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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Optimal non-negative resampling kernel

Sampling theory says the best kernel for resampling is $sinc(x)$ which is box function in frequency domain. Which kernel is optimal under constraint that the kernel is non-negative everywhere? More ...
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Variational principal question regarding functions that have a minimum at the origin under a restriction.

I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module. I looked at the function in the hint and noticed that for some ...
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Variational calculus … Prove that a linear functional $\varphi [h]$ cannot have an extremum unless $\varphi [h] \equiv 0$

From definitions in the book calculus of variations - Gelfand and fomin http://web.cs.iastate.edu/~cs577/handouts/variations.pdf I try prove that a linear functional $\varphi[h]$ cannot have an ...
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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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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*} ...
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Finite Element formulation of mixed BVP of Variational Problem

Suppose we are given the followin where $f$,$u$, $g$ are given functions: $-\Delta u = f$ in $\Omega$ $u=u_o$ on $\Gamma_1$ $\frac{du}{dn}=g$ on $\Gamma_2$ So in order for me to form the ...
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How to simplify the Euler-Lagrange equation of Brachistochrone in this way?

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 ...