# 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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### 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 ...
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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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### 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/...
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### Lagrange multiplier for PDE

I am dealing with fluid-structure interaction problem and I have a pde subjected to a constraint . I must use Lagrange multipliers but I don’t know how. Please, any one give a simple example for how ...
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### Beam equation is neccesary condition for minimum of specific functional.

The beam equation is given by: $Mu^{(4)}(x)+Nu''(x) = f(x) \; x \in [0,L]$ This is supposed to represent the bending of a beam of length $L$ when a sectional mass density $f(x) \ge 0$ acts on the ...
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### Functional form of an Ordinary Differential Equation

I was learning about Variational Method to solve the differential equation for the following differential equation: $$\frac{d^2u}{dx^2}-u=-x, 0<x<1$$ $$u(0)=0, u(1)=0$$ However, the textbook I ...
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### Similarity between the differential of functionals and functions

In the book Calculus of variations by I. M. Gelfand, the differential of a functional is defined in the following way, and its uniqueness is proven: However, it seems strange to me that the ...
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### An example for a stable harmonic map which is not a local minimizer

I am looking for an example of a harmonic mapping between compact Riemannian manifolds, which is stable but not a local minimizer of the Dirichlet energy. A harmonic map $f:M \to N$ is said to be ...
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### Solving PDEs using the Ritz method on variational calculus problem (Student questions)

I'm reading the book "Conduction Heat Transfer" by Vedat S. Arpaci. I'm currently at chapter 8 (I didn't read the rest of the book, though), which talks about the Variational Formulation - Solution by ...
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### Minimal Surface and Mean Curvature

In the frame of a course on instabilities, I am trying to prove that the soap film between two rings has the form of a catenoid. Since pressure is equal on either side of the film, we expect to have a ...
### Wondering how to get this analytical solution of $\text{E}\big(\log(f)\big)$, $f\sim$ Normal Distribution
I'm trying to arrive at the weak form from the following strong form for the Anisotropic Poisson problem: \begin{align*}-\nabla\cdot(\textbf{A}\nabla u) &= f \hspace{.5cm}\textrm{in }\Omega \\ ...