Galerkin Approximation
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Galerkin Approximation
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Re: Galerkin Approximation
#2I find the general idea of treating differential equations as infinitely dimensional linear systems quite powerful. I was first introduced to the concept while studying quantum mechanics, but applications are everywhere.
Re: Galerkin Approximation
#3I find the general idea of treating differential equations as infinitely dimensional linear systems quite powerful. I was first introduced to the concept while studying quantum mechanics, but applications are everywhere.
I first encountered it for deriving finite element methods for structural analysis, and later as a general method for PDEs. Sadly this treatment is about as inscrutable as my grad level PDE course.
Re: Galerkin Approximation
#4And, guess what? Since, the Galerkin approximation requires one to choose a basis that is appropriate to the problem at hand, we now have a deep learning solution too (since neural network learning is essentially equivalent to learning an adaptive basis).
It is called the Deep Galerkin Method [1]. In a nutshell, the method directly minimizes the L2 error over the PDE, boundary conditions and initial conditions. The integral is tricky though, and computed via a Monte Carlo approximation.
Re: Galerkin Approximation
#5Check out the rest of the guy's blog. I desperately wish there were more resources where complex mathematical concepts were first introduced without all of the proofs.
Re: Galerkin Approximation
#6This made as little sense to me as it did when I was talking the Finite Element Methods class during graduate school.
Still don't quite understand why you can't just use Runge Kutta methods to numerically solve these problems. I became quite good at manipulating the symbols to derive variational solutions while having absolutely no idea what any of it meant.
Re: Galerkin Approximation
#7This made as little sense to me as it did when I was talking the Finite Element Methods class during graduate school. Still don't quite understand why you can't just use Runge Kutta methods to numerically solve these problems. I became quite good at manipulating the symbols to derive variational solutions while having absolutely no idea what any of it meant.
Runge-kutta works when you’re given an initial condition and the derivative is with respect to just one variable (so you’re given f(t) just at t=0). What do you do when you’re given a boundary condition and the derivative is with respect to many variables? This.
Re: Galerkin Approximation
#8This made as little sense to me as it did when I was talking the Finite Element Methods class during graduate school. Still don't quite understand why you can't just use Runge Kutta methods to numerically solve these problems. I became quite good at manipulating the symbols to derive variational solutions while having absolutely no idea what any of it meant.
Runge-kutta works when you’re given an initial condition and the derivative is with respect to just one variable (so you’re given f(t) just at t=0). What do you do when you’re given a boundary condition and the derivative is with respect to many variables? This.
I mean, you can also just do finite differences. Of course Galerkin opens up the very “rich” design space of finite elements methods.
Re: Galerkin Approximation
#9And, guess what? Since, the Galerkin approximation requires one to choose a basis that is appropriate to the problem at hand, we now have a deep learning solution too (since neural network learning is essentially equivalent to learning an adaptive basis). It is called the Deep Galerkin Method [1]. In a nutshell, the method directly minimizes the L2 error over the PDE, boundary conditions and initial conditions. The i…
Why would you use the Monte Carlo method when the quasi-Monte Carlo method converges so much more quickly? I admit, I am a little biased, because I worked on some QMC stuff in grad school, but it works really, really well in practice.
Re: Galerkin Approximation
#10Earlier quoted context omitted.
Runge-kutta works when you’re given an initial condition and the derivative is with respect to just one variable (so you’re given f(t) just at t=0). What do you do when you’re given a boundary condition and the derivative is with respect to many variables? This.
I mean, you can also just do finite differences. Of course Galerkin opens up the very “rich” design space of finite elements methods.
Finite differences are simple in cubes and domains that can be mapped to one. Not so much in an arbitrary tetrahedral mesh.