/sci/ - Science & Math

Gay and useless. All the real research is done on FDEs now. Fractional Differential Equations. Get with the times, dummy.

>>14879569
Literally every topic posted is an active field of research. Stop trying to troll and derail

>>14879634
>REEEE if i disagree with you, you're trolling
thread was doomed from the onset due to a faggot op

You must be confused, OP. This is /sci/. We post racism and frogs here.

>>14880676
Apparently thats the case, and then you make a high effort post and the only posts are some dude trying to derail or whatever.
Damn I will honestly just stop coming here. Theres bound to be a good anonymous place for discussion somewhere in the internet

Is there any way to represent the entirety of starcraft as a solvable PDE?

>>14881190
The video game? What in the fuck is this question? Explain yourself

>>14879555
PDE, or
>How I stopped worrying and learned to love spectral theory
Or even
>how I stopped worrying and learned to love the computer

>>14879555
any galerkin chads here? weak forms gets me hard.

How do I into pde's? I know a lot of analysis (graduate measure/functional/etc) but my undergraduate PDE class seemed to just be "apply this very specific trick to this very specific PDE" times 50, that I have to remember, and it's put me off on enjoying the field.
If I take one or more graduate courses does this change into things I know (operators, spectral theory stuff etc) or do the tricks just get more convoluted?

>>14881684
Well I mean, the way we mostly handle PDEs is employing a trick to turn them into ODEs then proceed from there.

>>14879555
OP it’s late here so I will have to come back to this but bless you for making /pdeg/. I’ve got some SDE content I’ll post tomorrow

Oh also going to suggest Peter LAX’s Lin Alf and Funct Anal books as good books for (mathematical) applications of those topics

>>14881684
You do PDEs by applying spectral theory from functional analysis

>>14881311
this

>>14881190
Yes, it should be obvious from Weierstrass approximation theorem.

>>14881562
Read on modern theory, theres an spectacular connexion between FEM and modern PDE formulation for determining existence and uniqueness.

>>14881684
It does absolutely change into operator theory, the undergrad class you took is complete outdated

>>14881708
Please do post

>>14881713
Thanks

>>14882116
> Read on modern theory, theres an spectacular connexion between FEM and modern PDE formulation for determining existence and uniqueness.
It’s not a “connection”, FEM is a straightforward application of PDE theory. I’m honestly surprised learn FEM without learning PDEs these days, but I guess it’s so useful to the masses that it needs to be taught without it’s original context attached.

>>14882143
Just to elaborate: Finite Element Methods are what you get when you project solutions and evolutions of PDE to finite dimensional function subspaces. The basis functions of this finite dimensional function subspace are sometimes called “splines”. The projections are defined using inner products on function spaces (L^2 usually, maybe H^1 is used sometimes but I can’t remember at the moment), these inner products being defined with integrals of the product of two functions. Choosing the splines well allows the integrals defining these inner products to be computed quickly thanks to quadrature techniques in numerical/applied analysis. And as far as that goes, that’s pretty much the theory to create FEMs from PDEs. You can proof a lot of theorems about FEM solutions to PDEs by just taking results from PDE theory of solutions and then projecting those solutions to the Finite dimensional function subspace you are working with.

Anybody doing numerical stuff with PDEs - what language do you use? And is it coz it's forced upon you, coz old shit is written in that language, or do you really like using it?

>>14881190
No, because many things in Starcraft are quantised and quantised PDEs are generally unsolvable

>>14882143
>FEM is a straightforward application of PDE theory
I know, I just thought using the word "connexion" would be more illustrative.

>>14882167
You can actually prove existence and uniqueness in some cases using discretizations, via Cea's lemma and interpolation operators.
I can post a simple example when I get home, I have some notes on this somewhere.


>>14882280
I use julia for pde stuff, because it was forced on me and because I like it. Try it out, its great.

the dirac delta is just a neutrino irl

>>14882330
> You can actually prove existence and uniqueness in some cases using discretizations, via Cea's lemma and interpolation operators.
Good point, I believe Evans proves regularity of solutions of elliptical PDE using discretization techniques and he mentions in the section on the parabolic PDE could be proven the same way (though he uses something else, also the discretizations he used were not great for proving smoothness of the boundary, which is why he illustrated a different technique for parabolic PDE)

Thought the PDE bros would appreciate this:
https://cs.dartmouth.edu/wjarosz/publications/sawhneyseyb22gridfree.html

>>14883281
This is actually pretty insane, thanks for sharing

>>14882330
How are you using Julia for PDEs?

>>14883533
Not the Julia anon, but I heard it's great for massive number crunching.

>>14883533
Multiple ways, there are some good existing libraries on automatic differentiation, which is very performant given the language's abstract type system. So its great for nonlinear problems as it simplifies dealing with jacobians and so on.
Also the functional style the language uses allows you to reason about programming basically exactly the same as you reason mathematically, so writing code in julia is very organic to people like us.
Finally, the linear algebra stuff in julia is wicked fast and easy to use with Matlab style slicing for arrays and some pretty sophisticated functionality.
Bonus: unicode characters can be used in code so know instead of delta your variable name is actually [math]\delta[/math]

>>14884598
I meant how you use it specifically, like what application you are solving (FEM, Fluid dynamics, ...) I guess you are using GridAp or some similar package, right?

statistical systems with interchanging variables are more efficient

>>14882167
>. The basis functions of this finite dimensional function subspace are sometimes called “splines”.
No, they are called basis functions and are typically polynomials, not splines.

>>14884617
A spline is a piecewise polynomial function.
Something like
[eqn]f(x) = \begin{cases} 0 & \text{for } |x|>1 \\
1 - |x| & \text{for } |x| \leq 1 \end{cases} [/eqn]
is an example of a spline that's not a polynomial and this is exactly the kind of function that's used for 1-dimensional FEM.

>>14879555
Ok PDE chads, give it to me straight: How do I get into PDEs? What books are best and in what order? Is it just applying tricks and sorceries like solving infinite sums, integrals or ODEs? Is measure theory needed?

>>14884603
Ive done both FEM and FVM with it. Used it to solve both linear and non linear problems.
I am not using it professionally or at all right now but ive used both julia and python for pde numerics and julia is generally better.

>>14884689
Whats your background?

>>14882167
Bullshit

>>14884713
basically undergrad engineering with higher maths from self studying. Never had experience with PDEs and ODEs only linear ones. I was put off by the vast amount of techniques which mostly required rote memorization, but now I want to dive into it again. I will need measure theory, right?

>>14884733
Measure Theory is essential and so is Functional Analysis

The reason you need it is that the vector spaces [math]C^k[/math] of the k-times continuously differentiable functions are not complete which makes them behave badly under limits. Instead you need the Sobolevspaces [math]W^{k,p}[/math] instead which are Banachspaces on the other hand and solve the PDEs in them. Later you can use stuff like the Sobolev Embedding Theorem theorems from Functional Analysis to prove the weak solutions you got are solutions in the classical sense.

PDE theory is divided in the the theory of elliptic PDEs (basically time-independant PDEs) and Evolution Equations (time dependant PDEs) which are further divided into parabolic and hyperbolic PDEs.
You basically need to study elliptic PDEs at first.
Evolution Equations can usually be brought into the form
[eqn]\frac{\partial}{\partial t} u = Au + f[/eqn]
where [math]A[/math] is some operator that includes only spatial but no time derivatives. The solution of it reduces to showing that [math]A[/math] generated a C0-semigroup [math](e^{t A})_{t \geq 0}[/math] but to prove it for a given operator you need to study the spectral theory of the operator [math]A[/math] which requires you to study the PDE
[eqn] \lambda u = A u[/eqn]
which is an elliptic PDE again.

>>14884733
>>14884812
It is possible to do PDEs without measure theory and functional analysis, you will just learn special cases of the more general theory, for example instead of considering solution spaces with L^p initial conditions, you can work out solution spaces when initial conditions are smooth and compact without having to introduce measure theoretic concepts. Walter Strauss’s book does this and is aimed at undergrads, someone with an engineering background can probably do well with it.

>>14884812
I guess what you said can be found in the books mentioned above, although surprisingly none of them cover measure and integration. So then it's settled, measure theory first and only then PDEs. Any good books for lebesgue integration and/or measure theory you could recommend?

>>14884893
Royden is the text I see used most for this topic at both an undergrad and grad level. Though again check out Walter Strauss book if you read happy to work on the large subset of PDEs that doesn’t require measure theory.

>>14884891
>>14884893
Thanks for the recommendation, but I don't think it's fruitful to learn PDE without understanding integration theory. I've been evading lebesgue integration for too long and now is the time to put an end to it. I suppose one can still confidently get away with a riemannian or engineering treatment of PDE theory. I will nonetheless check your recommendation out, it looks like a fine introductory book.

>>14884893
For functional analysis check out linear functional analysis by Rynne and Youngson, when you're done with that you can dive into modern theory, for books on the subject look at op.

>>14884903
> Thanks for the recommendation, but I don't think it's fruitful to learn PDE without understanding integration theory. I've been evading lebesgue integration for too long and now is the time to put an end to it. I suppose one can still confidently get away with a riemannian or engineering treatment of PDE theory. I will nonetheless check your recommendation out, it looks like a fine introductory book.
That’s great! I would definitely encourage anyone to study measure theory that is interested, I know it’s a topic I really loved. Just wanted to make sure you weren’t giving yourself an unnecessary barrier to learning PDEs if that’s what you were interested in. One more thing I will say I like about Walter Strauss book is that it *is* rigorous, it’s in no sense a “plug and play without understanding” type book. If you do finish measure theory before moving on to PDEs, while Strauss is still good you might be ready for Laurence Evans’ text. Evans is the standard graduate level intro and reference for PDE. Evans has a chapter on Functional Analysis for when you’ll need that.

>>14884914
Lebesgue Integration has always been rather weird, it seems like everyone has different definitions of measures, outer measures and so on. Also some authors introduce lebesgue integration without measure theory at all (called the Daniell scheme).

>>14884914
>That’s great! I would definitely encourage anyone to study measure theory that is interested
This, if you want to study this and know it well, measure theory is a must.
Moreover measure theory underpins modern analysis, so knowing it will help you way beyond pdes.

>Evans has a chapter on Functional Analysis for when you’ll need that.
While true, its only an appendix and its meant as a quick reference. I suggest taking the time go thru a dedicated functional analysis book. Again, this will help you beyond understanding PDE, and its actually a very interesting topic.

As a note of encouragement, I myself have a bachelors in EE, granted, I did take a lot of math courses but I never got the chance to take courses in measure theory and functional analysis.
I am now doing a masters in applied and computational mathematics, and thus I had to learn measure theory and functional analysis on my own.
I did so and now im doing very very well and my understanding of the topics I am interested in has increased dramatically, as well as the amount of topics I can properly study now.
In short, even if you have an engineering background, you can definitely get into these topics and doing so will be very rewarding.

>>14884981
>>14884981
Yeah I believe the Daniel scheme is used in “baby” Rudin. I didn’t learn it that way though and I don’t know anyone who did.

>>14884655
not other anon, but mathematical splines were inspired by physical splines used in carpentry.
it's a word that has been used in slightly different ways in different contexts, and so isn't terribly useful when you want to be exact about what you are talking about.
i tend to avoid the word for the sake of clarity.

>>14884914
Shut up nerd

I didn't get an answer on /mg/, but what is the easiest, softball, baby introduction to Navier-stokes? I am a math undergrad, have only taken up to multivariable calculus.

>>14885452
why would you leap from undergrad straight into Nav-Stokes?

>>14885546
Its interesting.
I don't have any pretenses of making contributions, i just want to understand it.

>>14885452
Hi, are you wanting to solve them numerically, understand how they are derived, or work on their analytical properties?

>>14885564
The last two.
I've played a bit already with discrete navier stokes iterative methods in matlab, with textbook examples (computational mathematics, by White).

I can't seem to find a smooth introduction into them, and I feel with "tools" from calculus 3, I should be able to understand it somewhat now.

>>14879555
How can I get better intuition for PDEs? Coming from control theory perspective.

>>14885695
Intuition? PDEs are just differential equations dependent on multiple variables. In Controls you usually have one time dependent state and that's it. Imagine you want to control the distribution of heat on a plate, it is dependent on time AND the position. PDE control is a whole subfield, an active field of research but also not trivial at all. As for applications of PDEs in controls, look for the Hamilton Jacobi Bellman Equation, which is used frequently in optimal controls.

Any bros here that work with kalman filters?
I am in over my head.

>>14885711
Well, I do understand partial differential equations, and I also understand the state variables concerned, but when it comes to combining both understandings cohesively into an intuitive manner for the given problem, I get the fits. It's like these PD equations are black boxes and I'm doing "something" to get out "something". I mean, it does work in application, but I wanna have an intuitive understanding of the processes involved.

>>14885758
I can't follow. Maybe you could show a specific problem where your understanding fades.

>>14885647
>I feel with "tools" from calculus 3, I should be able to understand it somewhat now.
you'll want to understand analysis at least at the undergrad level first

>>14885695
PDEs are generally statements about physical symmetries like time reversal and rotational invariance (the latter is exactly what the Laplacian is expressing). Everything from there is either straight math or changing the problem a bit:
>perturbing the equation
>finding a weak solution rather than a strong one
>ignoring the fact that certain functions are normalizable because physicists don't know what generalized eigenfunctions are

>>14886511
Gotcha, so I still need more. Figured. Thanks!

we could have had /fag/ but no

>>14887365
I posted literally the only serious question in that thread but no one ever replied.
It okay, fag bros are also welcome in pdeg

B-b-based thread

not really PDE but what does it mean when we say a vector space is equal to it's double dual? The double dual consists of mappings so how can they both be equal? I know there is an isomorphism in finite dimensions. Is it just notational abuse?

>>14888630
You say its equal when there is an isometric isomorphism between the space and its dual because when such a mapping exists then there really isnt any way to distinguish between the 2 spaces.

>>14888630
>when we say
Should be “when someone says”

>>14881708
>>14882116
>SDE content
OK So I'm working on some SDE OC actually, the thing I had planned to post is on another computer at the moment, I will get it later but I wanted to start with something else for now. Here is "Cool SDE Facts Part 1/2"

Cool SDE Facts Part 1/2 : Cool PDE Facts first

The 1st order linear PDE is called "The Transport Equation". This would be your first PDE if you read, for example, p. 18 in Evans textbook (2nd Edition). The simplest case is the 1-D "intial value problem" which is to find [math]u(x, t)[/math] that has initial values [math]u(x, 0) = g(x)[/math] and its mass is "transported" by the dynamics defined, for some scalar [math]b[/math], by the PDE

[math] u_t + b u_x = 0 [/math].

One way to solve this is a general technique that applies to many first order PDE: "the method of characteristics". The motivation is you consider the path that mass takes from the starting time until whatever the current time is, and you try to reverse the process to understand how much mass is at your current location by reversing that evolution and tracing back to where you mass started at in the beginning. It's easier to understand it explicitly:

Fix [math]x_0[/math] in the domain of [math]g[/math]. Let [math]p(t)[/math] be the path the mass starting at [math]x_0[/math] travels in time, so [math]u(p(t), t) = g( p^{-1} (t) ) [/math].

continued...

>>14889117
...continued

Since [math]p(t)[/math] tracks the same mass traveling through time, [math]\frac{du(p(t), t)}/{dt} = 0[/math]. Also, just by computation, [math]\frac{du(p(t), t)}/{dt} = u_x p'(t) + u_t [/math]. Putting these together shows [math] u_x p'(t) + u_t = 0 [/math]
and so [math]p(t)[/math] is the trajectory satisfying [math]p'(t) = c[/math] and [math]p(0) = x_0 [/math], so that [math]p(t) = x_0 + ct [/math].
This gives the solution of [math]u[/math] to the transport equation: [math]u(x, t) = u( x_0 + ct , t) = u( x_0 , 0 ) = g( x_0 ) = g( x - ct )[/math].

This way of solving the transport equation is a general technique called "the method of characteristics" because the value of [math]u[/math] at a point [math]x[/math] at the current time [math]t[/math] was found by tracing the path back (the characteristic) of where [math]u(x, t)[/math]'s mass was back to where it came from in the initial condition.

This technique can be applied to a large class of first order PDE, see Section 3.2 of Evans 2nd Ed. for more information.

Can this technique be applied beyond first order PDE though? In a way, yes, that's why this is a lead in for a "Cool SDE fact"

>>14889117
...continued

Since [math]p(t)[/math] tracks the same mass traveling through time, [math]\frac{du(p(t), t)}{dt} = 0[/math]. Also, just by computation, [math]\frac{du(p(t), t)}{dt} = u_x p'(t) + u_t [/math]. Putting these together shows [math] u_x p'(t) + u_t = 0 [/math]
and so [math]p(t)[/math] is the trajectory satisfying [math]p'(t) = c[/math] and [math]p(0) = x_0 [/math], so that [math]p(t) = x_0 + ct [/math].
This gives the solution of [math]u[/math] to the transport equation: [math]u(x, t) = u( x_0 + ct , t) = u( x_0 , 0 ) = g( x_0 ) = g( x - ct )[/math].

This way of solving the transport equation is a general technique called "the method of characteristics" because the value of [math]u[/math] at a point [math]x[/math] at the current time [math]t[/math] was found by tracing the path back (the characteristic) of where [math]u(x, t)[/math]'s mass was back to where it came from in the initial condition.

This technique can be applied to a large class of first order PDE, see Section 3.2 of Evans 2nd Ed. for more information.

Can this technique be applied beyond first order PDE though? In a way, yes, that's why this is a lead in for a "Cool SDE fact"