Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.
Questions tagged [variational-analysis]
349 questions
13
votes
2 answers
Confused by Kullback-Leibler on conditional probability distributions
I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the use of the KL divergence in conditional…
Lodore66
- 235
11
votes
1 answer
If $f$ is proper, lsc, and $\frac{f(x) + f(y)}{2} = f^{**}\left(\frac{x + y}{2}\right) \implies x = y$, is $f$ necessarily convex?
Suppose $X$ is a real Hilbert Space and $f : X \to (-\infty, \infty]$ is a lower semicontinuous, proper function. Further, suppose $f$ satisfies the following, for all $x, y \in \operatorname{dom} f$: $$\frac{f(x) + f(y)}{2} = f^{**}\left(\frac{x +…
Theo Bendit
- 53,863
10
votes
2 answers
Can the geodesic equation be used to solve the Brachistochrone Problem?
Assume the initial condition is that a point mass starts at height $y_0$ . After descending to height $y < y_0$ , we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$ . Thus, the displacement element can be written as $$ds^2 = dx^2 + dy^2 = v^2…
Faraz Masroor
- 1,340
9
votes
0 answers
Looking for a rigorous treatment of the functional derivative the way it's used by physicists.
A lot of theories in physics can be derived from a variational principle: some action functional S on a space of e.g. field configurations $\phi$ is given, and the equations of the theory follow from the requirement that the action functional be…
Inzinity
- 1,956
8
votes
2 answers
Prove that the minimum of a functional doesn't exist
Prove that there is no smooth solution ho the minimization problem: $$\mathcal{L} (u)= \int_{0}^1 e^{-u'}+u^2 dx$$ Where the admissible space is $X =\{ u \in \mathcal{c}^2 [0,1] | u(0)=0, u(1)=1 \} $ UPDATE: GOAL: I am trying to define a sequence of…
Véronique
- 63
7
votes
1 answer
Does the functional have a minimizer?
Given the following functional with fixed parameters $a>0$ and $b>1$ , $$F[x;u,u']: = \int_{1}^b\left(x^3-\frac{ax}{u^2}\right)(u')^2~\mathrm{d}x$$ Suppose the space of functions satisfy $u(1) = u(b) = 1$ and $u\geq \frac{1}{x^2}$ . Does $F$ have a…
user31899
- 4,163
7
votes
3 answers
Variational Calculus - Derivation of Lagrangian Equation
While learning about the calculus of variations to look at the principle of least action, we arrive at a point where we want to minimise the following functional (or of similar form): $$S(y(x),y'(x),x) = \int_{x_{1}}^{x_{2}}…
7
votes
1 answer
Maximizing $\int_0^\infty (1+xy')^2e^y dx$ subject to $\int_0^\infty e^ydx = 1$
I'm trying to solve a calculus of variations-type problem, which requires finding the extrema of: $$\int_0^\infty (1+xy')^2e^y dx, $$ subject to the constraint that $\int_0^\infty e^ydx = 1$ . Intuitively, $e^y$ is the probability density function…
Fred Li
- 634
7
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0 answers
How to find $\operatorname{argmin}_{\int_{\Omega}\Delta u=0, u(z_1)=u_1,...,u(z_m)=u_m}{\|\Delta u\|}$?
Let $d\in\mathbb{N}$ and let $\Omega$ be a non-empty bounded arcwise connected open subset of $\mathbb{R}^d$ with regular boundary. Denote by $C^1(\bar\Omega)$ the set of differentiable continuous functions on $\Omega$ with uniformly continuous…
Bob
- 6,015
7
votes
1 answer
Show that a functional has no minimum in a given set
This is a problem from my calculus of variations class: Let $X=\{v:[0,1]\rightarrow\mathbb{R}$ of class $C^1$ , $v(0)=1$ , $\ v(1)=0\}$ , let $F:X\rightarrow\mathbb{R}$ be the functional $$F(v):=\int_0^1 (e^{v'(x)}+v^2(x))dx.$$ Integrate the…
Uskebasi
- 1,591
6
votes
0 answers
Minimizing elastic energy of shrinking balls
I am new to variational analysis and I am currently working in the following setup: We denote the $2$ -sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$ . In coordinates $(\theta,\varphi)$ , we have a metric on $S_r^2$ given by $$g_r=r^2…
Pink Panther
- 871
6
votes
2 answers
Courant (1943) and History of Finite Element Method
I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal Collocation Revisited” which has a brief section on…
L. Young
- 191
6
votes
0 answers
Does this functional satisfies the Palais-Smale condition?
Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^N$ , $\lambda\in \mathbb{R}$ be an eigenvalue of $-\Delta$ on the Sobolev space $H^1_0(\Omega)$ and $f\in L^\infty(\Omega\times\mathbb{R})$ such that $\forall x\in \Omega, t\mapsto…
Bob
- 6,015
6
votes
1 answer
Do there exist energy-minimizing immersions?
Let $M,N$ be $d$ -dimensional connected oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E_d:C^{\infty}(M,N) \to \mathbb{R}$ be the $d$ -energy, i.e. $$ E_d(f)=\int_M |df|^d \text{Vol}_M.$$ Set $E_{M,N}=\inf \{ E_d(f) \, |…
Asaf Shachar
- 26,337
6
votes
0 answers
All the symmetries of the Dirichlet energy are conformal
It seems to be "folklore" knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps. Specifically, I have found this nice proof for the following claim: Proposition: Let $M,N$ be oriented $n$-dimensional Riemannian…
Asaf Shachar
- 26,337