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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
10 views

In finite element method for second order elliptic problem with neumann boundary value, is the solution weakly satisfies the boundary conditions?

For example, let's consider the problem \begin{equation} -\Delta u+u = 0 \end{equation} For $g\in H^{\frac{1}{2}}((D)$, where $D$ is the domain, assume that $u$ is the solution of \begin{equation} (u,...
GeneLIU's user avatar
  • 53
0 votes
0 answers
6 views

A inequality involving an exponential non linearity of a PDE

Let $f : \mathbb{R} \to \mathbb{R}$ a continuous function and suppose there exists $\alpha_0 > 0$ such that $$ \lim_{|s| \to +\infty} \frac{|f(s)|}{e^{\alpha |s|^{N/(N-1)}}} = 0, \quad \forall \...
Lucas Linhares's user avatar
0 votes
1 answer
31 views

Derivative in information/rate theory

I am trying to understand how to obtain the following derivation w.r.t. $p(\hat{x}|x)$ but cannot for the life of me find out how they do. $$\frac{\partial}{\partial p(\hat{x}|x)} \left( \sum \sum p(x)...
idlatva's user avatar
  • 155
1 vote
1 answer
35 views

Converting differential calculus problem to variational calculus problem

I was reading The Finite Element Method for Engineers by Huebner, and read that solving for $\phi$ in the following differential equation $\frac{d^2\phi}{dx^2}=-f(x)$, $\phi(a)=A$, $\phi(b)=B$ is ...
user1308395's user avatar
0 votes
0 answers
9 views

Derivation of Evidence lower bound and KL diveregence in VAE. I cannot understand the independence condition.

In the article "An Introduction to Variational Autoencoders" written by Diederik P. Kingma, there are this equation in page 18. \begin{align} \log p_{\theta}(x) &= \mathbb{E}_{q_{\phi}(z|...
user14096975's user avatar
0 votes
0 answers
13 views

Strongly monotone operator implies coercivity

I read from the book "Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I" (P. 156) and paper "Finite-dimensional variational inequality and nonlinear ...
zzgsam's user avatar
  • 107
0 votes
1 answer
73 views

Functional Derivative Rate distortion theory

Can someone help me to see where the problem of the functional derivative below is? Minimize the functional: $$ F[p(\hat{x}|x)] = I(X; \hat{X}) + \beta \sum_{x \in X}\sum_{\hat{x} \in \hat{X}} p(x,\...
J.doe's user avatar
  • 173
-1 votes
1 answer
47 views

Why is the Brachistochrone cycloid flipped

While solving the Brachistochrone problem, I setup a variable $T$, $$T=\int\dfrac{\sqrt{1+(y')^2}}{\sqrt{2gy}}$$ Then employing the Euler-Lagrange equations (Beltrami's identity): $$\dfrac{dy}{dx}=\...
Anirudh Yamunan Govindarajan's user avatar
0 votes
0 answers
12 views

Proof of intractability of computing the prior distribution in bayesian inference

In this paper: https://www.tandfonline.com/doi/epdf/10.1080/01621459.2017.1285773?needAccess=true I do not understand why there are K^n terms in the integral in Equation (9) instead of n^K. If we have ...
StackExchanger's user avatar
5 votes
1 answer
189 views

Optimality condition inspired by subdifferential of square root: $y\in \text{argmin}(g(x)-a^Tx ) \Rightarrow y\in \text{argmin}(g^2(x)-2g(y)a^Tx).$

Let $f:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ be a proper convex lower semicontinuous function. Suppose that $f$ is bounded by below, and for simplicity that $\inf f = 0$. Set $\varphi:\mathbb R^d\to\...
Alberto's user avatar
  • 243
0 votes
0 answers
34 views

The explaination of Einstein-Hilbert action in mathematics-wise

I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This ...
PermQi's user avatar
  • 509
1 vote
1 answer
50 views

Find the first variational equation of the following perturbed IVP: $\dot{x}=x^2+\varepsilon, x(0)=x_0,\varepsilon\geq 0$.

I want to show that the first variational equation of the system is $$\dot{\phi_1} = 1 +\frac{2x_0}{1-x_0t}\phi_1(t),\qquad\phi_1(0)=0$$ And solve the IVP approximately up to Order $O(\varepsilon^2)$. ...
designerresearch44's user avatar
0 votes
0 answers
34 views

Is a Forward KL Divergence assumed in the original Diffusion Model Objective and if so, why?

I'm currently getting into Diffusion Models and Variational Inference in general. I've studied Lilian Weng's Blog on VAEs , where the importance of using the Reverse KL $$D_{KL}(q_{\Phi}(z|x)||p_{\...
macfeuchtner's user avatar
2 votes
1 answer
41 views

Step in derivation of Hadamard's First Variational Formula for Hermitian Matrices

This arose in trying to understand the details of a proof of Hadamard's first variational formula in Terence Tao's "Topics in Random Matrix Theory". Suppose $A$ is an $n \times n$ Hermitian ...
J.B.R's user avatar
  • 117
0 votes
0 answers
43 views

Finding the variational formulation of the following transport equation with boundary condition u'(1) + u(1) = 1

I'm studying variational problems and finite element method, and I'm trying to solve the following equation with boundary conditions: for $u: ]0,1[ \to \mathbb{R}$, $-((1 + x)u')'= x,$ for $x \in ]0,1[...
Amsel's user avatar
  • 11
0 votes
0 answers
24 views

Analysis of a parametric quadratic game

I am solving a game theory problem for N players. At each step, each player solves a projection-based gradient descent $\operatorname{proj}_{X}\left(x_i^{(k)}-\eta F\left(x_{i}^{(k)}\right)\right)$ ...
zzgsam's user avatar
  • 107
0 votes
0 answers
21 views

Use of the Kullback-Leibler divergence in variational bayes and deep learning

I am trying to grasp the asymmetry of the KL-divergence from the point of variational approch and deep learning. Deep learning seeks $q$ by minimization of $KL(p||q)$ and variational Bayes seeks $q$ ...
Avec's user avatar
  • 21
1 vote
0 answers
30 views

Problem with variation of calculus for noether

In the variational calculus, we discuss general case variation where endpoints vary as well. (i.e $x_0$ and $x_1$ vary as well). By using some calculations, we end up with: $\delta J = \left[\frac{\...
Giorgi's user avatar
  • 55
1 vote
1 answer
65 views

First order variation of length of geodesics

On a Riemannian manifold, a geodesic $\gamma_{pq}$ between two points $p$ and $q$ locally minimizes the length between these two points. I would generally expect this to imply that if $\delta \gamma_{...
pseudo-goldstone's user avatar
0 votes
0 answers
72 views

Does Weak convergence in $W^{1,p}(\Omega)$ implies almost everywhere of the gradient $\nabla$

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$ and $1<p<N$. Let $\{u_n\}$ in $W_0^{1, p}(\Omega)$ be such that $$ u_n \rightharpoonup u \quad \text { in } W_0^{1, p}(\Omega) . $$ Then, ...
kilose2013's user avatar
1 vote
1 answer
36 views

A statement regarding tangent cones and polyhedricity and variational inequalities (need help with a proof)

I'm reading this thesis on page 182. We work in a Hilbert space $H$, and $y \in H$ solves the variational inequality: $$y \in K : \langle Ay-f, v-y \rangle \geq 0 \; \forall v \in K$$ for some given ...
StopUsingFacebook's user avatar
0 votes
0 answers
32 views

Help with Variational Calculus & Leibnitz Rule

Let $f: (0,\infty)\to \mathbb{R}$ be convex and lower-semicontinuous with $f(1)=0$ and $\mu$, $\hat{\mu}$ be two probability distributions on a measurable space $\mathcal{X}$ which are absolutely ...
pablopez's user avatar
1 vote
0 answers
53 views

Applying Variational Approximation [duplicate]

I am trying to solve a problem involving variational approximation, where the task is to calculate a value $C$ such that $$C > \frac{\int_{-\infty}^{\infty} |f'(y)|^2 dy}{\int_{-\infty}^{\infty} \...
Newbie's user avatar
  • 11
0 votes
0 answers
17 views

Variational integral problem with corners

Consider the integral $$ J(y) = \int_{x_1}^{x_2} \frac{y(x)}{(y'(x))^2} + (y'(x))^2dx\,, \ \ \ \ y'(x) \neq 0 $$ Show that for this problem corners are possible. (Hint: The corner conditions are ...
ddddd's user avatar
  • 1
3 votes
0 answers
65 views

Transform a differential equation into Hamiltonian form

I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov. Exercise 33.4.1: Consider the differential equation \begin{...
user 1234's user avatar
  • 425
1 vote
0 answers
97 views

Riesz representation theorem in the derivative of functional.

Definition 1 (Functional derivative): Given a function $f \in \mathcal{F}$, the functional derivative of $F$ at $f$, denoted $\frac{\partial F}{\partial f}$, is defined to be the function for which: $$...
Elio Li's user avatar
  • 567
0 votes
0 answers
36 views

Example of empty Clarke subdifferential for function lipschitz over a closed convex set

If $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}$ is a proper lower semi-continuous function that is Lipschitz continuous over $\text{dom}(f)$, where $\text{dom}(f):=\{x\in\mathbb{R}^n:f(x)<+\...
William's user avatar
  • 997
0 votes
0 answers
37 views

Why is th coerciveness of a bilinear form important for making sure a variational problem is well-posed?

Hello, I've come across the idea of the coerciveness of a bilinear form in the context of variational problems. Could you help me understand in simpler terms why this property is so important for ...
user134613's user avatar
1 vote
1 answer
210 views

A question about the derivative of functional.

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds, in this paper they defined $$ E:=\left\{u \in H^m(M): \int_M u d \mu_g=0\right\}, $$ and $$\|u\|:=\left(\...
Elio Li's user avatar
  • 567
0 votes
0 answers
21 views

Papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional.

Can you recommend me some papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional. I glance over some methods including mountain pass theorem and ...
Elio Li's user avatar
  • 567
3 votes
2 answers
146 views

Minimizing a functional subject to boundary conditions

I want to find a smooth function $y : [0,1] \to \mathbb{R}$ that minimizes $$ S = \int_0^1 (y(x)y'''(x) + 3 y'(x) y''(x))^2 dx $$ subject to the constraints / boundary conditions: (i) $y(0)=1$, (ii) $...
André's user avatar
  • 135
0 votes
0 answers
17 views

Obtain a vector function that minimizes "gas cost"

I'm struggling with an optimization problem: Suppose you have the amount spent on gas from 0 to a time T as $u=\int_0^T|f| dt$, where $\textbf{f}=m(\ddot{x}, \ddot{y}(t),\ddot{z}(t)+k)$ with k a ...
Eduardo Kuri's user avatar
0 votes
1 answer
83 views

Equivalent Formulations of Variational Problems

This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or ...
Small Deviation's user avatar
1 vote
1 answer
76 views

Deriving Lagrangian for common a class of PDEs

I am interested in constructing a Lagrangian for a PDE of type $$ u_t(t,x) - F(u,u_t,u_x)=0 $$ such that for some functional $\quad I[u] = \int D[u]\mathcal{L}[u,u_t,u_x]$ its associated Euler ...
user3166083's user avatar
0 votes
0 answers
34 views

On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry

On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways: curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
Inzinity's user avatar
  • 1,733
0 votes
0 answers
62 views

Is the indicator function of a closed set prox-bounded?

The indicator function of a closed set $C \in \mathbb{R}^n$ is defined as bellow: $$ \delta_{C}(x)= \begin{cases} 0 \quad \quad\,\, x \in C\\ + \infty \quad x \notin C\\ \end{cases}. $$ Question Is ...
Saeed's user avatar
  • 175
1 vote
1 answer
43 views

sequential compactness of saddle-point lagrangians

In minimization problems, say $\min_{u \in X} F(u)$, $X$ metric space, one of the steps of the Direct Method is the compactness of minimizing sequences: show that for any sequence $u_n$ such that $\...
mps94's user avatar
  • 21
1 vote
1 answer
126 views

Finite element method in polar coordinates

I need to solve the following problem $$ -\dfrac{1}{r}\left( \dfrac{d}{dr} \left( r \dfrac{du}{dr} \right) \right) = f(r), ~~ r \in (0, 1)$$ $$ u'(0) = 0, ~~~ u(1) = 0 $$ with the finite element ...
Stanislav Morozov's user avatar
1 vote
0 answers
59 views

A Question on Calculus of Variation

Find the extremal of the Functional $J[y(x)]=\int_{0}^{1}(2e^{y}-y^{2})dx$ subject to $y(0)=1$ and $y(1)=e$. Using Euler's equation we get the answer $y=e^{y}$. My question is how to check the ...
Shash's user avatar
  • 87
2 votes
1 answer
64 views

Extending the fundamental lemma of the calculus of variations so that the integral is proportional to the endpoint of the integrand

Consider the analytic and bounded function $f:\mathbb{R}_{\leq x_0}\rightarrow \mathbb{R}$ with bounded first derivative such that $\alpha\in\mathbb{R}$ and \begin{equation} \alpha\varphi''(x_0)f(x_0)=...
Jean Daviau's user avatar
2 votes
1 answer
60 views

Proof of Theorem 6.14 in "Variational analysis (Rockafellar, Wets)"

I am currently reading "Variational analysis" by Rockafellar and Wets, and I am trying to understand the proof of Theorem 6.14 about normal cones to sets with constraint structure. But, I ...
kumegon's user avatar
  • 85
1 vote
0 answers
19 views

Prove that $u(x_i)=u_h(x_i)$ in variational discretization

Suppose $f \in L^2(0,1)$ and $u \in H^1_0$ is the solution to the following problem: \begin{equation} \int_0^1 u'(x)v'(x) dx=\int_0^1 f(x)v(x)dx ~~~~~~~~~~~~~~~~~~~~(1) \end{equation} for all $v \in ...
Thomas345's user avatar
0 votes
0 answers
36 views

Variational discretization: proof that $u(x_i)=u_h(x_i)$

Suppose $f \in L^2(0,1)$ and $u \in H^1_0$ is the solution to the following problem: \begin{equation} \int_0^1 u'(x)v'(x) dx=\int_0^1 f(x)v(x)dx ~~~~~~~~~~~~~~~~~~~~(1) \end{equation} for all $v \in ...
Andreas804's user avatar
1 vote
0 answers
31 views

Cylindrical surface with maximum area?

Imagine we have a parametic surface given by: $$ \Phi(r, \theta) = \begin{pmatrix} r \cos{\theta} \\ r \sin{\theta} \\ f(r) \\ \end{pmatrix} $$ $$ r \in \mathbb R^+, \; \; \; \ \theta \in [0, 2\pi] $$...
Álvaro Rodrigo's user avatar
1 vote
1 answer
112 views

Weak solution is strong solution

I came across the following statement: If $V=\{v \in C^1[0,1])~ |~ v(0)=0\}, ~~f \in C([0,1])$ and $u \in C^2([0,1])$ then any solution to \begin{equation} u(0)=0 ~~\text{and}~~ \text{for all}~~ v \...
Andreas804's user avatar
0 votes
0 answers
141 views

How does a vector field act on a differential form?

In the definition of the Euler-Lagrange operator (2.6, I. Anderson), the total differential vector fields $D_{i}, D_{ij}$ act on differential forms. How does that work? Feels like Lie derivatives to ...
Jian's user avatar
  • 29
2 votes
1 answer
122 views

weak variational formulation of Poisson equation with some restrictions.

The variational formulation of Poisson equation $$ \begin{cases} -\Delta u = f & \text{in } \Omega \\ u = 0 & \text{on } \partial \Omega \end{cases} $$ is that: Find $u\in H_0^1(\Omega)$, such ...
Chandler's user avatar
  • 445
0 votes
0 answers
34 views

What is the point of the Jacobi accessory equation?

In Where does Jacobi's accessory equation come from? Chappers gives two derivations of the Jacobi accessory equation. From the first, it is clear that the Jacobi accessory equation is a means to ...
Seb Ellwood's user avatar
0 votes
1 answer
123 views

variational form of fourth order elliptic equation

Given $$ \alpha\Delta^2y+y=\alpha^{1/4}y_d-\alpha^{5/4}\Delta(f+u_d), \Omega $$ $$ y=0, \alpha^{1/2}\Delta y+\alpha^{3/4}(f+u_d)=0 $$ I am trying to verify the regularity result as $$ \alpha^{1/2}|y|_{...
79999's user avatar
  • 157
2 votes
1 answer
79 views

Solve variational equation using the Fourier transform

Let $u_0: \Omega \subset \mathbb{R}^2 \to \mathbb{R}^2, n \geq 1 $, be a continuous function and the functional $E$ given by $$ E(u) = \int_\Omega \dfrac{1}{2} \| u(x) - u_0(x) \|_2^2 dx - \dfrac{\...
Esteban's user avatar
  • 45

1
2 3 4 5
7