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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About the equivalence of two definitions of topological linking

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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28 views

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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1answer
22 views

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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57 views

Simple exercise with differentials

Given the diferentials $$ \begin{equation} d'Q=K(x-x')d'x \end{equation} $$ $$ \begin{equation} d'Q'=K(x'-x)d'x' \end{equation} $$ where $K$ is a constant, I need to show that $$ \begin{equation} \...
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1answer
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Why minimize $ \Vert u_1 + u \Vert_1 $ in this Finite Element Analysis variational problem?

$\textbf{The problem reads:}$ Let $\Omega$ be bounded with $\Gamma: = \partial \Omega$ and let $g:\Gamma \rightarrow \mathbb{R} $ be a given function. Find the function $u\in H^1(\Omega)$ with ...
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33 views

Functional derivative of gradient of cross product

I need to find functional derivative of the following function with respect to $\eta$ $F = \int[(n\times \nabla\eta) + (m\times \nabla\eta)]^2dr$ Where, n and m are vectors and constant, $\eta$ is a ...
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1answer
42 views

Properties of the $\frac{1}{2}x^2$ function relevant for variational analysis in mathematical physics

There is a theorem in mathematical physics that, by the looks of it, hinges on the nature of the function $f(x)=\frac{1}{2}x^2$ This question is about examining properties of the $f(x)=\frac{1}{2}x^2$ ...
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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 $x, y \in \operatorname{dom} f$: ...
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Differentiate an energy containing integral in a region to derive curve evolution

I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/ Here we want to minimize the energy where $p$ is ...
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78 views

Variational formulation of a parabolic equation

I have the following problem $$ \dfrac{\partial u}{\partial t} - \Delta u + F(u)= f(x,t) $$ where $u \in L^2([0,T[;H^1(\mathbb{R}^n))$ with $\partial_t u \in L^2([0,T[;(H^1(\mathbb{R}^n))')$ I want ...
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76 views

Understanding Arnold's definition of “differentiable”

I'm looking at Arnold's Mathematical Methods of Classical Mechanics at the beginning of chapter 3 p 55 which defines when a functional is differentiable. Slightly paraphrasing and skipping a couple ...
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34 views

Variational methods

What happens if I use the free parameters in variational methods in a non-linear manner? I have this question in front of me and I am not sure how to answer it.
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minimization problem of integral functional for a given function

Let $\Omega \subset \mathbb{R}^{n}$ ($n\geq 2)$ be a bounded domain satisfying an exterior ball condition for any $\xi_{0} \in \partial\Omega$ and $\forall x \in \overline{\Omega},\, u_{\xi_{0}}(x) = \...
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94 views

$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

$f,g$ are differentiable non-negative functions on $[a,b]$ with $ \int_{a}^{b}\sqrt{f(t)}dt\geq\int_{a}^{b}\sqrt{g(t)}dt $. So do we have that $\int_{a}^{b}f(t)dt\geq\int_{a}^{b}g(t)dt$ ? Does ...
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54 views

Best path in vectorfield

Consider a vectorfield on the surface of a sphere, which is a (continuously differentiable) function representing the wind on earth. Now, let's say I want to calculate the path of an airplane such ...
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1answer
48 views

Mean field approximation

I am a Japanese student studying machine learning and stuff like that. When I have studied Mean Field Approximation HERE, I got question. In that post, equation (14) is as follows: $$ L[q_1,...,q_m] -...
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12 views

Bounded-length path that maximizes work in non-conservative field

Given a "sufficiently well behaved", but not necessarily conservative, vector field ${\bf E}$ in $\mathbb{R}^3$ defined in a bounded domain $\Omega$, what is the maximum value of $$ \int_C {\bf E} \...
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Finding a specific set of functions. (Please, read the description)

The task is to find the sequence of functions $\lbrace f_n \rbrace _{n\geqslant1} \subset \mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ (where $\mathbb{K}\mathbb{C}^{1} \left[ 0;1 \right]$ is a set of ...
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60 views

Equivalent variational formula for eigenvalues

First we define Rayleigh quotient as $$R(u)=\frac{\int_M|\nabla u|^2 dV_g}{\int_M |u|^2d V_g}$$ We can deduce that the first nonzero eigenvalue of Laplacian operator is $$\lambda_1=\inf\{R(u): \int_M ...
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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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286 views

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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241 views

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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2answers
134 views

How to get differential equation of Fermat Principle

I've found this file https://www.colorado.edu/physics/phys3210/phys3210_sp15/Notes/NDSolve_Fermat.pdf And it says path of beam from Fermat's Principle can solved from $$\dfrac{n'}{n}=\dfrac{y''^2}{1+...
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1answer
85 views

Why is a weak variational inequality *equivalent* to a strong variational inequality for a continuous, monotone operator?

Consider a closed set $Q$, and a continuous monotone operator $g: Q \rightarrow E^*$. Consider the "weak" variational inequality: $$\mbox{find } x^* \in Q: \, g(x)^T (x-x^*) \geq 0, \, \forall x \in Q....
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Is the trapezoid rule integral operator $A$ a non singular matrix?

Let $A$ be the matrix of a linear bounded integral operator $A : L^2 \rightarrow L^2$, discretized by the trapezoid rule defined on the function space $BV([0, L])$, $L \in \mathbb{R}$. So, if $u \in ...
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52 views

$\lambda = \max_{\mathbf{x}}\frac{\mathbf{x}^{T}\mathbf{A}\mathbf{x}}{\mathbf{x}^{T}\mathbf{x}}$ for non-negative matrices $A$?

Let $A$ be a non-negative irreducible matrix. By the Perron-Frobenius theorem, the eigenvalue of max. absolute value $\lambda$ is positive and has an eigenvector of all positive entries. Is it true ...
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64 views

Definition of fundamental eigenvalue and sign of fundamental eigenfunction.

Consider the following integral eigenvalue equation $u = \lambda Ku$, where $K\colon L^2(F)\longrightarrow L^2(F)$ is a symmetric, compact, self-adjoint operator with positive, continuous kernel of ...
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50 views

Is there always a non-trivial homotopy class of maps which minimizes the Dirichlet energy?

Let $M,N$ be $d$-dimensional oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E:C^{\infty}(M,N) \to \mathbb{R}$ be the Dirichlet energy, i.e. $$ E(f)=\int_M |df|^2 \text{Vol}...
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1answer
146 views

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 \...
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12 views

First variation of coordinate transformation

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth coordinate transform in $\mathbb{R}^3$. Also, let $\mathcal{E} = C(\mathbb{R}^3)$ to be a space of real, scalar-valued functions on $\mathbb{R}^...
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1answer
32 views

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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72 views

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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1answer
19 views

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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1answer
54 views

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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1answer
107 views

Existence and uniqueness of weak solutions to the homogeneous biharmonic equation.

I am currently trying to prove boundedness and coercivity of the bilinear operator $a \colon H^2_0(\Omega) \times H^2_0(\Omega) \to \mathbb{R}$ defined by $$ a(u, v) = \int_\Omega \Delta u \Delta v \,...
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66 views

Stationary Condition of Variational Iteration Method

Would you kind help me? I am an undergraduate student and i am studying about variational iteration method (VIM) by Ji Huan He. But, when i study about the stationary condition of VIM, there is a ...
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115 views

Calculus of Variation - Minimize a Functional

I am struggling with a basic question from a homework assignment regarding Calculus of Variation. I would love if somebody could get me started in the right direction, as I am pretty lost and the ...
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400 views

Is Euler-Lagrange equation necessary and sufficient for minimization in a variational problem?

To the best of my knowledge, the function that minimizes of the integral posed in calculus of variations must also satisfy the Euler-Lagrange equations. In other words, the Euler equations are a ...
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3answers
74 views

integrating a line with a changing slope

I'm trying to figure out the following: You're at a point on a graph $(x_0, 1-x_0)$, and you have to obey the following rules: You're allowed to move down as much as you wish, "banking" the amount ...
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1answer
66 views

Is the normal cone well defined?

I am reading Mordukhovich's Variational Analysis and Generalized Differentiation, and the first definition there is that of a $\epsilon-$normals Given a Banach space $X,$ and $\Omega\subseteq X,$ ...
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71 views

Minimize Variational Function over Normalized Distributions

I am looking at finding literature to be able to minimize the following variational problem: Minimize, $$ \mathcal{F}\left[p(y|x)\right] = I(X;Y) + \beta \ \mathbb{E}_{p(x,y)}\left[ d(x,y) \right] $$...
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1answer
48 views

Finding consumption function which maximizes utility

I can across this question in my applied real analysis textbook that I'm having trouble with. It asks us to consider the utility function $U(C) = \sqrt{e^{-rt}C}$. I'm supposed to find the consumption ...
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68 views

Solving $\frac{d^2 \theta}{d x^2} - m^2\theta = 0$ using the Ritz method

I'm trying to solve the following ODE using the Ritz method: $$\frac{d^2 \theta}{d x^2} - m^2\theta = 0$$ With the boundary conditions $$\frac{d\theta}{dx}\Bigg{|}_{x=0} = 0$$ $$\theta(1) = \...
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1answer
574 views

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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2answers
107 views

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 ...
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1answer
53 views

Wondering how to get this analytical solution of $\text{E}\big(\log(f)\big)$, $f\sim$ Normal Distribution

I am reading variational inference for gaussian process modulated poisson processes and find the result (19) is unclear about its source. I am wondering how they get that. The equation is shown here \...
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119 views

Anisotropic Poisson equation strong to weak form

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 \\ ...