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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187 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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651 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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77 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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164 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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1answer
125 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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232 views

Reparameterization Trick in VAE

I was reading on variational auto-encoders https://wiseodd.github.io/techblog/2016/12/10/variational-autoencoder/ and am unable to understand how the function below is generated. Based on my limited ...
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12 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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22 views

(Stochastic Variational inference) What is the expectation of natural parameter with respect to approximate density?

Apologize if this is a stupid question. I am working with Stochastic Variational inference of regression having both local hidden variable (z) and global hidden variable ($\beta$), and $x$ is the ...
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16 views

Understanding a variational calculus argument in the Hamiltonian setting

On pg. 102 of No-Nonsense Classical Mechanics, the author assumes that $S$ is a minimum so that when we add $\epsilon$ to $q$ and $\tilde{\epsilon}$ to $p$, we obtain that $S$ is equal to which ...
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21 views

Extending the domain of a lower semicontinuous function

Let $D$ be a closed nonempty set in $\mathbb{R}^{m}$. Let $f: \mathbb{R}^{m} \times (0, \infty) \to \overline{\mathbb{R}}$ be a lower semicontinuous function such that $f(u, r) \nearrow \delta_{D}(u)$ ...
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6 views

Lower Limits at infinity and growth properties

I am trying to prove the following theorem from "Variational Analysis"by Rockafellar and Wets: Following the proof guide I have come up with this proof: Setting $\bar{\gamma} = \liminf_{|x| ...
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24 views

Expand this integral: $L(q) = \int \prod_{i=1}^{K} q_{i}(\theta_{i}) log\left( \frac{p(\theta,x)}{\prod_{i=1}^{K} q_{i}(\theta_{i})} \right) d \theta$

Suppose we have the following integral: $$ L(q) = \int q(\theta) log\left( \frac{p(\theta,x)}{q(\theta)} \right) d \theta$$ Letting $q(\theta) = \prod_{i=1}^{K} q_{i}(\theta_{i})$, we have: $$ L(q) = \...
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5 views

Nonexpansive mapping and resolvent

Suppose that $T$ is a maximal monotone mapping on Hilbert space, so that $$ \langle Tx - Ty, x - y \rangle \geq 0, $$ for all $x, y$. Then let $R = (I + T)^{-1}$ denote its resolvent. Suppose that $...
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9 views

Constructing a sequence of scalar to “merge” two sequences

I am currently trying to prove Proposition 1.4 in Variational Analysis by Prof. Boris Mordukhovich (Springer, 2018), namely Let $\Omega_1 \subset \mathbb{R}^n$, $\Omega_2 \subset \mathbb{R}^n$ and $...
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10 views

Infinitesimal variation of level sets of a function

Let $X$ be a complex manifold, and let $f:X\to \mathbb C$ be a holomorphic function. Fix a constant $C>0$. Assume $$ Y_t:=\{x\in X\mid |f(x)-t|^2=C\}$$ is smooth for all $t\in[0,\epsilon]$. Is ...
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16 views

A quote from Euler and the relationship with Variational Analysis

Namely, because the shape of the whole universe is most perfect and, in fact, designed by the wisest creator, nothing in all of the world will occur in which no maximum or minimum rule is somehow ...
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8 views

Comparison theorem for variations inequalities

I have the following system $h(x,t) \geq g(x,t)$ $h_t(x,t) + \frac{1}{2}h_{xx}(x,t)\leq0$ $(h(x,t)-g(x,t))(h_t(x,t) + \frac{1}{2}h_{xx}(x,t)) = 0$ Is there anything we can say about the free ...
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25 views

Variational Inequalities and Convex functions

I am having troubles in understanding the proof of the following proposition. Proposition If $f(x)$ is a convex function and $x^*$ is a solution to VI($\nabla f$,$K$), then $x^*$ is a solution to the ...
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45 views

Jensen's inequality and convex Lagrangian

I was reading some lecture notes, and there was a following example that I didn't quite understand. If we have a following variational problem: $ \int_{a}^{b}f(u'(x))dx$ where the Lagrangian $f$ is a ...
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106 views

Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{...
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17 views

Why $p(x|z)$ is assumed to follow multivariate Gaussian distribution in Variational Autoencoder?

In VAE paper including Kingma's paper, they assumed $p(x|z)$ to be a Gaussian/Bernoulli distribution according to data type. $p(x|z)$ as a decoder functions to map the latent value $z$ to the ...
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25 views

Why both infima are in fact minima?

Let $L:\mathbb{T}^d\times\mathbb{R}^d\mapsto\mathbb{R}$ be a Lagrangian over the $d$-dimentional standard torus times $\mathbb{R}^d$. Let $\alpha >0$, the infinite horizon optimal control problem ...
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15 views

Laplace Equation on a circle with homogeneous Dirichlet boundrary

This is actually originally a Physics problem involving variational calculus, but I've come this far and beyond this point it's only mathematics thus I ask here! To begin with, I've derived a ...
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11 views

How inner product is equal to this expression

Basically i want to prove ($ \left\vert u\right\vert ^{q-2}u$ , for $q\geq2$) is strongly monotone. But i can't understand first expression. $\left\langle \left\vert u\right\vert ^{q-2}u-\left\vert v\...
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23 views

Why : $\int_{\Omega}\operatorname{div^{2}\vec{u}}dx=\int_{\Omega }(\nabla u)(\nabla u)^{t}dx$

Isee this steps in book PDE : I'm going to understand it : Steps : $$u\in\mathbb{R^{N}}$$ And $\Omega \subset\mathbb{R^{N}}$ $$K=\int_{\Omega}\operatorname{div^{2}\vec{u}}dx=\int_{\Omega }\...
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14 views

Minimizing a functional with local constraint.

The problem that I am trying to solve is finding extremum of the integral: $\int|\vec{Q}(x,y)|^\alpha dxdy$, subject to a constraint: $\nabla\cdot\vec{Q}(x,y)=R(x,y)$. Now, if the constraint was ...
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40 views

Checking Euler-Lagrange equation in Sobolev spaces

I'm working on a problem in variational calculus. However, I don't feel completely sure of what I'm doing. Consider $Q:=(0,1)\times(0,1)\subseteq \mathbb{R}^2$ and define the following functional $$...
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17 views

Mean Field Assumption - Minimize KL Divergence

$Q_{1} = $ { $q(x): q(x) = \prod_{i=1}^{N} q_{i}(x_{i}) $ } where $q_{i}(x_{i})$ is a distribution which only depends on the $i^{th}$ variable $x_{i}$ of x. p(x) $\propto 1 - \prod_{i-1}^{N} x_{i} $, ...
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57 views

Problem with finite element method

Task picture Hey! I need some help with this task! Task: Derive variational formulation, finite element formulation, and time-marching scheme for problems (1a) - (1d). I know how to solve the ...
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18 views

Eulerian Variation of Four Velocity ??

Problem is Given in the Image, Iam unable to understand the way $\delta u^{\alpha}$ is calculated here, please help in this regard
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55 views

Finite Element Method

Hey! I need some help with this task! Task: Derive variational formulation, finite element formulation, and time-marching scheme for problems (1a) - (1d). Where should I start or how do I solve this?...
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23 views

Calculus of variations with a change in parameter after some time.

$$\text{fun}:=\int_0^\infty e^{-rt}\left[u(c'(t))+w-p\,c'(t)\right]e^{-b\,c(t)}\,dt. $$ I am maximizing this objective function with some function c(t). As an extension, I want to get a solution when ...
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24 views

Calculus - variation iteration method

Something I don't understand The linear differential equation of first order $$u' + a(t)u = b(t) $$ Its correction functional can be written in the form $$u_{n+1}(t) = U_n(t) +\int_0^t \lambda \{\...
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55 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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37 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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37 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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33 views

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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29 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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85 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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55 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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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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467 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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87 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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21 views

About variational calculus conmutative

I need some help with the development of an expression, it is: $$\delta\left\{\frac{\partial^3F(q^1,q^2,t)}{\partial t\partial q^1 \partial q^2}\right\}$$ with each of $q^1,q^2$ are functions of $t$. ...
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73 views

Compute a epsilon normal set

Could you please help to me to compute the following $\epsilon$-normal set: Given $\epsilon>0$, how to compute the $\epsilon$-normal set of $C:=[2,\infty)\times \Bbb{R}$ at the point $(2,0)$. Thank ...

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