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

Calculating the inverse of planar flows

I am trying to find a way to calculate the inverse of a planar flow. In general, I understand that with normalizing flows, one can simply go from one distribution to the other with the change of ...
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Weakly lower semicontinuity on weighted Sobolev space

Let $W^{1, p}(\Omega; \mathbb{R})$ be the standard Sobolev space and $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. If $f\colon\Omega\times\mathbb{R}\times\mathbb{R}^2$ is a Caratheodory ...
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41 views

Proof for equality of energy for heat equation.

Given: Heat equation: $(1)\begin{cases}\frac{\partial u}{\partial t }-\frac{\partial^2 u}{\partial x^2}=f\quad pp. \text{in}\quad \Omega\times]0,T[\\ u=0\quad pp. \text{in}\quad \partial\Omega\times]...
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homogeneity of an equation

I am a student studying Michael Struwe's Variational method book. In this book, he said using the homogeneity of $$-\Delta u+\lambda u=u|u|^{p-2},$$ a solution of problem $$\begin{array}{lcl} -\Delta ...
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what is variational over-pruning

I come across this term "variational over-pruning", as I was reading about variational inference. But I didn't understand it, and I couldn't find a proper simplified answer. Can anyone ...
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1answer
47 views

Calculating the variation of a scalar field along the two parameter variation $\alpha(t; r, s) = \exp_{x(t)}(rV(t) + sW(t))$

I'm currently reading about a problem regarding second variations of some functional defined on a Riemannian Manifold $M$ equipped with the Levi-Civita connection, and am confused about how to express ...
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relation between total derivative and directional derivative, Gateaux and Frechet derivative

Consider a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$. Given the total derivative $df(x,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}^m$, we can get the directional derivative $df(x,h)$ for every ...
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34 views

A problem about calculus of variations

I am reading PRML by Bishop, I encounter the following variational calculus problem. I want to get a function $y(\textbf{x})$ minimize the average loss given by following: $$ \mathbb{E}[L] = \iint \{y(...
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Are the solutions of the Euler Lagrange equation OPTIMAL

my question is are the solutions of the Euler -Lagrange Equation always OPTIMAL , in the sense of the best solution for example for the 'Brachistochrone' problem could exist a piecewise linear ...
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HJB equation with first order derivative in H

I am trying to solve the following HJB equation: $$ 0 = \delta_tH + \sup_{\delta}\{\lambda e^{-\kappa \delta} [ \delta + h(t,q-l(\delta))-h(t,q)]\} $$ when trying to solve the $\sup$ for $\delta$, I ...
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Chain rule involving Fréchet derivative

Suppose that $P(w)$ is a probability density function with support $w \in [0,\infty)$ and $G = G[P]$ is a functional satisfying $G[P] \in [0,1]$. I saw a paper used a chain rule of the following form: ...
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204 views

Variational inequality or partial differential inequality

My understanding is that variational inequalities involve a scalar product and that equations are solved using duality relations. How come the following is a variational inequality? It rather looks ...
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62 views

weak variational formulation of Poisson equation with Dirichlet boundary conditions

I have given the Poisson equation with Dirichlet boundary conditions \begin{cases} -\Delta u & = f & \text{in} &\Omega \\ \quad u & = g & \text{on} & \partial\Omega \end{cases}...
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128 views

Existence of a global or local minimizer of $F(u) = \int^{b}_{a}{u'(x)^2 + \arctan(u(x)) dx}$

I am trying to solve the following problem: Let $f : [a,b] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x,u(x),u'(x)) = u'(x)^2 + \arctan(u(x))$ or $f(x,z,p) = p^2 + \...
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38 views

Solution for $F = \int^{b}_{a}{f(y'(x)) dx} \mbox{ min }$ with $\frac{d }{dp}f(p) = 0$ for $f(x,z,p) = f(p)$ and $f$ is convex.

I tried to solve the following problem, but somehow my ideas seem to a dead end: For $D := \lbrace y \in C^1([a,b]; \mathbb{R}) |y(a) = y_1 , y(b) = y_2 \rbrace$ we define \begin{equation*} F(y) = \...
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35 views

Can I move arbitrary function inside integral?

I have a term $\delta g\int_V dV$, where $\delta g$ is the variation of a continuous function g(x,y,z) and $V$ is an arbitrary volume of integration over some physical 3D body. Moreover, $\delta g=0$ ...
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31 views

A transforms converts an unbounded sequence into bounded

Recently, I read a paper(here the link) about variational methods. It said that Then we study a maximizing sequence $\{(u_k,v_k)\}$ of the functional $I(u,v)$. Such a sequence may be unbounded. Using ...
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First variation and Gâteaux derivatives

I have two questions about the relation between Gâteaux and functional derivatives. In calculus of variation textbooks, the first variation is usually defined as follows. Let $V$ be a 'function space' ...
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1answer
42 views

Extremal of $J[y] = \int_{0}^{1}e^x \sqrt{1 + y'^2} dx$ where $y \in C^2[0,1]$

This is a problem from a mathematical contest. We have to find the form of the extremal for the variational problem $J[y] = \int_{0}^{1}e^x \sqrt{1 + y'^2} dx$ where $y \in C^2[0,1]$. The answer ...
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1answer
85 views

Natural boundary conditions for a variational problem when $~f~$ is a function of higher order derivatives of $~y~$

I am not sure about the boundary conditions for a variational problem when $~f~$ is a function of higher order derivatives of $~y~.$ i.e. if we have: $~f(x,y,y',y'')~$ then what is the natural ...
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Obstacle Problem Help

I am attempting to solve the chapter 1 problems in Avner Friedman's Variational Principles and Free Boundary Problems, for independent study. There are no solutions posted anywhere that I can find, ...
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Extremal of the variational problem $J[y] = \int_{0}^1 (y'(x))^2 dx$ with the conditions $y(0)=0$, $y(1) = 1$ and $\int_{0}^1 y(x)dx =0$

I want to solve the variational problem $$ J[y] = \int_{0}^1 (y'(x))^2 dx $$ with the conditions $y(0)=0, y(1)=1$ and $\int_{0}^{1}y(x)dx = 0$. Using Euler's condition (for an extremal): $$ \frac{\...
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Variational calculus on Lie algebra valued one forms - Chern-simons theory

The action in the Chern-Simons theory is given as: $$S=\frac{k}{4\pi}\int_M \text{Tr}(A \: \wedge \:dA + \frac{2}{3}A \: \wedge \:A \: \wedge \: A). $$ The wikepida page gives the euler lagrange ...
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34 views

Given an ODE then regard it as an Euler-Lagrange equation, how to find a functional relative to it?

Recently, I have been studied about Lp Minkowski problem. I met some confusion. The equation is an ODE like that $$u_{xx}+u=\frac{g(x)}{u^{p+1}}, \quad x\in\mathbb{R} , \quad p\geq 0.$$ $g(x)$ is a ...
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60 views

Characterization of $H^{1/2}(\partial \Omega)$ norm?

I can't understand why $\|g\|_{H_{\partial}^{1/2}} = \|u\|_{H^1} $ where $u$ solves the problem \begin{equation*} - \Delta u + u = 0, \quad \Omega\\ u|_{\partial \Omega} = g \end{equation*} The weak ...
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Missing all superscripts in bayesian inference ELBO

I'm a student studying bayesian inference. When I read this paper(https://arxiv.org/pdf/1312.6114.pdf), I had one question in eq2. $$log_{p_\theta}(x^{(i)}) = D_{KL}(q_\phi(z|x^{(i)})||p_{\theta}(z|x^{...
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374 views

Prove that the minimum of a functional doesn't exist

Prove that there is no smooth solution ho the minimization problem: $$\mathcal{L} (u)= \int_{0}^1 e^{-u'}+u^2 dx$$ Where the admissible space is $X =\{ u \in \mathcal{c}^2 [0,1] | u(0)=0, u(1)=1 ...
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33 views

Expectation of $-\frac{1}{2}( x - \mu )^2$

I tried to extend the expectation below from https://zhiyzuo.github.io/VI/ where it is a Variational Inference topic \begin{align} \dfrac{\partial}{\partial \phi_{ij}}~\text{ELBO} & \propto \...
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Variational formulation of the transportation of fluid in a capillary tube.

Let the following boundary problem: \begin{eqnarray*}\label{eq1} -\frac{d}{dx}\left[c_1 (x)\frac{du}{dx}(x) \right] + c_2(x)\frac{du}{dx}(x)&=&f(x); \quad \forall x\in\Omega=(0,1),\\ u(x)&=...
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49 views

Integral of orthogonal functions with a differential operator

I have an infinite set of functions $\{q_n\}_{n\in\Bbb N}$ which can be used to expand functions $q(\tau)$ from a given class as series, i.e. $$ q(\tau)=\sum_{n=0}^\infty c_nq_n(\tau) $$ and whose ...
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46 views

Is the lagrangian density convex if the lagrangian is convex?

Let $L = \dot{q}^T M(q) \dot{q} + V(q)$, i.e., the lagrangian has a quadratic form and hence is convex w.r.t to the velocities, considering that $V(q)$ plays the role of a constant. And now let the ...
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17 views

How to understand $\int q_j \left\{ \int \ln p(X,Z)\prod_{i\neq j}q_iDZ_i \right\}dZ_j =\int q_j \ln\tilde{p}(X,Z_j)dZ_j$

In PRML chapter 10, equation 10.6 It derive $\int q_j \left\{ \int \ln p(X,Z)\prod_{i\neq j}q_iDZ_i \right\}dZ_j = \int q_j \ln\tilde{p}(X,Z_j)dZ_j$ where $\ln\tilde{p}=\mathbb{E}_{i\neq j}[\ln p(X,...
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33 views

Finite Element Method and implementation

I'm studying this problem: $$ \begin{equation} \left (\mathcal P\right ) \quad \begin{cases}\label{P} -u''(x) = f(x), \quad \text{sur} \quad I = ]0,1[\\ u'(0) = 1 \\ u(1) = 0 \end{cases} \end{equation}...
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22 views

Variation w.r.t metric of Electromagnetic Tensor

In this link https://www.theoretical-physics.net/dev/relativity/fields.html#equation-stress-energy-tensors they state that: $$ \frac{\delta (F_{\alpha\beta} F^{\alpha\beta} )}{\delta g^{\mu\nu}} = \...
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Doubt in derivation of Euler-Lagrange equations for image registration

I am reading this paper "On the Metrics and Euler-Lagrange Equations of Computational Anatomy" and it contains a derivation of the LDDMM equations for image matching. I am having trouble ...
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37 views

Catenary solution of minimum surface area of revolution

To get a curve having minimum surface area of revolution passing through end points ( $x_1$, $y_1$) and ($x_1$,$-y_1$), we got the equation: $$ \DeclareMathOperator{\arccosh}{\operatorname{arcosh}} y =...
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2D Isoperimetric Problem in finite plane

I am working on an isoperimetric problem that appear in our project. Specifically, I want to find a curve with minimal length that encloses a constant area in a 2D plane. Which for an infinite plane ...
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1answer
39 views

Functional derivative of a complex fourier sum

I'm trying to parse a derivation of a lagrange-multiplier fourier technique, and can't quite grasp an intermediate step. The author sets up $$ e_x(z,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty}...
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35 views

Variational Calculus Optimization Problem with x-independent Lagrangian Solution via Beltrami Identity

Optimize $$\int_0^1 y^2 (y')^2 dx$$ subject to $y(0)=0, y(1)=1$ For x-independent Lagrangians it is easier to use Beltrami Identity: $$ F-y'{\frac{\partial F}{\partial y'}}=C $$ I have used that ...
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Existence of a plane curve with a given curvature $\kappa(s)$ that passes through two given points $P_1, P_2$ and arc length $L$.

As the title, says, is it possible to find a curve (i.e. prove its existence) whose curvature is equal to a given function $\kappa(t)$, begins in $P_1$, ends in $P_2$, and has arclength $L$? The ...
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39 views

A homework question on compact embedding of Sobolev spaces

I have encountered the following issue on a homework question. There seems to be a gap in my knowledge and I cannot answer it. For the set up, let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain ...
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19 views

Optimization/variational calculus in the space of kernel functions?

I have a "does this area of math exist and what are the right Google search terms?" question below: Take $\mathcal K$ to be the space of kernel functions on a compact domain $\Omega\subseteq\...
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Variational method and the weighting function

Why the variational function, while writing the weak form, is called the weighting function? Is there any specific region behind it?
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28 views

variational calculus and Lax Milgram

$$ \begin{cases} -u''(x) + u(x) = f, \quad \text{on} \quad I = (0,1)\\ -u'(0) + \alpha u(0) = 0 \\ u(1) = 0 \end{cases} $$ given v a function with a compact support we can write: $$ -\int_{0}^{1} u''(...
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Calculus of variations for a Lagrangian functional

I'm trying to understand eq. (2.2) in the following excerpt taken from this lecture notes (p. 10): I'm sure that this result is rather trivial, but I've got a hard time to follow the notation. Since ...
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47 views

Purpose of the “functional derivative”

I'm trying to understand the concept of "functional (or variational) derivatives". I'm familiar with Gateaux and Fréchet derivatives as well as the pushfword between manifolds. The ...
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1answer
50 views

How to minimize the KL divergence with respect to fixed parameters?

I read the LDA paper multiple times but I'm having trouble with the following. Let's say I define a LDA model as: For each doc $m$: Sample topic probabilities $\theta_m \sim Dirichlet(\alpha)$ For ...
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1answer
44 views

KL divergence theorem

Given two distributions $p(x)$ and $q(x)$, I can find the similarity measure between them using $KL(p||q) = -\sum_xp(x) log \frac{q(x)}{p(x)}$. Now I'm defining mutual info as $I(X,Y) \equiv KL(p(x,y) ...
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14 views

Variational derivative in time varying domain?

Consider a spatial domain $\Omega$ enclosed by a boundary $\partial \Omega$. Denote by $\zeta\in\Omega$ the spatial variable and by $x=x(\zeta,t)$ the state variable of the dynamical system decribed ...
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33 views

Convert differential equation to variational statement

I know how to convert a differential equation with constant coefficients to variational form. But the question in the picture has non-constant coefficients. How do people get around that? Any ...

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