Convergence of gradient flow
$begingroup$
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 the simple case of $f : mathbb{R}^n rightarrow mathbb{R}$, where we know that $f$ obtains a global minimum somewhere. No assumption of convexity, we allow for arbitrarily many local minima and/or multiple global minima, saddle points, etc. It is fine if part of the conditions on the function require an assumption on the nature of these critical points (e.g. that $f$ has no degenerate saddle points, or something like that). I know that gradient flows are quite general, and PDE's can in many cases be cast as gradient flows on functionals, but I'm not interested in this level of abstraction at the moment - I'm thinking more about nonlinear optimization.
I've done some looking around on the internet, but most discussions get into Morse theory, with which I have no experience, and so I've had some difficulty understanding the jargon. In addition to an answer or simple categorization of the relevant functions (if one exists), I would much appreciate a reference where I could read more about this (and in particular cite at some point) - it's okay if the reference is a bit more technical, or is on Morse theory. It's very possible that the tools of Morse theory are necessary to answer this type of question
ordinary-differential-equations differential-geometry riemannian-geometry gradient-flows
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
$endgroup$
add a comment |
$begingroup$
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 the simple case of $f : mathbb{R}^n rightarrow mathbb{R}$, where we know that $f$ obtains a global minimum somewhere. No assumption of convexity, we allow for arbitrarily many local minima and/or multiple global minima, saddle points, etc. It is fine if part of the conditions on the function require an assumption on the nature of these critical points (e.g. that $f$ has no degenerate saddle points, or something like that). I know that gradient flows are quite general, and PDE's can in many cases be cast as gradient flows on functionals, but I'm not interested in this level of abstraction at the moment - I'm thinking more about nonlinear optimization.
I've done some looking around on the internet, but most discussions get into Morse theory, with which I have no experience, and so I've had some difficulty understanding the jargon. In addition to an answer or simple categorization of the relevant functions (if one exists), I would much appreciate a reference where I could read more about this (and in particular cite at some point) - it's okay if the reference is a bit more technical, or is on Morse theory. It's very possible that the tools of Morse theory are necessary to answer this type of question
ordinary-differential-equations differential-geometry riemannian-geometry gradient-flows
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
$endgroup$
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15
add a comment |
$begingroup$
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 the simple case of $f : mathbb{R}^n rightarrow mathbb{R}$, where we know that $f$ obtains a global minimum somewhere. No assumption of convexity, we allow for arbitrarily many local minima and/or multiple global minima, saddle points, etc. It is fine if part of the conditions on the function require an assumption on the nature of these critical points (e.g. that $f$ has no degenerate saddle points, or something like that). I know that gradient flows are quite general, and PDE's can in many cases be cast as gradient flows on functionals, but I'm not interested in this level of abstraction at the moment - I'm thinking more about nonlinear optimization.
I've done some looking around on the internet, but most discussions get into Morse theory, with which I have no experience, and so I've had some difficulty understanding the jargon. In addition to an answer or simple categorization of the relevant functions (if one exists), I would much appreciate a reference where I could read more about this (and in particular cite at some point) - it's okay if the reference is a bit more technical, or is on Morse theory. It's very possible that the tools of Morse theory are necessary to answer this type of question
ordinary-differential-equations differential-geometry riemannian-geometry gradient-flows
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
$endgroup$
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 the simple case of $f : mathbb{R}^n rightarrow mathbb{R}$, where we know that $f$ obtains a global minimum somewhere. No assumption of convexity, we allow for arbitrarily many local minima and/or multiple global minima, saddle points, etc. It is fine if part of the conditions on the function require an assumption on the nature of these critical points (e.g. that $f$ has no degenerate saddle points, or something like that). I know that gradient flows are quite general, and PDE's can in many cases be cast as gradient flows on functionals, but I'm not interested in this level of abstraction at the moment - I'm thinking more about nonlinear optimization.
I've done some looking around on the internet, but most discussions get into Morse theory, with which I have no experience, and so I've had some difficulty understanding the jargon. In addition to an answer or simple categorization of the relevant functions (if one exists), I would much appreciate a reference where I could read more about this (and in particular cite at some point) - it's okay if the reference is a bit more technical, or is on Morse theory. It's very possible that the tools of Morse theory are necessary to answer this type of question
ordinary-differential-equations differential-geometry riemannian-geometry gradient-flows
ordinary-differential-equations differential-geometry riemannian-geometry gradient-flows
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
edited Jan 23 at 19:22
The Wind-Up Bird
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
asked Jan 23 at 19:08
The Wind-Up BirdThe Wind-Up Bird
240125
240125
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15
add a comment |
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3084920%2fconvergence-of-gradient-flow%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3084920%2fconvergence-of-gradient-flow%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
If there is an $x_{infty} in mathbb{R}^n$ with $displaystyle lim_{t rightarrow infty} x(t) = x_{infty}$, then we must have $nabla f(x_{infty}) = 0$, i. e. $x_{infty}$ must be a stationary point of $f$. Isn't that right? This doesn't answer the question, but just to clarify.
$endgroup$
– Christoph
Jan 24 at 5:47
$begingroup$
I would also say that convergence/divergence depends a lot on the initial value of $x$, not just on $f$.
$endgroup$
– Christoph
Jan 24 at 6:24
$begingroup$
For a sufficient condition, see Hale and Massatt's Asymptotic behavior of gradient-like systems. Also, see Thm. 4.3 in Hale's Topics in Dynamic Bifurcation Theory, or Hale and Raugel's Convergence in gradient-like systems with applications to PDE.
$endgroup$
– user539887
Jan 24 at 11:15