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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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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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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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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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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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formula for arc length.

Today in a variational principles lecture we were shown the following: If we have two points on the plane, namely, $(x_1,y_1)$ and $(x_2,y_2)$ and we have a curve $y(x)$ s.t. $y(x_1)=y_1$ and $y(x_2)...
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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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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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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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Find Extremum of Functional with No Initial Conditions

I have the following functional $ J(x)=\int\limits_{a}^{b}[{y(x)^2 + \dot{y}(x)^2 + 2 y(x) e^x}] dx $ and I want to find the extremum of it. So I compute the Euler-Lagrange Equation and I get the ...
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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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Minimizing elastic energy of shrinking balls

I am new to variational analysis and I am currently working in the following setup: We denote the $2$-sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$. In coordinates $(\theta,\varphi)$, ...
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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 ...
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Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations

I am having a difficult time understanding Jacobi's necessary condition for weak extrema of functionals. Graphics and detailed explanations would be helpful. I am following the following two texts: ...
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Euler-Lagrange equations from a complex-valued Lagrangian

I've been looking without success for references describing a generalization of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” ...
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Calculus of Variations (Gelfand & Fomin): Proof of Euler's Equation for Constrained Variation

I am in Section 2.12.1 of Calculus of Variations by Gelfand & Fomin. I am attempting to follow the proof of the Euler equation for Constrained Variation (Theorem 1, pg. 42). However, I'm confused ...
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Maximizing a nested sum of infinitely many variables (discrete variational calculus)

I am wondering if there is a way of finding a neat closed form solution for a series of numbers $r_t$ that satisfies the following. This is supposed to give the optimal strategy of an economic agent ...
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Calculus of Variations (Gelfand & Fomin): Proof of Functional Dependence Identity

I'm on page 40 of the book (Section 2.10 - Variational Problems in Parametric Form). It states that for $\int_{t_0}^{t_1}F\left(x,y,\frac{\dot{y}}{\dot{x}}\right)\dot{x}dt=\int_{t_0}^{t_1}\Phi\left(x,...
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Is the Clarke Subdifferential always defined for Lipschitz continuous functions?

According to various websites, for some function $f:X\to R$ we can define a map $$ D(x,v):= \lim_{y\to x} \sup_{t\searrow0} \frac{f(y+tv)-f(y)}{t} $$ and the Clarke Subdifferential (Def 1) is $$ \...
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Interior solution of a Variational Inequality in Hilbert Space

I'm trying to understand the proof of the following claim: Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$...
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Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates

I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ...
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The explicit expression of $\frac{δF}{δP}$

I'm writing simulation code of ferroelectric domain, and there is a math problem that I can't solve. The expression of $F$ is $$ F = \frac{|\vec{k} \cdot \vec{P}(\vec{k})|^2}{k^2}. $$ $\vec{k}$ is a ...
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Why is the epigraph of Moreau-Yosida Regularization a projection of a convex set?

The Moreau-Yosida Regularization is given by \begin{equation} f_\mu(x) = \inf_y \left( f(y) + \frac{1}{2\mu} \| x - y \|^2 \right). \end{equation} We know that $L(x, y) = f(y) + \frac{1}{2\mu} \| x -...
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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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Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
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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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With the solution of the Euler-Lagrange equation prove the following equation

Let $\Omega \subset \mathbb{R}^n$ and $F=F(z,p):\Omega \times \mathbb{R} \times \mathbb{R}^n$ be smooth and independent of $x\in\Omega$. Let $u$ be a solution of the Euler-Lagrange equation of $\...
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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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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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How to treat an equation of the form $-\Delta u=G\cdot \nabla u+f(u) ?$

There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear ...
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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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Minimum expected value over all probability functions

Find the minimum value of $E[X]$ over all probability density functions f(x) satisfying the following three constraints: (I) $f(x) = 0 $ for $x \leq 0$ (II) $ \int_{-\infty}^{\infty} f(x) dx = 1 $ ...
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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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1answer
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How to take derivative of integral of function?

I'm reading a textbook where it forms a Lagrangian function $$ L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$ But how do you take the derivative of this thing? The ...
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Optimization in Banach space: Find functions that minimize the supremum of a convex operator.

Let $D \subset \mathbb{R}^n$ be compact. Denote by $C(D, \mathbb{R}^n)$ the space of continuous functions from $D$ to $\mathbb{R}^n$. Let $K$ be a real, symmetric, positive-definite $n \times n$ ...
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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
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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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Does this functional satisfies the Palais-Smale condition?

Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^N$, $\lambda\in \mathbb{R}$ be an eigenvalue of $-\Delta$ on the Sobolev space $H^1_0(\Omega)$ and $f\in L^\infty(\Omega\times\mathbb{R})$...
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Characterisation of the rate distortion function: issue with functional derivative

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)}{...
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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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1answer
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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 ...
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1answer
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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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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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26 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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40 views

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 ...