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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15 views

Producing term with Lie derivative from variational principle

Is it possible to obtain variational principle (Lagrangian) such that the equations of motions contain Lie derivative? For example, if $g$ is a standard metric tensor in Euclidean space $E^3$ and we ...
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32 views

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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29 views

Minimal divergence-free projection onto unit vector field

I'm attempting to use a variational method to find a critical vector field. I want to find the local components of a given vector field, $\bf{v}$, that are as "close" to $\bf{v}$ as possible, but are ...
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Existance of Solution in Neumann Variational Inequality

I was working on the following question and got stuck (Chap 2, #10 Kinderleher and Stampaccia) Let $\Omega$ be a bounded domain with a smooth boundary $\partial \Omega$ and suppose given $f \in H^{-...
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50 views

Does functional satisfies Palais-Smale condition?

Check if the functional $f(u)=\int_0^{1/2} u^2(x)dx$ satisfies the Palais-Smale condition on the Hilbert space $L^2([0,1],\mathbb{R})$. We have definied the Palais-Smale condition as follows: $f$ ...
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59 views

How is the Lagrangian function homogeneous in the velocities?

I'm working through The Variational Principles of Mechanics by Cornelius Lanczos and in chapter 6, section 10, he says that the Lagrangian function is homogeneous in the velocities. $L_1 = L(q_1,......
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40 views

Intuition behind Pohozaev identity

I'm wondering if the Pohozaev identity: $$n\int_\Omega\int_0^{u(x)}f(t)\operatorname{d}t\operatorname{d}x-\frac{n-2}{2}\int_\Omega u(x)f(u(x))\operatorname{d}x=\frac{1}{2}\int_{\partial\Omega}\left|\...
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1answer
380 views

Confused by Kullback-Leibler on conditional probability distributions

I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the ...
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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*} ...
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1answer
60 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 ...
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20 views

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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31 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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85 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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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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80 views

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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22 views

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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68 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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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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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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113 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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74 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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127 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 \\ ...
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110 views

Finding stationary point of the functional

Find the stationary point of the functional $$ J[y]=\int \left( x^2y'^2+2y^2 \right) dx $$ where $y(0)=0, y(1)=2.$ My Solution: E-L equation: $x^2y''+2xy'-2y=0.$ This is also Cauchy-Euler equation. ...
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24 views

Relation between two types of pseudomonotone function

As far as I know, there are two different definitions for pseudomonotone function. The first definition was introduced by H.Brezis in 1968: Pseudomonotone function in the sense of Brezis: Let ...
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121 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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1answer
49 views

Determining a variational formulation of $u^{(4)} = f$ with $3$-rd order BC.

Below is a problem from a recent exam and I have some questions about it. Given the boundary value problem:$$\dfrac{\partial^4 u}{\partial x^4} = f,\quad f\in L^2(0,1)$$ $$u(0) = u''(0) = u'(1) = u''...
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2answers
45 views

Weak derivative of $u(x)=|x|$ not belong in $W^{1,p}(-1,1)$

Consider the functión $u \in W^{1,p}(-1,1) $, defined by $u(x)=|x|$, we know its weak derivative is $$g(x)=\left \{ \begin{matrix} 1 & \text{if }x\in(0,1) \\ -1 & \text{if } x \in (-1,0) \...
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105 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 ...
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144 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
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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66 views

Understanding the proof for the existence of solutions of a variational inequality (by F. E. Browder)

I am trying to understand the proof of Theorem 1 in this paper (see page 784 for the proof). I understand that by the finite intersection property there exists an element $u_0$ in the weak closure of $...
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1answer
99 views

An $\mathcal{L}^1$ optimization problem in function space

Let $f: (-\infty,\infty)\to [0,\infty)$ be a smooth map. Then, how to solve the following optimizatin problem w.r.t. $f$? \begin{align} \mathrm{minimize}_f&~~\int_{-\infty}^{\infty} |f(x) - \exp(-...
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23 views

Existence of Geodesics

Consider a unit sphere in $\mathbb{R}^3$. How can we prove that any two points on the sphere can be joined by a minimizing geodesic? More generally, under what conditions does the same statement hold ...
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36 views

computing variational lower bound

I am trying to re-derive the variational lower bound for the binary logistic regression that is obtained in the paper by Saul & Jordan 1999 and it is given in equation 22 $$\langle \ln(1+e^z)\...
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27 views

Understanding the proof of the deformation lemma

Let $V$ be a Banach space, $V\supseteq N\supseteq N_{2\rho}\supseteq N_\rho \supseteq N_\delta$. Let for every $u\in V$, $\phi_u:[0,\infty[ \to V$ be a solution of the initial value problem $$\begin{...
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57 views

Neumann problem with zero average sobolev space

Let us considère the following Laplace-Neumann problem $-\Delta u=0$ with homogenuous boundary condition of type neumann, i:e $\frac{{\partial u}}{{\partial n}} = 0$. The variational formulation is ...
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25 views

Variational Formulation - inhomogeneous

I'm not sure how to get started with the following. Consider, $- \Delta u=f$ in $\Omega$ $u=u_o$ 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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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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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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Euler-Lagrange equations square integrable function

(A) I am trying to minimise the integral of the form $I=\int_{-\infty}^{\infty} F(x,y(x),y'(x)) dx $ (1), $y(-\infty)=y(\infty)=0$ (2) with the contrain that $y(x)$ is square integrable $\int y(x)^{...
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9 views

Equality of function with respect to functional energy

Let $u : \mathbb{R}\times[0,T]\to\mathbb{R}$ and $u\in C([0,T],H^{1}(\mathbb{R}))\cap C^{2,1}(\mathbb{R}\times(0,T))$. Define a continuous functional $J : H^{1}(\mathbb{R})\to\mathbb{R}$ with $J[u(t)]=...
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27 views

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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17 views

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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36 views

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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20 views

A variational problem involving integral

Let $W\gt 0,\,T\gt 0$ be fixed. Let $R(f)$ be an arbitraty function in $L^2(-W,W).$ Define $$\alpha_{R}:=\frac{\int_{-W}^W\,df^\prime\int_{-W}^W\,df^{\prime\prime}\frac{sin\,\pi T(f^\prime-f^{\prime\...
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29 views

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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1answer
20 views

showing a different form of the Euler Lagrange Equation provided $f\in{C^2}$ and $y'\ne{0}$

Suppose that $f(x,y(x),y'(x))$ is s.t $f\in{C^2}$ and $y'(x)\ne{0}.$ I am trying to show that the Euler-Lagrange equation $\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'}=0$ ...
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31 views

Is this function differentiable

Define the following function \begin{eqnarray} f(Q_p) &=& \max \quad x^T(Q-Q_p)x +\mbox{trace}(Q_p X) \\ &&s.t. \quad (x, X) \in \mathcal{P} \end{eqnarray} where $\mathcal{P}$ is ...
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28 views

Given a smooth family of strictly convex norms in $\mathbb{R}^n$ ($n\geq 2$), what can we say about the family of dual norms?

Let $U\subset{\mathbb{R}^n}$ be an open set (may has compact closure) and $F:U\times\mathbb{R}^n\rightarrow\mathbb{R}$ such that $F(p,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ is, for each $p\in{U}$, ...
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1answer
33 views

Calculus of Variations by Charles Fox: Question on Statement in Section 2.4

Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...