I am reading an arxiv paper that references high dimensional PDEs. I have a picture from the text below. The claim is about " In filtering and optimal control, we are easily interested in PDEs occurring in hundreds or thousands of dimensions – here,classical methodology can rarely be employed"
I am familiar with conventional PDE which are discretized spatially and in time, such as fluid dynamics simulations, or climate models, etc. So those PDEs are discretized in say $x, y, t$, or in $x, y, z, t$. I was trying to understand what is meant by these high dimensional PDEs that are in 100s or 1000s of dimensions.
I think my confusion is really just about what this terminology is used of high dimension? I understand the fundamental of optimal control as well. In an airplane control simulation, the aircraft is still moving over a 3 dimensional geographic space, but the pilot has a set of control actuators that guide the aircraft to its target, given some constraint Lagrangian. That control function is of course infinite dimensional, but in practice we will discretize the function and then use something like direct transcription to find the unknown control function.
So is the "high dimensional" pde, really just referring to that unknown control function at each discretized location. Is it as simple as that, or am I missing something.
I know almost nothing about control theory, so I could be missing something.
When talking about PDEs in hign dimensions, one usually refers to the dimension of the domain of the function, so in place of $x,y,z,t$, you could have $x_1,\dots,x_N,t$. There are examples of such PDEs that are physical and occur in some areas. The main example is many-body quantum mechanics. A very ‘’’simple’’’ case is to model the evolution of a wave function $\psi\colon \mathbb R^{3N}\times\mathbb R\to\mathbb C$ representing $N$ elementary particles in the $3D$ space under the action of a potential. The PDE is simply the Schrödinger equation, which after normalizing some comstants to be $1$, can be written as $$ i\partial_t\psi=-\sum_{j=1}^{3N} \partial_j^2\psi+V\psi, $$ where $V \colon \mathbb R^{3N} \to\mathbb R$ is the potential that represents the interaction between the particles and external forces. There could be many instances of high-dimensional PDEs in the non-quantum world, and the authors mention high dimensional PDEs pop up in control theory, but I can’t think of another example at the moment. I am pretty sure this is what they are referring to, although I have to say I have a good background in PDEs but I know nearly nothing of control theory and FEM.
Edit: thus you can see that the main problem with the finite element method is that if you work in $N$ dimensions and, say, you want to use a grid with $10$ nodes for each dimension, you end up with $10^N$ nodes, which becomes easily impossible to fit in a computer if $N$ is something like 100.