# 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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### 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 $$-\Delta u+u = 0$$ For $g\in H^{\frac{1}{2}}((D)$, where $D$ is the domain, assume that $u$ is the solution of (u,...
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### 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 ...
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### 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|...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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