&ndash;Kirchoff identity</h2>

<h2>Numerical experiments</h2>

<p>We consider the scattering of points sources by a kite of size \(\lambda/2\) centered at \(2\lambda + 2\lambda i\) for \(k=2\pi\) (wavelength \(\lambda =1\)). We compare the results obtained for the near-field matrix \(N\), the imaginary near-field matrix \(I\), and the cross-correlation matrix \(C\). For \(N\) and \(I\), we take \(J=80\) equispaced co-located sources and receivers on the circle of radius \(5\lambda\),
<p>In progress.</p>

<!--<p>We consider the scattering of points sources by a kite of size \(\lambda/2\) centered at \(2\lambda + 2\lambda i\) for \(k=2\pi\) (wavelength \(\lambda =1\)). We compare the results obtained for the near-field matrix \(N\), the imaginary near-field matrix \(I\), and the cross-correlation matrix \(C\). For \(N\) and \(I\), we take \(J=80\) equispaced co-located sources and receivers on the circle of radius \(5\lambda\),
$$
\boldsymbol{x}_j = 5\lambda e^{i\theta_j}, \quad \theta_j = \frac{2\pi}{J}(j - 1).
$$
For the matrix \(C\), the \(L=80\) random sources are located on the circle of radius \(50\lambda\),
$$
\boldsymbol{z}_\ell = 50\lambda e^{i\theta_\ell}, \quad \theta_\ell = \frac{2\pi}{L}(\ell -1+ \beta_\ell),
$$
where \(\beta_\ell\) is drawn from the uniform distribution on \([0,\beta]\) with \(\beta=0.1\), and we measure at the points \(\boldsymbol{x}_j\). To simulate noisy measurements, we add some white noise of amplitude \(5\times 10^{-2}\) to each matrix; we call the noisy matrices \(N_\delta\), \(I_\delta\), and \(C_\delta\). Finally, we probe the medium on a \(100\times100\) uniform grid on \([-6\lambda,6\lambda]\times[-6\lambda,6\lambda]\). Here are the results.</p>
where \(\beta_\ell\) is drawn from the uniform distribution on \([0,\beta]\) with \(\beta=0.1\), and we measure at the points \(\boldsymbol{x}_j\). To simulate noisy measurements, we add some white noise of amplitude \(5\times 10^{-2}\) to each matrix; we call the noisy matrices \(N_\delta\), \(I_\delta\), and \(C_\delta\). Finally, we probe the medium on a \(100\times100\) uniform grid on \([-6\lambda,6\lambda]\times[-6\lambda,6\lambda]\). Here are the results.</p>-->

<hr>
<h4>Blog posts about inverse scattering</h4>

<p>2024 &nbsp; <a href="2024-01-28.html">The linear sampling method for data generated by small random scatterers</a></p>
<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for 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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2 changes: 1 addition & 1 deletion _site/index.html
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Expand Up @@ -102,7 +102,7 @@ <h2>and welcome to my website</h2>

<div class="blog-post-thumb">
<div class="blog-post-title">
<h3><a href="2024-02-04.html">The linear sampling method for data generated by small random scatterers</a></h3>
<h3><a href="2024-02-04.html">The linear sampling method for small random scatterers</a></h3>
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<span><i class="fa fa-date"></i>February 4, 2023</span>
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2 changes: 1 addition & 1 deletion blog.html
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Expand Up @@ -98,7 +98,7 @@ <h1>Blog</h1>

<div class="col-md-offset-1 col-md-10 col-sm-12">

<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for data generated by small random scatterers</a></p>
<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for small random scatterers</a></p>
<p>2023 &nbsp; <a href="2023-10-02.html">Strongly singular integrals over curved elements</a></p>
<p>2023 &nbsp; <a href="2023-02-12.html">Finite-boundary element coupling in MATLAB</a></p>
<p>2022 &nbsp; <a href="2022-12-30.html">Image matting with examples in Python</a></p>
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2 changes: 1 addition & 1 deletion index.html
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Expand Up @@ -102,7 +102,7 @@ <h2>and welcome to my website</h2>

<div class="blog-post-thumb">
<div class="blog-post-title">
<h3><a href="2024-02-04.html">The linear sampling method for data generated by small random scatterers</a></h3>
<h3><a href="2024-02-04.html">The linear sampling method for small random scatterers</a></h3>
</div>
<div class="blog-post-format">
<span><i class="fa fa-date"></i>February 4, 2023</span>
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2 changes: 1 addition & 1 deletion research.html
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Expand Up @@ -161,7 +161,7 @@ <h4>Papers</h4>

<h4>Blog posts</h4>

<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for data generated by small random scatterers</a></p>
<p>2024 &nbsp; <a href="2024-02-04.html">The linear sampling method for 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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