# 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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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 ... 1answer 55 views ### 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 ... 2answers 93 views ### Numerical compuation of functional derivative I was wondering if it is possible to numerically compute $$\frac{\delta L}{\delta f}$$ where$L$is a functional of the function$f$known only in a discrete number of point by numerical computation ... 1answer 25 views ### Variational Formulation - inhomogeneous I'm not sure how to get started with the following. Consider,$- \Delta u=f$in$\Omegau=u_o$on$\Gamma$I need to find a$u \in V(u_o)$such that$a(u,v)=(f,v)\forall v\in H^1_o$where$...
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If $E$ is a Banach space, denote with $I$ the identity map of $E$, denote the space of continuous functions from $E$ to $E$ with $\operatorname{C}(E,E)$ and denote the space of homeomorphism from $E$ ...
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### Minimizing a nonlinear functional over finite elements

I am looking for a reference (an url, a book, or a paper) that could help me in discretization and minimization of the following cost function $J\left(\omega\right)$, over 3D tetrahedrons (finite ...
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### Two-way anova or One-way anova

I have to tested multiple reinforcement learning systems and record their performances [test-accuracy] based on two categories of [system-type] and [communication-level]. The system-type can be either ...
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### Reparameterization Trick in VAE

I was reading on variational auto-encoders https://wiseodd.github.io/techblog/2016/12/10/variational-autoencoder/ and am unable to understand how the function below is generated. Based on my limited ...
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### Terminology / reference request: Calculus of variations with constraints on the family of curves

I am interested in variational problems of the following form: $$\max J(y) \quad\text{such that}\quad y\in C,$$ where $J(y)=\int_0^b f(x,y,y')\,dx$ is a functional and $C$ is a specified family of ...
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### Lagrange Multipliers for Bingham Flow

I want to prove existence and uniqueness of Lagrange multipliers of the following variational inequality (which is derived from Bingham fluids) \begin{equation*} \int_\Omega \nabla u \cdot \nabla (v-u)...
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### Permutation and combinations using chairs?

After reading and watching a lot about permutation, combination and variation i still don't understand them fully. So i have two questions: How many ways are there to position 5 people on 10 chairs? ...
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### Variational Equation in Perturbation Theory Chapter of Arnold *Geometric Methods in ODE*

In the preface to Chapter 4 Perturbation Theory" in Arnold's book Geometric Methods in the Theory of Ordinary Differential Equations available for preview here https://www.springer.com/gp/book/...