# unconventional linear and bilinear forms

I'm wondering if the following is possible to express easily in UFL. I'll write it in math/latex style.

Let's say my domain is an interval x \in [a,b]. Now let u = u(x) \in R^n and v = v(x) \in R^n. These are n-vectors whose elements depend on x. And let A = A(x) \in R^n x R^n be a matrix whose elements depend on x.

I would like to work with linear forms such as

\int_a^b u^T v dx

and bilinear forms such as

\int_a^b u^T A v dx.

The matrix A might be a "mesh function" that returns an nxn matrix for each mesh point.

Reading through the FEniCS book -- particularly Chapter 17 -- I can't tell if such a thing is possible.

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Jan Blechta
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 Revision history for this message Paul Constantine (paul-g-constantine) said on 2013-01-24: #1

Oh. I also want to be able to take du/dx \in R^n.

 Revision history for this message Martin Sandve Alnæs (martinal) said on 2013-01-24: #2

What is n?

Martin
Den 24. jan. 2013 05:26 skrev "Paul Constantine" <

> Question #219998 on UFL changed:
>
> Oh. I also want to be able to take du/dx \in R^n.
>
> --
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 Revision history for this message Paul Constantine (paul-g-constantine) said on 2013-01-24: #3

I have systems with n=3 and n=3 billion. I'll start wherever.

 Revision history for this message Marie Rognes (meg-simula) said on 2013-01-24: #4

Yes, this is perfectly feasible and not so unconventional (at least for moderate n): Here's some suggested DOLFIN code (replace VectorFunctionSpace with VectorElement etc to get UFL)

from dolfin import *
mesh = UnitIntervalMesh(20)

n = 30
V = VectorFunctionSpace(mesh, "CG", 1, dim=n)
u = TrialFunction(V)
v = TestFunction(V)

a = inner(u, v)*dx

Q = FunctionSpace(mesh, "DG", 0)
f = Function(Q)
A = as_matrix([[f for i in range(n)] for i in range(n)])

a = inner(A*u, v)*dx

 Revision history for this message Martin Sandve Alnæs (martinal) said on 2013-01-24: #5

As Marie says, but for 3 billion forget it and find another approach. Is
that even possible? You have a O(n^2) computation there...

Martin
Den 24. jan. 2013 07:51 skrev "Paul Constantine" <

> Question #219998 on UFL changed:
>
>
> Paul Constantine is still having a problem:
> I have systems with n=3 and n=3 billion. I'll start wherever.
>
> --
> You received this question notification because you are a member of UFL
> Team, which is an answer contact for UFL.
>
> _______________________________________________
> Post to : <email address hidden>
>

 Revision history for this message Paul Constantine (paul-g-constantine) said on 2013-01-24: #6

Thanks! That's tremendously helpful. This comes from a recent idea for parallel-in-time simulations of chaotic dynamical systems. See, e.g., http://arxiv.org/abs/1211.2437

In this case n=3 would be something like the Lorenz system, while n=3e9 would be the spatially discretized, linearized forward and adjoint operators from a Navier-Stokes model at high Reynolds number (turbulent flow). Obviously, the latter is out of reach in practice. But that's what we put on proposals. :) However, something like the Kuramoto-Sivashinsky equation would be the next step after Lorenz. In this case, the matrix A would still be a spatially discretized operator, but its physical dimension would be 1 and n=100 would be a reasonable discretization.

Maybe there's a way to write it in UFL that would let DOLFIN do the spatial discretization automatically. Can you express one partial derivative for a v=Function(V) where V=FunctionSpace(<2d mesh>)?

 Revision history for this message  Jan Blechta (blechta) said on 2013-02-24: #7

> Can you express one partial derivative for a v=Function(V)
> where V=FunctionSpace(<2d mesh>)?

 Revision history for this message Paul Constantine (paul-g-constantine) said on 2013-02-24: #8