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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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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41 views

A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
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12 views

How to get 3d direction vector of a 2d function?

If I have a function $f(a,b)$, where I can plot it on a 3d graph, using $a$ and $b$ as $x,y$ coordinates and $f(a,b)$ as the $z$ coordinate (i.e. ...
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8 views

Normal Gradient z component is too small?

I am trying to draw arrows on a graph to show the normal gradient of a function. By gradient, I mean the arrows should follow the surface of the function, not being perpendicular to it. For example ...
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28 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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21 views

Gradient of expectation of e times its conjugate

Assuming e is a complex series of data, why equation below holds for it's gradient? $$ \bigtriangledown E\{e(n) e^*(n)\}= 2E\{ \bigtriangledown(e(n))e(n)^* \}$$ Where $\bigtriangledown $ is the ...
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4answers
28 views

Direction of gradient vector

How do we know that gradient always points in direction of greatest increase of function and not the greatest decrease?
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16 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
44 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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30 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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42 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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128 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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17 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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126 views

Gradient flow and the heat equation

The Dirichlet energy functional in $(L^2(\mathbb{R}^n),\mathcal{L}^n) $ is $$ D(u) := \dfrac{1}{2} \int_{\mathbb{R}^n} \|\nabla u(x)\|^2 \ dx.$$ How can I show that the gradient flow of the Dirichlet ...
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1answer
83 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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366 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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1answer
91 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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225 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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165 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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103 views

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
107 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
103 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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216 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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206 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
62 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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1k views

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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223 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 ...