Okay, so welcome everyone.

For this week and also for next week, we will have Professor Abir Iseles from the University

of Cambridge.

And for today, he will be speaking about fast approximation on the real line.

Please, Professor Iseles, we're looking forward to your talk.

So you can share your slide, I think.

You are now co-host, so I think it should be possible to share your slide.

Okay, so let me...

No, I don't think I have the...

Oh, yes.

Yes.

Okay.

Maybe you can try full screen, maybe.

Exactly.

Okay.

Yeah, perfect.

Thanks.

So this is, first of all, this is work that I've done with two of my colleagues.

Karen Luong, who is my current research student in Cambridge, and Marcus Webb in Manchester.

And this is a work that has been motivated by work in solving equations of quantum mechanics.

I can explain why.

But I want to start, actually, from a very, very broad setting, and the setting of solving

differential equations.

So we have differential equations that are naturally defined in RD for some D, and many

of them are defined in the entire space.

So what is known as a Cauchy problem.

That once we want to compute them practically, typically we need to restrict the domain to

a finite domain, because we need to use a finite number of degrees of freedom.

Finite difference methods, finite element methods, finite volume methods, and so on.

The hint is in the word finite.

And so how to reduce everything to a finite domain?

And there are long chapters of applied analysis exactly to this end.

How to reduce problems on infinite domain to a finite domain.

There is the Dirichlet-to-Neumann map, the absorbing boundary conditions, reflecting

boundary conditions, perfectly matched layers.

And then there are various tricks simply saying, okay, from now on I'm assuming that the boundary

conditions are periodic because the solution is so small in the boundary that we don't

care.

Or whatever we can set boundary conditions, we set them, and many people do it without

any further theory, and sometimes it is very wrong.

There's one exception.

There is one family of numerical methods that can be applied directly in RD, and still we

can retain a finite number of degrees of freedom, and these are spectral methods.

So very, very briefly for those of you who haven't encountered spectral methods, one

slide introduction.

So suppose that we have a PDE LU equal F, a boundary value problem is given in X, for

every X in some domain omega, with boundary conditions on the boundary of omega, and the

solution evolves in some sub-Olef space.

Suppose that we have an autonormal system defined in the same sub-Olef space of omega,

it is autonormal in L2, it is complete in L2, and it is consistent with boundary conditions.