“There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man’s life to know them the little new that each man gets from life is very costly and the only heritage he has to leave.” - Ernest Hemingway (More…)
News #
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Posts #
Paper Reading - Optimal coefficients for elliptic PDEs (2512.08431)
This paper gives a clear, fairly complete picture of how to optimally choose the coefficient $a(x)$ (think “material quality”) in an elliptic PDE, with compliance as the main model and then a general optimal control formulation. Problem Setup # Considering the boundary value problem: $$ -{\rm div}(a(x)\nabla u) = f \quad\text{in } \Omega,\qquad u=0 \text{ on } \partial\Omega, $$ where $\Omega$ is a bounded domain, $f$ is a given load, and $a(x)$ is the design variable.
Paper Reading - Optimal sources for elliptic PDEs (2509.01521)
Introduction # The authors study how to “best choose” a source term $f$ in a Poisson-type equation $$ -\Delta u = f \quad\quad\text{in }\Omega,\quad u = 0\text{ on }\partial\Omega, $$ so that a given performance measure (a cost functional) is optimized. The twist is that the source itself is the control, and it can be subject to various constraints (size, bounds, sign, etc.). This makes the problem sit at the intersection of optimal control, shape optimization, and regularity theory.
Restriction and extension
Considering a smooth compact hyper-surface $\mathcal{S}$ in $\mathbb{R}^d$ with surface measure $d\sigma$. Given $f \in L^1(\mathbb{R}^d)$, the Fourier transform defined as follow: $$ \begin{equation} \hat{f}(x) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ which by Riemann-Lebesgue is a bounded, continuous function vanishing at infinity. Since $\hat{f}$ is continuous on $\mathbb{R}^d$, by the Rimann-Lesbegue lemma its restriction to the compact hyper-surface $S \subset \mathbb{R}^d$ is is well-defined pointwise. Specifically, the restriction $\hat{f}\mid_{S}: S \rightarrow \mathbb{C}$ is the continuous function given by $$ \begin{equation} \hat{f}\mid_{S}(\sigma) = \hat{f}(\sigma) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ for each $\sigma \in S$. This is bounded (as $\hat{f}$ is bounded) and can be integrated against the surface measure $d\sigma$ on $S$.