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Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) an …

Let $\tau>0$ and $d\in\mathbb N$. Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\r …

Let $\tau>0$; $d\in\mathbb N$; $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\mat …

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta …

I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and Geo …

I'm trying to understand the intuition behind shape optimization using Hadamard's method. Please consider the following simple example: Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\math …

Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $$\mathcal A:=\{\Omega\subseteq\mathbb R^d:\Omega\text{ is bounded and open},\overline\Omega\subseteq U\text{ and }\partial\Omega\text{ is of clas …

Let $d\in\mathbb N$; $U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$ $E:=\bigcup_{\Omega\i …

I've started reading about shape optimization. Most of the concepts I've encountered so far (such as the shape derivatives of domain and boundary integrals and the corresponding) seem to be complex, b …

Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to …

Let $(\Omega,\mathcal E,\mu)$ be a probability space, $k\in\mathbb N$, $$W:=\left\{w:E\to[0,\infty)^k:\sum_{i=1}^kw_i=1\;\mu\text{-almost surely}\right\},$$ $G$ be a finite nonempty set and $a^{(g)}:E …

I'm trying to solve a saddle point problem of the following form: Let $(E,\mathcal E,\lambda)$ be a measure space; $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$ $W$ be …

Let $H$ be a $\mathbb R$-Hilbert space and $F:H^2\to\mathbb R$. Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$? Since the question is rather abstract, feel free to impose a …

I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a fin …

Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find. Let $(E,\mathcal E,\lam …