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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17 views

What's the equation of the gradient descent path in an elliptical field from a starting point (x,y)?

The following figure is generated from $f(x,y)$ = $x^2\over36$ + $y^2\over 16$. The black trajectory is the gradient descent trajectory from $(5,5)$ to $(0,0)$. Is there an equation for this ...
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13 views

Gradient flow remains on manifold

Consider a gradient flow $x(t) = -\nabla f(x)$ with the property that for some manifold $M$, $x(0) \in M$ and for all $x \in M$, $\nabla f(x) \in T_x(M)$ where $T_x(M)$ denotes the tangent space at $x$...
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16 views

Spherical Gradient

I was reading some physics when I read this particular paragraph: enter image description here I understand that along $\hat \theta $, $ds=rd\theta$ and also the preceding argument about $dr$ but I ...
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Computing $L^2$ Gradient Flow for a “compactness energy” over a weighted graph.

I'm reading the following paper on applying Mean Curvature flow to gerrymandering: https://www.math.ucla.edu/~majaco/papers/gerrymandering.pdf The setting is that we have a weighted graph of vertices,...
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15 views

Gradient of Column Vector

I saw the gradient of a function, $f(x,y) = \| x - y \|^2$ as $\nabla f_x = 2x^T(x-y)$ where $x$ and $y$ are column vectors. I really would love to know the Mathematics behind this. Also, I thought ...
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24 views

Two Questions: How do I compute this Directional Derivative and interpret the given contour plot for critical points?

So I am completely lost right now and don't exactly know what my professor is asking me. For Problem 3) I figured if I am applying Directional Derivative twice on a function f, then: $ D_u(D_uf) ...
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33 views

How do you call a function that is the gradient of a function?

Let us consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that there exists a real-valued function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ that satisfies $\nabla F(x) =f(x)$. How would you ...
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11 views

What is the gradient at a variable wrt another if it is only partially undefined?

I have come across the following example: Take $$ \text{ReLU}(x) = \text{max}(0,x) \\ f(a, b) = (r(a*0+b*1)-1)^2+(r(a*1+b*1)-0)^2 $$ I am trying to find $f_a'(1,0)$ and $f_b'(1,0)$. The problem here ...
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18 views

How to prove that a smooth vector field on a compact manifold with boundary that is interiorly directed is positively invariant?

I have begun self-teaching some basic concepts on dynamical systems for a paper I am reading. There are a few claims that I am having trouble finding proofs for online. They are intuitively obvious, ...
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67 views

Gradient of $g(t):=f(tx+(1−t)y)$

I want to find the gradient of $g(t)=f(tx+(1−t)y)$, where $f$ is a single valued function, but I'm unable to do so. My approach to this: $$g\prime(t)=<\nabla f((tx+(1−t)y),x−y>$$ $$g\prime\prime(...
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105 views

Singularities in conservative vector fields

I am trying to compute $\oint\limits_C m(x,y) dx + n(x,y) dy$ where $F(x,y)=\langle m(x,y), n(x,y) \rangle = \langle \frac{\cos{(\ln{(xy)})}}{x}, \frac{\cos{(\ln{(xy)})}}{y} \rangle $ and $C$ is the ...
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24 views

About push-forwards of distributions

Consider being given a random variable $X$ and $f$ is a bijective function. Suppose $p_X$ and $p_{f(X)}$ denote the p.d.fs of $X$ and $f(X)$. Can someone kindly explain as to when is the following ...
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32 views

Divergence theorem with some assumptions

Problem: Let $S$ be a closed oriented surface in $\mathbb{R}^3$ and $D$ be the (simple connected) region enclosed by $S$. Let $f$ be a smooth function such that $f(x,y,z)\neq 0$ in $\mathbb{R}^3$. ...
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Slope of lower semicontinuous geodesically convex functional

Let $(X,d)$ be a complete separable metric space and let $E: X \to \mathbb{R} \cup \{+ \infty\} $ be a lowersemicontinuous functional s.t $$D(E):= \{ x \in X \mid E(x) < +\infty\} \ne \emptyset.$$ ...
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20 views

For differentiable functions, bounded on the unit disc, how large can $\inf \{|\nabla f(x)|: |x| \leq 1 \}$ get?

Let $B_1(0) := \{(x,y) \in \mathbb{R}^2: x^2 + y^2 \leq 1\}$ be the unit disc in $\mathbb{R}^2$. What is the infimum of all $k$ such that for all $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ which are (...
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180 views

Is a path which decreases a function in the quickest way a gradient flow?

Let $U \subseteq \mathbb{R}^n$ be open, and let $F:U \to \mathbb{R}$ be a smooth function. Fix a point $p \in U$, and suppose that $\nabla F(p) \neq 0$. Let $\alpha(t)$ be a $C^{\infty}$ path ...
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74 views

Reeb foliation of the plane and the Palais-Smale condition

Definition. Let $X$ be a manifold. A smooth map $f:X\to \mathbb R$ is said to satisfy the Palais-Smale condition over $y\in \mathbb R$ if whenever $y_n\to y$, any sequence $x_n\in f^{-1}(y_n)$ such ...
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19 views

How to compute the derivative of a function whose parameters are on a high-dimensional sphere?

$F(w)$ is a function whose parameters $w=(w_1,w_2,...,w_n) \in \mathbb{R}^n$, and $\Vert w\Vert_2 = 1$. Actually I am considering the gradient flow $\frac{dw}{dt} = -\frac{dF(w)}{dw}$ s.t. during ...
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44 views

Prove that: $\nabla f(x,y)=\frac{\partial f(x,y)}{\partial \vec{u}}\vec{u}+ \frac{\partial f(x,y)}{\partial \vec{v}}\vec{v}$

Let $ f $ be differentiable and be $\vec{u}$ and $\vec{v}$ two vectors of $\mathbb {R}^2$, unit and orthogonal. Prove that: $$\nabla f(x,y)=\frac{\partial f(x,y)}{\partial \vec{u}}\vec{u}+ \frac{\...
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210 views

Are gradient flows the quickest way to minimize a function for a short time?

Let $F:\mathbb{R}^n \to \mathbb{R}$ be a smooth function, and let $p \in \mathbb{R}^n$. Let $\alpha(t)$ be the solution to the negative gradient flow of $F$, i.e. $$ \alpha(0)=p, \, \, \dot \alpha(t)...
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48 views

Chain rule on gradient of two vectors (Momentum Equation)

I am confused about the application of the chain rule with the following equation: $$\boldsymbol{\nabla} \cdot (\rho \mathbf V \mathbf V)$$ where: $\mathbf V = u \hat {\mathbf i} +v \hat{\mathbf j}...
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39 views

Which way does the gradient point?

I'm extremely confused about which way the gradient points. If given the equation $z = x^3+y^3-6xy$, I could calculate the gradient at point (1, 2, -3) by rearranging the equation to $f(x,y,z) = x^3+y^...
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42 views

Finding the entropy

Let $M$ be a compact manifold and $F:M\to \mathbb{R}$ Morse function. Let $\phi_t$ be a flow generated by $F$ in the following way: $\frac{d\phi _t}{dt}=-\nabla F(\phi _t)$. Let $f=\phi _1$. Find $h(f)...
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102 views

State-of-the-art techniques for maximizing a differentiable function with multiple local maxima

Suppose I want to find the global maximum of a differentiable function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ with multiple local maxima, where $n$ is on the order of $100$. I have access to both $...
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48 views

Field line dynamics of gradient fields on $S^2$

I am reading parts of the book: Hasselblatt B., Katok A. (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press. In Exercise 1.6.2 we are asked to find a compact, ...
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46 views

Are levelsets of a scalar field without critical points a foliation?

Let $f$ be a differentiable scalar field on an $n$-dimensional Riemannian manifold $X$ without critical points, i.e. $\nabla f \neq 0$ everywhere on $X$. (Assuming $X$ has properties as required to ...
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Homotopy of integral curves of a gradient field preserves levelsets (?)

Given a differentiable scalar field $f$ on a Riemannian manifold $X$ (with properties as required) I would like to formulate a homotopy of maximal integral curves $\gamma, \tilde{\gamma}$ of the ...
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256 views

Tricky surface integral of vector field

We have the embedded surface $S= \{(x,y,z)\in \mathbb{R}: z = e^{1-(x^2 + y^2)^2}, z>1\}$ and the vector field $\mathbf F:\mathbb{R}^3\to \mathbb{R}^3; (x,y,z)\mapsto (x e^{y^2}, 2ye^{x^2}, 5-3z) $....
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441 views

Solution of a coupled gradient system?

Suppose $U \times V \subset \mathbb R^n \times \mathbb R^n$ is an open set and $\phi, \psi: U \times V \to \mathbb R$ are two $C^{\infty}$-smooth functions. Furthermore, for each $y \in V$, $\phi_y: U ...
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50 views

Homotopy type of transversal submanifolds through deformation

Let $A,B \subset M$ be two transversal submanifolds of a compact manifold $M$. It seems rather intuitive that if $A$ and $B$ are deformed (say smoothly) in a way that they remain transversal to each ...
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34 views

Why is the gradient summed in a computational graph for an operation with split output?

I was looking at back propagating a gradient through a computational graph, and all makes sense aside from when a node has multiple outputs. Take the following function: $$f(x) = (x+3)(x+2)$$ Which ...
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69 views

Morse deformations for a family of functions on a manifold with boundary

Let $F_s : M \to \mathbb{R}$, $s \in [0,1]$ be a family of smooth functions on a compact manifold $M$ with boundary $\partial M$. Suppose that for any $s \in [0,1]$, $0$ is a regular value of the ...
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59 views

Existence of a function with $||grad f||>\epsilon$

I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
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37 views

Legendre-Fenchel conjugate: $\dot x \in \operatorname{dom}(\Psi^*), -\operatorname{D}\mathcal E(x) \in \partial \Psi^*(\dot x)$

Consider $\mathcal E: \Bbb R \to \Bbb R$ lower semi-continuous with $\mathcal E(x) \to \infty$ if $|x| \to \infty$ and the equation $\dot x = -\nabla \mathcal E(x)$. Apparently, this differential ...
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73 views

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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61 views

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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38 views

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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1answer
80 views

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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1answer
147 views

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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99 views

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
43 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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242 views

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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1answer
154 views

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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127 views

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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48 views

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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600 views

How to solve $\dot{x} = \frac{f(x)}{\|f(x)\|}$?

How to solve the following ODE? $$\dot{x} = \frac{f(x)}{\|f(x)\|},$$ where $x : \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f : \mathbb{R}^n \to \mathbb{R}^n$ ...
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1answer
265 views

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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1answer
42 views

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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37 views

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 compact ...
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44 views

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 ...