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 ($\...
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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 ...
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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 ...
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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 $P = P' \circ T \circ P''$, ...
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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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