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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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Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
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Multivariate Chain Rule with Derivatives in Intermediate Functions

I have a function $$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$ where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
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Integral by parts on a scalar field over a curve

Let $M$ be a compact and boundaryless Riemannian manifold. Take $f\in C^\infty(M)$ and let $T_s(x)=\exp_x(s\nabla f(x))$ be its gradient flow. I have proven for my specific case that $$\int_0^1\Delta ...
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Derivative of a gradient flow $T(x)=\exp_x(\nabla\psi(x))$

Consider a gradient flow $$T(x)=\exp_x(\nabla\psi(x))$$ on a Riemannian manifold $M$, with $\psi\in C^\infty(M)$. What is the derivative of this flow? I mean, if $X\in\mathfrak{X}(M)$, who is $$dT_p(...
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How to verify whether a direction is the steepest descent, in multi-variables case?

Consider the following energy function $$-\sum_{i<j\in[n]}\cos(\theta_i-\theta_j)$$ where $\theta_i\in\mathbb{R}$, for $\forall i\in[n]$. At any vector $\vec\theta$ that is not all-equal vector ($\...
chloe's user avatar
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Morse flow: cancelling handle pairs away from deformation retract

Given a smooth manifold (not closed, maybe with boundary) $M$ in $R^n$, take a section with a hyperplane $H$ of some dimension $d$. Assume that $M$ has $M\cap H$ as deformation retract. For example, a ...
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Discretization Error of Mirror Descent

It is well known that for sufficiently differentiable functions $f$ and small $\eta>0$ the iterate given by gradient descent $$ x_{k+1}=x_k-\eta \nabla f(x_k)$$ is within $\mathcal O(\eta^2)$ of ...
Small Deviation's user avatar
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60 views

Flow is defined using a $C^1$ function is $C^1$

I have a flow defined by the initial value problem: $$\frac{d}{dt}y_t(x)=f_t(y_t(x)), \quad y_0(x)=x$$ where $f_t:\mathbb{R}^k\rightarrow\mathbb{R}^k$. I know the above problem has a unique solution ...
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Using Gronwall to prove bi-Lipschitz

I am working through a proof which, for fixed $x \in \mathbb{R}^k$ considers an initial value problem of the form: $$\frac{d}{d t}u_t(x)=v_t(u_t(x)), \quad u_t(0)=x$$ where $u_t:\mathbb{R}^k \...
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How do we show that narrow convergence is only needed on a spanning uniformly dense subset

At the beginning of Chapter 5 of Ambrosio's Gradient Flows book, he introduces the idea of a narrowly convergent sequence of measures as... "A sequence $(\mu_n) \subset \mathcal{P}(x)$ is ...
IdenticallyEulerian's user avatar
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Existence of projection $P$ equivalent to $P' \circ T \circ P''$ for projections $P', P''$ and smooth translation $T$?

A projection in the linear algebraic sense is a linear map $P$ such that $P^2 = P$. I'm interested in knowing when there is guaranteed to exist a projection $P$ such that $P = P' \circ T \circ P''$, ...
Tanishq Kumar's user avatar
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1 answer
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Are linear interpolation curves on Wasserstein spaces absolutely continuous?

Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric. Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$ ...
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(Explaination of) Proof of Global Stable Manifold Theorem in Audin & Damian's book

I am reading Michèle Audin and Mihai Damian's book $\textit{Morse Theory and Floer Homology}$ and I stick at one sentence regarding Global Stable Manifold Theorem which states $W^s(a)$(that is, the ...
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Gradient flow of Nesterov accelerated gradient methods

I am reading a nice paper [1] that gives a differential equation for NAG methods. The updating rules of NAG are: $$x_k = y_{k-1} - \eta \nabla f(y_{k-1}) \tag{1}$$ $$y_k = x_k + \frac{k-1}{k+2}(x_k - ...
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L2 gradient solution time

There is something I don't understand. Imagine I want to solve : $$Min_{u}\int_{B(0,1)} \left| \nabla u(x) \right|^{2}+F(u(x))dx$$ It's L2 gradient flow is given by : $$\partial_{t} u =2 \Delta u - F^{...
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1 vote
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Help with an article on convex-splitting method

I trying to simulate gradient flow $$\frac{\partial u}{\partial t}=-\nabla_x F(u)$$ where F(u) is (in my case): $$F(u)=2u^4-u^2$$ for u $\in [-0.5,0.5]$. Since the equation is non-linear, I would like ...
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5 votes
1 answer
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Solving $\partial_t \gamma_t(x) = - \gamma_t(x) + \frac{\gamma_t''(x)}{\gamma_t'(x)^2}$, a nonlinear PDE on quantile functions

While pondering Wasserstein-2 gradient flows of the Kullback-Leibler divergence functional $\text{KL}(\cdot \mid \nu)$, where $\nu \sim \mathcal N(0, 1)$ is the standard normal distribution (yes, I ...
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About minimizing dynamic of a convex function

Suppose that we have a strictly convex function $f: \mathbb R \rightarrow \mathbb R$ that admits a unique minimizer $x^*$ and $f(x^*)=0$. Let $x_0< x^*$, clearly $f$ is decreasing on $[x_0,x^*)$ ...
Jeffrey Jao's user avatar
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Structure of the solutions of the matricial ODE: $\dot U(t) = -U^2 - BU$.

Let $A$ and $B$ be a fixed $n \times n$ positive semi-definite matrices. Given an $n \times n$ matrix $U$, define another $n \times n$ matrix by $F(U) := U^2 + U B$. Consider the ODE $$ \dot U(t) = -F(...
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Preconditioned gradient flow keeps boundedness

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a real analytic function with $\inf_{x\in\mathbb{R}^n} f(x) > -\infty$. If we know the solution $x:[0,\infty)\rightarrow\mathbb{R}^n$ to the vanilla ...
William's user avatar
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Bounded gradient flow solutions without compact sublevel sets

A standard result in the theory of dynamical systems states that for a continuous system $(*)$ defined by $\dot{x}(t) = F(x)$, where $ F : \mathbb{R}^n \rightarrow \mathbb{R}^n $ locally Lipschitz, if ...
Andreea M's user avatar
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1 answer
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Finding the Direction of Steepest Descent in a Temperature Field Restricted to a Plane

my question is: Consider a temperature function in space given by $$R(x,y,z) = \frac{1}{(x-1)^2 + (y-1)^2 + (z-3)^2 + 1}$$ A bird is located at the point $(-1, 1, 3)$ and wants to minimize radiation ...
Saiko's user avatar
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2 votes
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125 views

Examples of a gradient flow

Suppose we have a gradient flow in $\mathbb{R}^n$ : $$\frac{d}{dt}x(t)=-\nabla F(x(t)), \qquad x(0)=x_0.$$ where $F : \mathbb{R}^n \to \mathbb{R}$ and $x : \mathbb{R}_+ \to \mathbb{R}^n$. What are ...
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How to apply Gronwall's inequality to get the desired claim?

Let $H\colon U\to\mathbb{R}$ with $U\subset\mathbb{R}^n$ be smooth and suppose $H$ has a unique minimum $u_\infty\in U$. Moreoever, suppose that for some $\lambda>0$, we have that $H$ is strong ...
selector's user avatar
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5 votes
2 answers
154 views

closed-form Newton flow of tanh(ln(1+x^2))

The differential equation for the Newton flow $z (t)$ of $f (t)$ is given by \begin{equation} \dot{z} (t) = - \frac{f (z (t))}{\frac{d}{d t} f (z (t))} = - \frac{f (z (t))}{\dot{f} (z (t))} \end{...
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5 votes
2 answers
159 views

Is the Lambert W function the Newton flow of the exponential function?

Is this right? The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$. The continuous Newton's method ...
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130 views

Advection reaction equation is solved by projection of solution of continuity equation

Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation $$ \label{eq:CE} \tag{CE} \partial_t \mu_t = \...
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1 vote
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Finding the gradient of the squared $L_2$ matrix norm during constrained optimization

I'm reading this paper for a university project, and I'm currently stuck on part B.2 of section 3 where the goal is to minimize $$ \frac{1}{N}\sum_ne^{-\rho y_nf(V;x_n)}, $$ with respect to the ...
Jaime Zamora's user avatar
1 vote
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Is Divergence without a dot product symbol a gradient on vector?

Divergence operating on a vector field ($\mathbf{u}=[u, v, w]\in\mathbb{R}^3$) outputs a scalar field: $\nabla\cdot \mathbf{u} = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\...
MerelyLearning's user avatar
3 votes
1 answer
122 views

Stationary distribution of gradient-biased random walk

I'm looking for a result that I suspect should be fairly standard and well-known to probabilists and statistical physicists (and perhaps numerical/financial analysts). The result in question should ...
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How to show that $\int_0^\infty (\partial_t \phi(x_t, t)+\left\langle \dot x_t, \nabla_x \phi(x_t, t)\right\rangle) \mathrm d t=0$?

Let $u :\mathbb R^d \to \mathbb R$ be smooth and $v := \nabla u$ its gradient. We fix $x_0$ and let $x_t$ be the solution of $\dot x_t = v(x_t)$ started from $x_0$. Let $\rho_t := \delta_{x_t}$ for ...
Akira's user avatar
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3 votes
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What does the equation $s=-\nabla \cdot(\rho \nabla u)$ mean?

I'm reading about gradient flows in Wasserstein space in this note. Let $\mathcal{P}_2 (\mathbf{R}^d)$ be the set of probability measures with finite second moment. In [Ott01] the tangent space $T_\...
Akira's user avatar
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3 votes
0 answers
91 views

Is there a Stokes-like theorem for gradient of a vector?

Cosidering the usual central field in physics: \begin{equation} \mathbf{V}=\frac{\hat{r}}{\mathbf{r}^2}. \end{equation} I can get that divergence and curl (and therefore vorticity) are zero when $...
Erick Pastén's user avatar
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1 answer
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Gradient flows: is this a typo in a discretization scheme?

I'm reading below theorem from this lecture note. Theorem 4.5. Let $H$ be a Hilbert space over $\mathbb{R}$. If $\varphi: H \rightarrow \mathbb{R}$ is differentiable and convex, then for every $u \in ...
Akira's user avatar
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4 votes
1 answer
76 views

When do solution to differential equations belonging to the same parametric family intersect?

General Problem I am interested in studying whether solutions to the Fokker-Planck equation: \begin{equation} \tag{1}\label{fp} \frac{\partial p(x, t)}{\partial t} = \textrm{div}(-p(x, t)\nabla\log q(...
pglaser's user avatar
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2 votes
0 answers
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Gradient flows: Stability of minimizers in infinite dimensions

Suppose I have an $L^2$ gradient system $\dot{x}=-\nabla V(x)$, where $V:H\rightarrow\mathbb{R}$ is an analytic potential on an infinite dimensional Hilbert space $H$. Suppose also that I have ...
sobol's user avatar
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0 answers
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Proof of equivalence of solutions to initial value problem given same gradient

I have two functions $x(t)$ & $y(t)$ I have $x(0) = y(0)$ and that $\dot{x} = f(x,y)$ & $\dot{y} = g(x,y)$ I have that when $x = y$, $\dot{x} = \dot{y}$. I want to show that $x(t) = y(t)$, I ...
Governor's user avatar
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1 answer
51 views

Distance between solutions of two first-order ODEs

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a twice-continuously differentiable function, and $\varphi:\mathbb{R}\to\mathbb{R}$ be a smooth concave function. For a given initial condition $x_0\in\mathbb{R}^...
Kas's user avatar
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1 vote
1 answer
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How to calculate the gradient of $\mathbf{x}^T\mathbf{W}^2(\mathbf{W}^2)^T\mathbf{x}$ w.r.t. $\mathbf{W}$?

I need to calculate the gradient of $\mathbf{x}^T\mathbf{W}^2(\mathbf{W}^2)^T\mathbf{x}$ w.r.t. $\mathbf{W}$. Here is what I have tried. Let $A=W^2$, then the form reduces to \begin{align*} \frac{\...
suineg's user avatar
  • 385
4 votes
1 answer
224 views

Gradient flow around degenerate critical point

Consider a $C^2$ function $f \colon \mathbb{R}^n \to \mathbb{R}$ and suppose that $\nabla f(0) = 0$ and $\nabla^2 f(0) = 0$. Suppose also that $0$ is an isolated critical point (that is, there is a ...
Sap's user avatar
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2 votes
1 answer
414 views

Heat equation as gradient flow of Dirichlet energy

I am looking for a reference which rigorously explores the heat equation as a gradient flow of the Dirichlet energy (say in $L^2$ ? or some other inner-product space). I don't know this literature ...
trenkoir viske's user avatar
5 votes
0 answers
181 views

Linearization of Gradient Flow

As someone who has only "theoretical" knowledge in Riemannian geometry, I have a hard time trying to wrap my head around how to actually compute the so called "linearization" of a ...
Nuke_Gunray's user avatar
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0 votes
1 answer
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Deduce stability from strict convexity of gradient systems

Given a twice differentiable function $f(x):\mathbb R^n\rightarrow \mathbb R$, and its corresponding gradient system $$\dot x=-\nabla f(x)$$ My question is: If $f(x)$ is strictly convex, can we ...
chloe's user avatar
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3 votes
1 answer
119 views

Globally asymptotic stable gradient system has unstable point

Given a gradient system $$\frac{d\theta_1}{dt}=-\sin(\theta_1-\theta_2)$$ $$\frac{d\theta_2}{dt}=-\sin(\theta_2-\theta_1)$$ The system is a gradient system since $$\frac{d\vec \theta}{dt}=-\nabla V(\...
chloe's user avatar
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-1 votes
2 answers
60 views

Find counterexample of gradient system with non globally convex energy function has globally asymptotic stable equilibrium point?

Given $$\dot x=-\nabla f(x)$$ and suppose it has an equilibrium point $x=0$. It is known that if $f(x)$ is globally convex, then $x=0$ is globally asymptotic stable. I am interested its converse: can ...
chloe's user avatar
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1 vote
1 answer
126 views

Prove stability by convexity of energy function of gradient system

Given a 1-dimensional gradient system which has one equilibrium point $x=0$ $$\frac{dx}{dt}=-f'(x),x\in R$$ where $f'(x)$ denotes $\frac{df(x)}{dx}$. Let $f(x)$ be a convex function, i.e. $f''(x)>0$...
chloe's user avatar
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4 votes
1 answer
637 views

The lyapunov function of gradient system

Given a dynamical system $$\frac{dx}{dt}=-\nabla f(x)$$ which $x=0$ is the only equilibrium point, i.e. $-\nabla f(x)|_{x=0}=0$. I am reading this tutorial, and it states: $f(x)$ is a lyapunov ...
chloe's user avatar
  • 1,052
1 vote
1 answer
257 views

Does the Hessian matrix of energy function of a gradient system have to be positive semidefinite when the system has one globally stable point?

Given a gradient dynamical system $$\frac{d\theta_i}{dt}=f_i(\theta_1,\cdots,\theta_n),\forall i\in\{1,\cdots,n\},$$ where $$\frac{\partial G}{\partial \theta_i}=f_i(\theta_1,\cdots,\theta_n),$$ where ...
chloe's user avatar
  • 1,052
1 vote
1 answer
1k views

Find flux of the vector field $\vec A$ through the surface of a sphere with radius R and center on the origin

The vector field is given by the spherical coordinates $\vec A = kr^3\,\vec {e}_r$, where $k$ is a constant and $r=\sqrt {x^2+y^2+z^2}$. I thought about using the Gauss-Ostrogradski theorem, where the ...
bia's user avatar
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3 votes
1 answer
1k views

What do you call third order derivative matrix and what does it geometrically signify?

The first-order derivatives matrix is known as Jacobian, gives the gradient of the graph. Similarly, the second-order derivatives matrix is Hessian, which gives the curvature of the plot. What next? i....
Sachin's user avatar
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