# Questions tagged [variational-analysis]

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.

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### Explicit Solution to Euler Lagrange Equation of Shortest Distance between two Points

So if $f(t)=(x(t),y(t))$ with $f(0)=a$ and $f(1)=b$, I should minimize $L(f)=\int_0^1{|\dot{f(t)}|}dt$. I get a jumble of equations when solving the Euler Lagrange equations with respect to $t$. Would ...
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### Fundamental Lemma of Variational Calculus for Closed Surface Integrals

The fundamental lemma of the calculus of variations says that if the integral of a function times a test function over an open region is equal to zero for all test functions that vanish at boundary of ...
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### Legendre Transforms

I want to show that the Legendre Transform of a function $g(x) = kf(x-x_0) + m$ for $m$, $k$ constants is equal to $g^*(p) = kf\left(\frac{p}{k}\right)+p \cdot x_0+m$ where $f^*$ is the Legendre ...
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### Graphical Convergence

Suppose that $U$ is a dense subset of $\mathbb{R}^d$, what is an example of a lower semi-continuous map $f:\mathbb{R}^d\rightarrow \mathbb{R}$, for which $f|_U$ converges graphically to $f$? Note: ...
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### 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 ...
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In Elements of Information Theory, I can't figure out how the functional derivative $\frac{\delta J}{\delta q(\hat{x}|x)}$ for J(q) = \sum_x \sum_{\hat{x}} p(x)q(\hat{x}|x)\log{\frac{q(\hat{x}|x)}{... 0answers 51 views ### Error is the Projection on the Noise - Does the intuition carry over from least squares? My question concerns a generalization of the following: Background It is a well known that the least squares problem (on some Hilbert space) \begin{align*} \|AX-Y \|_2^2 \to \min_{A}! \end{align*} ... 0answers 26 views ### Finite Element formulation of mixed BVP of Variational Problem Suppose we are given the followin wheref$,$u$,$g$are given functions:$-\Delta u = f$in$\Omegau=u_o$on$\Gamma_1\frac{du}{dn}=g$on$\Gamma_2\$ So in order for me to form the ...
I already know that in the Brachistochrone problem, we have Euler-Lagrange equation: $$\frac{1}{2y}\sqrt{\frac{1+y'^2}{y}}+\frac{d}{dx}[\frac{y'}{\sqrt{y(1+y'^2)}}]=0$$ To solve this equation, we ...