Add links to research topics

about.html

@@ -138,7 +138,7 @@ <h3>Education</h3>
 
                     <h3>Awards</h3>
 
-                    <p>2020 &nbsp; Fatured in SIAM Journals Top Three Most Cited Papers <a href="https://sinews.siam.org/Details-Page/siam-journals-top-three-most-cited-papers">November 2020 list</a></p>
+                    <p>2020 &nbsp; Featured in SIAM Journals Top Three Most Cited Papers <a href="https://sinews.siam.org/Details-Page/siam-journals-top-three-most-cited-papers">November 2020 list</a></p>
 
                     <p>2017 &nbsp; <a href="https://siamukie.wordpress.com/2017/05/22/sc-oxford/">Best talk</a> at the 9th Oxford University SIAM Student Chapter Conference</p>
 

research.html

@@ -100,9 +100,18 @@ <h1>Research</h1>
 
                     <h3>Research interests</h3>
 
-                    <p>My general research interests lie in <a href="https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations">numerical methods for PDEs</a> and <a href="https://en.wikipedia.org/wiki/Integral_equation">integral equations</a>, <a href="https://en.wikipedia.org/wiki/Inverse_problem">inverse problems</a>, <a href="https://en.wikipedia.org/wiki/Approximation_theory">approximation theory</a>, and <a href="https://en.wikipedia.org/wiki/Machine_learning">machine learning</a>. More generally, my background is in <a href="https://en.wikipedia.org/wiki/Numerical_analysis">numerical analysis</a> and <a href="https://en.wikipedia.org/wiki/Computational_science">scientific computing</a>.</p>
+                    <p>My general research interests lie in <a href="https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations">numerical methods for PDEs</a> and <a href="https://en.wikipedia.org/wiki/Integral_equation">integral equations</a>, <a href="https://en.wikipedia.org/wiki/Inverse_problem">inverse problems</a>, <a href="https://en.wikipedia.org/wiki/Approximation_theory">approximation theory</a>, and <a href="https://en.wikipedia.org/wiki/Machine_learning">machine learning</a>. More generally, my background is in <a href="https://en.wikipedia.org/wiki/Numerical_analysis">numerical analysis</a> and <a href="https://en.wikipedia.org/wiki/Computational_science">scientific computing</a>. I highlight, here, my contributions to the following four topics:</p>
 
-                    <h3>Inverse acousting scattering</h3>
+                    <ul>
+                    <li><a href="#section1">Inverse acoustic scattering;</a></li>
+                    <li><a href="#section2">Boundary element methods;</a></li>
+                    <li><a href="#section3">Neural network approximation theory;</a></li>
+                    <li><a href="#section4">Spectral methods.</a></li>
+                    </ul>
+
+                    <hr class="solid">
+
+                    <h3 id="section1">Inverse acoustic scattering</h3>
 
                     <h4>Background</h4>
 
@@ -155,7 +164,7 @@ <h4>Blog posts</h4>
 
                     <hr class="solid">
 
-                    <h3>Boundary element methods</h3>
+                    <h3 id="section2">Boundary element methods</h3>
 
                     <h4>Background</h4>
 
@@ -165,7 +174,6 @@ <h4>Background</h4>
                     $$
                     based on the single-layer potential; \(G\) is the Green's function of the Helmholtz equation in 3D. Once the integral equation is solved for \(\varphi\), the solution \(u\) may be represented by the left-hand side for all \(\boldsymbol{x}\in\mathbb{R}^3\setminus\overline{D}\). From a numerical point of view, such integral equations are particularly challenging for several reasons. First, when \(\boldsymbol{x}\) approaches \(\boldsymbol{y}\), the integral becomes singular and standard quadrature schemes fail to be accurate&mdash;analytic integration or carefully-derived quadrature formulas are required. Second, the resulting linear systems after discretization are dense. For large wavenumbers \(k\), only iterative methods can be used to solve them (with the help of specialized techniques to accelerate the matrix-vector products such as hierarchical matrices). In this respect, the use of high-order schemes may be helpful in enlarging the interval of feasible wavenumbers.</p>
 
-
                     <h4>Results</h4>
 
                     <p>In 2022, I presented algorithms for computing weakly singular and near-singular integrals arising with high-order boundary elements. I utilized the continuation approach in a novel way that simplifies implementation, decreases computational costs, and carefully handles both singular and near-singular integrals&mdash;existing methods are designed for singular integrals but are not accurate in the near-singular case. More generally, the paper utilizes a variety of numerical tools (e.g., Newton's method, transplanted Gauss quadrature). The resulting method is applicable to any 3D object, including resonant cavities and thin layers. These algorithms were extended to the strongly singular case in 2023. Implementations in C++ and Python can be found in the <a href="https://gitlab.labos.polytechnique.fr/leprojetcastor/fembem">fembem</a> and <a href="https://github.com/Hadrien-Montanelli/singintpy">singintpy</a> packages.</p>
@@ -209,7 +217,7 @@ <h4>Blog posts</h4>
 
                     <hr class="solid">
 
-                    <h3>Neural network approximation theory</h3>
+                    <h3 id="section3">Neural network approximation theory</h3>
 
                     <h4>Background</h4>
 
@@ -257,7 +265,7 @@ <h4>Blog posts</h4>
 
                     <hr class="solid">
 
-                    <h3>Spectral methods</h3>
+                    <h3 id="section4">Spectral methods</h3>
 
                     <h4>Background</h4>