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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ANSML - Proving of the matrix identity $\nabla_AtrABA^TC = CAB+C^TAB^T$

(ANSML is a tag I would like to use for Andrew Ng's Stanford Machine Learning - 2008) In this course, there were four matrix identities that I would like to prove. \begin{align} \nabla_a \text{tr}AB ...
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1answer
41 views

Find the largest range of values of the step size α for which the algorithm is globally convergent

Consider a fixed-step-size gradient algorithm applied to the following function f. $$ f(x)=1+2x_1+3(x_1^2+x_2^2 )+4x_1 x_2$$ Find the range of values of the step size α for which the fixed step ...
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36 views

Flows on a compact smooth manifold.

Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values ...
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3answers
92 views

Is the gradient of a convex paraboloid pointing up or down?

Intuitively the gradient is the vector pointing to the maximum rate of change. But this can be either up or down. How would the gradient point on this surface?
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1answer
53 views

Problem in Gradient operator and Kronecker delta function

I have this expression $$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$ Where $r$ is a distance. I tried this, but encountering manipulations of $\delta_{ij}$ with $\hat{r_i},\hat{r_{j}}$ and still ...
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34 views

Problem about Gradient Vectors

You are at the point corresponding to $(2,3)$ in a valley whose elevation at a point $(x,y)$ is given by the function $$h(x,y)=\frac{(x+y)}{(1+x^2)}$$ (Give your answers as vector for the following ...
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2answers
67 views

Show that $\nabla(fg)=f\nabla g+g\nabla f$ [closed]

This looks very much like the product rule to me. However, is this technically a valid answer to the question? How is it best answered?
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27 views

Find the maximum magnitude of the gradient

$e^{-(x^2 + y^2)}$ Trying to find the maximum magnitude of the gradient of the above. Have obtained the gradient vector $(-2 x e^{(-x^2-y^2)}, -2 y e^{(-x^2-y^2)})$ Not sure of next step besides ...
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1answer
44 views

How to find where the magnitude of the gradient of a function is maximized?

How to find where the magnitude of the gradient of ${{e}^{-({{x}^{2}}+{{y}^{2}})}}$ maximizes? I managed to calculate the magnitude - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
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1answer
23 views

Gradient as Normal Vector Problem

Here's my problem: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I know how to use the ...
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24 views

Gradient Vectors as Normal Vectors: How to Find D in Form Ax + By + Cz = D?

I've got a question that looks like this: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I ...
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0answers
16 views

Creating contour and gradient map

I have a requirement where, i have been given data set against X, Y Coordinate of a plane. This value is temperature at a point x,y. Now i am suppose to draw a graph with gradient color which displays ...
2
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0answers
28 views

notation question: What is $\nabla^{l}$ where $l\in \mathbb{N}$

Does $\nabla^{l}$ where $l\in \mathbb{N}$ mean anything? Could it mean dimension? We do have $\nabla^{2}=\Delta$, but I've never seen $\nabla^{3}$ . Found it in page 17 ...
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26 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
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19 views

Energy Function for Optimization with Time-Dependent Inputs?

I am working through a paper on energy functions for optimization and having some trouble understanding the notation. The author derives an E function for a neural network that is a function of both ...
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1answer
15 views

Reconstructing a 1D curve from normals/tangents?

I have a series of unit normals/tangents that are sampled at a regular intervals along the x dimension but I do not have their heights/y-component. For example: I would like to integrate the ...
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1answer
86 views

Norm of the gradient of function $f$ on manifold $g(x)=c$

Let $g,f:\mathbb{R}^2 \to \mathbb{R}$, $M=g^{-1}(c)$. Let say that we manage to write $f(x,y)=f_{*}(x)$ for $x\in M$. When I was calculating the square of norm of $$\nabla f_M (x,y)=\nabla ...
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38 views

how to find the maximum and minimum value of the directional derivative using Lagrange Multiplier Method?

I want to prove that the maximum value of $\frac{df}{ds}$ is $\left|\nabla f\right|$. To maximize $\frac{df}{ds}$ given by $\nabla f=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial ...
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0answers
23 views

Direction of gradient

If $\nabla f(1,0,0)=(-Ae^{bt},0,0)$ Is the direction of greatest increase at $f(1,0,0)$ $(-x,0,0)$? As long as $A$ and $e^{bt}$ doesn't change the sign? Since it says direction, I just normalise the ...
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1answer
25 views

Find the co-ordinates of the point on the curve

Calculate the points on the curve $y=(1-x)^4$ at gradient = -4 I solve little bit $\frac{dy}{dx} = 4(1-x)^{4-1}\cdot \frac{d}{dx} (1-x)\\ = 4(1-x)^3 \cdot (-1 )$ the gradient is =-4 so I put ...
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1answer
54 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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2answers
77 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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56 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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17 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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2answers
157 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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1answer
148 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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1answer
43 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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2answers
92 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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1answer
52 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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52 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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1answer
60 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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2answers
42 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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1answer
53 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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1answer
333 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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1answer
33 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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1answer
109 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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1answer
760 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 ...
3
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1answer
135 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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1answer
394 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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2answers
295 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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114 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 ...
2
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2answers
157 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 ...
2
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1answer
128 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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4answers
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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
254 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
478 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] ...
2
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1answer
67 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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2answers
2k 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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2answers
243 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 ...