An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.

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23 views

Gradient calculation for Matrix

I have the following: $$b^T a^{-1} b$$ what is the gradient wrt to $a$. $a$ is matrix and $b$ is vector. Basically I should take the derivative with respect to $a$. Is it correct that it equals to: ...
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21 views

Gradient of Frobenius norm

Given an optimization $J(Z)$ as follows: $$ J(Z) = \|A-ZUZ^T\|_F^2. $$ Why the gradient of it is : $$ \frac{\partial J}{\partial Z} = (ZU^TZ^T-A^T)ZU + (ZUZ^T-A)ZU^T.$$ Can you show in detail? 3ks.
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17 views

Maximum rate of change along which curve?

The temperature $T(x,y)$ at points in the $xy$-plane is given by $T(x,y)= x^2 -2 y^2$. An ant wishes to cool off as quickly as possible. Along what curve through $(2,1)$ should the ant move in order ...
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8 views

How to derive the gradient formula for the Maximum Likelihood in RBM?

I am learning RBM (restricted Boltzmann machine) for deep learning. The log-likelihood of RBM is given as : and its gradient w.r.t. the parameter is: I don't understand how is the gradient derived ...
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9 views

What is an extragradient method?

I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia ...
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41 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
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117 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
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34 views

How to find all stationary points of $ \alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2$

Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is $$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$ where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the ...
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56 views

gradient flow -cahn hilliard

hello $$$$ I am trying to find explanation how to derive cahn hilliard equation: $$ u_t =\Delta (w'(u)-\epsilon ^2 \Delta u)$$ as gradient flow of energy functional $$ : E[u]=\int ...
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44 views

Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...
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42 views

Gradient flows of functionals on manifolds

I'm reading some literature of ricci flows and on my way through it I quickly stumbled upon the gradien flow - one of the geometric flows. After searching on other books I found rigor definition of ...
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179 views

Stroke Width Transform

I have asked question related to stroke width transform http://stackoverflow.com/questions/22425545/stroke-width-transform-opencv-using-python I did not under stand the math behind the answer given ...
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1answer
57 views

Flow of a particle in dynamical system

Suppose I have the linear system $\dot{x}=Ax$, with $A=\left[ \begin{array}{cc} -1 & 0 \\ 0 & 2\\ \end{array}\right]$. I know that the phase portrait of the linear system has a saddle in ...
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36 views

Finding local minimum under constraint

How to find the minimum of $f(x) = ||x-\mu||^2$, where $\mu = (1, 1)$ and $< x, \mu > = 0$ (the inner product is $0$)?
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51 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
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197 views

gradient flow programming (matlab)

Consider $$f(x,y)= x \ y \ e^{-x^2-y^2}$$ I need to find the gardient flow of f starting at the point (0,1) then (2,4). I know that this entails solving the follwing ODE $$\dot X(t)= - \nabla ...
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31 views

“mass creating” flux in a conservation equation

$\phi: \mathbb{R} \to \mathbb{R}$ is a given, smooth gradient field and I come from the equation $\frac{\partial}{\partial t} u(x,t) = \text{div}(\phi(x,t))$ for $x\in \mathbb{R}$, $t\geq 0$ and some ...
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37 views

Laplacian of Backward heat kernel

I am reading about mean curvature evolution. The situation is, we have immersions of n-dimensional hypersurfaces $$\mathbf F: M^n \rightarrow \mathbb R^{n+1}$$ such that $$\frac{d}{dt}\mathbf ...
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1answer
99 views

$\dot u(t) = - \nabla V(u(t)) $ unique solution if $V$ is convex

I found this statement in a book I am reading: If $V: \mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable and convex, then the differential equation $$\dot u(t) = - \nabla V(u(t)) $$ has a ...
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557 views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
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105 views

Minimization problem as PDE

In the article "An Image Interpolation Scheme for Repetitive Structures" Luong, Ledda and Philips propose the following approach to denoising digital image. They consider that regularized total ...
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316 views

Hamiltonian for Geodesic Flow

I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where $$H = \frac{1}{2}g^{ij}p_i p_j$$ but I am stuck. Could somebody show ...
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203 views

error for Conjugate gradient method

Suppose A is a real symmetric 805*805 matrix with eigenvalues 1.00, 1.01, 1.02, ... , 8.89,8.99, 9.00 and also 10, 12, 16, 36 . At least how many steps of conjugate gradient iterations must you take ...
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What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
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2answers
133 views

Non-degenerate solutions to constant Hamiltonian flow

As I'm trying to work my way through Dietmar Salamon's "Notes on Floer Homology", I'm having trouble with the very first exercise. Let $(M, \omega)$ be a compact symplectic manifold. Let $H$ be a ...
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1answer
117 views

Gradient of the image

I'm trying to make a gradient flow for an image. For a test I made a small image 3x3 pixels with a black pixel in the middle. I found how to compute the direction of the gradient for one point given ...
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Why is gradient the direction of steepest ascent?

$$f(x_1,x_2,...x_n):R^n \rightarrow R$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}e_1 +\ ... +\frac{\partial f}{\partial x_n}e_n$$ which is a vector. Reading this ...
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1answer
236 views

Monge Ampere and Calculus

I am learning about mass transportation theory and the Monge-Ampere equation, to transport a function $f$ toward $g$ by a change of variable $T$. In particular, in order to solve for : $$ \min \int ...
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1answer
417 views

Gradient flow of a surface

I found the following definition in a book (S. Osher, R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces", p. 140): [the context is reconstruction of surfaces from unorganized point sets] ...
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1answer
65 views

Decreasing function in the context of gradient flows

I'm studying the lecture notes by Philippe Clément about Gradient Flows in Metric Spaces. Now the following problem arises ($X$ is a Hilbert space): Definition: Let $\phi:X \to (-\infty, \infty]$ ...
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Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
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234 views

Gradient flows in metric spaces

What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...