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Questions tagged [gradient-flows]

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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Textbook Recommendations: Solving Systems of Matrix ODEs

This is in reference to the works of Trendafilov whose approaches to multivariate statistical problems boil down to solving a dynamical system involving matrices. Question: Can anyone suggest a book/...
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The gradient of a scalar function

I found this definition of gradient of scalar function $\Phi$: $\nabla \Phi = (g^{ij}\partial_{j}) \vec{g_{i}}$ And I know: Metric tensor of spherical coordinates $g_{11} = 1$ $g_{22} = r^2$ $g_{...
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How is the replicator dynamic a gradient flow of the Fisher information metric?

I am trying to understand how the replicator dynamic can be derived as a gradient flow of the Fisher information metric (aka Shahshahani metric). I have a question about understanding a particular ...
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How to using chain-rule to calculate the gradient in flow chart?

I have an data flow chart as follow The $a$ and $x_1,x_2,x_3$ are vector, $W$ is the matrix Output is the $$y = ((aW+x_1)W +x_2)W+x_3)$$ How to use chain rule to compute $\frac{dy}{dW}$? My ...
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Mazur's lemma without Hahn-Banach theorem/axiom of choice?

In the development of gradient-flow theory (in Hilbert-space $H$), we soon stumble on the question whether the function $u \mapsto \varphi[u]:=\frac{1}{2}\|u\|^2+I[u]$ -where $I:H \to \mathbb{C}$ is ...
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Least Energy Path, Contour Following, around Hills toward Goal

I have a matrix of elevation values which could be said represents $h(x,y)$. I can obtain contours using this function that are like sides of hills, and I have a starting point and an end point. How ...
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1answer
30 views

Inequality for gradients under different metrics

I would like to know if the following holds: Let $(M,g_1)$ be a smooth Riemannian manifold, $f:M \longrightarrow \mathbb{R}$ a smooth function with gradient $\nabla^1f$ and $x:\mathbb{R} \...
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Convergence of gradient flow

I'm interested in a simple property of the gradient flow $x'(t) = - \nabla f(x)$: under what conditions on $f$ does the gradient flow converge to a stationary point? In particular, I'm interested in ...
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Asymptotic behaviour of gradient flows for $t \to \infty$

I have often heard about the asymptotics of gradient flows converging to some "equilibrium point" as $t \to \infty$. This concept has come to my ear by word of mouth multiple times and is often ...
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gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
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Level sets of $C^1$ function on Riemannian manifold

Suppose a continuously differentiable function $f$ on a Riemannian manifold $M$, $f:M\rightarrow \mathbb{R}$, has a set of local minimum points $S$ with $f(x)=f(y)$, for all $x,y \in S$. Is it ...
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Taking divergence of the gradient of a scalar field that depends only on the position vector in $\mathbb{R}^3$.

The scalar field $f$ depends only on $r=|\underline{\mathbf{r}}|$ which is the position vector in $\mathbf{R}^3$ and I need to calculate the quantity, $$\nabla \cdot \nabla f$$ i went about ...
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why is my expectation of where the critical points are disagreeing with my graph?

given the parametrised function $$f_t(x,y,z)=z^3+tz-x^2+y^2$$ We know that that the gradient vector field is $\nabla f_t=(-2x,2y,3z^2+t)$ Now I originally thought that the critical points were at ...
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Clarification on Theorem 4.1 Milnor's lecture on h-cobordism theorem. Why The set $K_p$ is compact?

I need a clarification about a passage in the statement of Theorem $4.1$ of Milnor's Lectures on the H-cobordism theorem Suppose that for some choice of gradient-like vector field $\xi$, the ...
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a question about gradient

Let $f(x,y)=\ln\|\mathbf r\|$ where $\mathbf r=x\mathbf i+y\mathbf j$. Show that $\nabla f=\frac{\mathbf r}{\|\mathbf r\|^2}$. I attempt to calculate $\nabla\mathbf r$. But I have no idea how to ...
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Conditions for gradient dynamical systems

In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. 74, no. 1, pp. 199–206, 1961, four conditions on a vector field $X$ on a compact manifold $M$ are given that are sufficient for ...
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Help understanding how to compute Laplace operator?

I'm working on a diffusion simulation, and I am struggling with how to calculate the Laplace operator for my density function as it applies to Flick's second law. Conceptually I believe it is clear: ...
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Cauchy-Riemann Reparametizing level curves to gradient flow lines Using Picard-Lindelof

Problems in question here How can I prove that level curves of function V can be reparametrized to gradient flow lines of U? Any help would be appreciated and if anybody has any tips for part 3 would ...
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1answer
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$\int_V \left( g\nabla^2f-f \nabla^2g \right) dV=\int_S \left( g\nabla f-f \nabla g \right) \cdot u_n \, dS$ not depend on time

I' m wondering why the following relationship, known as Green's identity, doesn't depends on time. Let $f(x,y,z,t)$ and $g=\frac{e^{ikr}}{r}$ so $$\int_V \left( g\nabla^2f-f \nabla^2g \right) dV=\...
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(Negative) Gradient and Orientability of its flow.

Before asking my question, I put the necessary definitions and some context. If you are used with Morse Theory, you can skip the text within [[[...]]]. [[[Let me first define what I mean by gradient ...
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1answer
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Solving for policy gradient in reinforcement learning

I have been trying to learn about policy gradients in regards to reinforcement learning. I have run into this equation. $$ \nabla_\theta J(\theta) = E\left[\nabla_\theta \log\pi_\theta (s,a)Q^{\pi_\...
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1answer
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Finding a metric to write a vector field as a gradient of a given function

Let $M$ be a smooth manifold, and $f$ a smooth function with an isolated local minimum at $p$. Furthermore, let $X$ be a vector field vanishing at $p$ such that for some neighborhood $U$ of $p$, $df_q(...
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How do you know what the units of force are measured in when calculating work with a line integral?

I use an example from Lamar University they have a work function F(x) = kx and determined the spring constant k to be 400 so the work is equal to the integral of 400xdx, And the answer is 200x^2. ...
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Does normalized gradient flow preserve level sets?

Suppose that $f\colon \mathbb R^n\to \mathbb R$ is a $C^1$ function such that $\nabla f(x)\ne 0$ for all $x\in \mathbb R^n$. Consider the initial value problem $$ \begin{cases} \dot{u}=-\frac{\nabla ...
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1answer
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Is a level set more than one level curve?

What prompts me to ask this is i am trying to prove the gradient is perpendicular to the level set. I am using w = x^2 + y^2 . When I graph this I get a cup shaped figure in three dimensions and ...
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$H^{-1}$ norm and $H^{-1}$ gradient flow

I know how to calculate the $L^2$ gradient flow. But I also noticed some books mentioned the $H^{-1}$ gradient flow. For instance, the Dirichlet Energy $$E(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 ...
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Does the gradient point to the direction of greatest increase or decrease or the direction of greatest increase?

This is going to take a bit of explanation. I am not sure how to ask this question. The math lecture said the gradient points to the direction of greatest increase and if you want to know the ...
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2answers
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Why is the gradient vector zero inside a circular level curve

I've been trying to find an awnser to my question but can't seem to find it. Say we have a level curve graph with a circular level curve. Then, inside there is a critical point. I understand in the ...
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How can I solve the gradient here?

My teacher taught us what is the gradient of a function and what it means. But in this equation: df = ∇f dl , I dont know how to solve ∇f. My teacher put this Can anyone please teach me how to solve ...
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understanding the gradient of a curve through derivative

Let's say I have a curve of the form $$y = x^2$$ therefore the gradient is $$\dfrac{dy}{dx} = 2x$$ My question is : does this mean that the value of the gradient at $x = 3$ is $6$ or the change of $...
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Drawing Vector Fields (beginner)

I have $f(x,y) = xy$ which has the Gradient Vector Field F$(x,y) = (y,x)$ I believe the correct way to draw individual vectors is to map points into the formula, for example at point $(1,-3)$ we will ...
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Let $\phi : X \to (-\infty, \infty]$ be proper, l.s.c., bounded. Show: $\varphi_n(y) := \inf_{z\in X} \{\phi(z) + nd(y,z)\} \to \phi$.

I am currently working through a paper on Gradient Flows in Metric Spaces by Philippe Clément and need to verify the following. Let $(X,d)$ be a complete, metric space and $\phi : X \to (-\infty, \...
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Gradient flow of functional in a riemannian manifold

I somewhere read about gradient flows of functionals in a Riemannian manifold. I want to learn about them. A quick google search did not turn up anything useful. So i am looking for references where ...
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Determining the set of functions that share a minimum in a vector flow

Let $U = \{x \in \Bbb R^n : ||x-p|| < \varepsilon\}$ be the open ball with center p and $V: U \to \Bbb R^n$ a vector field with the property that starting at any point in $U$ it will eventually ...
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Modelling a gradient flow subject to geometric constraints

Part of the John ellipsoid problem asks the following. Let $K\in\mathbb{R}^n$ be a convex, symmetric (w.r.t origin), closed set. Find the inscribed ellipsoid of largest volume. The existence of such ...
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Circulation around a self-intersecting curve

Is computing circulation around this curve sensible? In a circuit, electric current flows through intersecting paths - but I haven't seen an instance where it flows within a closed, self-intersecting ...
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(RESOLVED) Nonsense circulation in a conservative vector field

Please refer to image below (plot link), Consider the vector field F. It is conservative - thus, the circulation for all simple closed curves (except those through the origin) is zero. Yet, visually,...
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Motion in 2D video sequence in computer vision [closed]

I would compute the gradient of a video sequence. I have $x$, $y$, $t$ dimensions so I would obtain $dx$, $dy$, $dt$ referred to the all video sequence. I know that I have to use a kernel filter to ...
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Weakening the Hypothesis in a question involving an invariant set of a differential equation flow

Consider $f_1,f_2,V\in \scr{C}^1(\Bbb{R}^2;\Bbb{R})$. Suppose that (H1) $\langle (f_1,f_2),\nabla V\rangle\leq 0$ in $\Bbb{R}^2$; (H2) $\nabla V(x)\neq 0$ for all $x\in \Bbb{R}^2$. ...
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If $p_0$ is a strict local maximum of $g$, then it is a center of the Hamiltonian System $p'=H_g(p)$

This is supposed to be an ordinary differential equations class exercise: Let $g\in \scr{C}^2(\Bbb{R}^2;\Bbb{R})$ and consider $$H_g(x,y)=\begin{pmatrix}\partial_yg(x,y)\\ -\partial_xg(x,y)\end{...
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Using Flows to prove homogeneity of Connected Manifolds

Show that if $X$ is a connected manifold then for any $x,y\in X$, there exists a diffeomorphism $h$ sending $x\xrightarrow{h}y$. This is what I have so far: Given a connected manifold $X$, for any $x,...
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Parallel Transport Equations

I have a question about parallel transport that I'm very confused about and would appreciate some help. The question reads: What vector field $X$ on the unit 2-sphere in $\mathbb{R}^3$ has ...
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Lagrange - intuitive understanding of formula for multiple constraints

Say there was a function $f(x, y, z)$ constrained by $g(x, y, z) = c$. I understand intuitively/spatially/visually why at the max or min, $\nabla f$ and $\nabla g$ are parallel. (I saw a very good ...
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Find the equation of the tangent to the curve

Find the equation of the tangent to the curve: $$y = e^x +1$$ At the point: $$(1, e+1)$$ My process: $$Gradient: y' = e^x$$ Tangent: $$y-(e+1) = e^x(x-1)$$ $$y= xe^x-e^x+e+1$$ I don't ...
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How can entropy be concave in time?

At this point in a lecture by Cedric Villani, prof. Villani talks about his work revealing that "if entropy is a concave function of time, then the Ricci curvature is non-negative." But wait, how can ...
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Normal of Implicit Surface

I am trying to understand the reasoning behind finding what the normal of an implicit surface is. I found this article on the web: http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node27.html ...
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An analogue of Willmore flow for Gaussian curvature

Consider an embedded surface in $\mathbb R^3$ homeomorphic to a disk. If we wanted to find a similar surface that is minimal, i.e. has mean curvature $H=0$ everywhere, we could hold the boundary fixed ...
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Minimizing geodesic distance on cylindrical manifold

I will try to explain my question via a trivial analogy. Let $a$,$b$ be two points in 2D Euclidean space. I would like to make point $a$ closer and close to $b$. Let $L=||a-b||^2=(a_x-b_x)^2+(a_y-...
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1answer
78 views

Gradients “backward flow” calculation rulls

I am reading article Hacker's guide to Neural Networks. . Becoming Becoming a Backprop Ninja section The quote: lets just use variables such as a,b,c,x, and refer to their gradients as da,db,dc,...
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763 views

Why does the gradient point at direction of maximum slope only?

I'm just learning about gradient and I'm a bit hazy about this: In the above diagram, the length of the line represents magnitude of slope (change along z) in that direction. According to my ...