I’m trying to implement a stochastic filter for a system based on a time-dependent PDE (heat equation). The PDE depends on an unknown potential that I eventually want to estimate. For this, I need the derivative of the solution at a time t with respect to the potential. How do I get this?

I know that what I am looking for is basically the solution to the tangent linear equation, but I don’t want to implement this by myself, because I assume that dolfin-adjoint is able to do this much better.

```
from fenics import *
from fenics_adjoint import *
T = 50.0 #time of evaluation
N_t = 200 #number of time steps
dt = T / N_t #time step size
N_s = 64 #space mesh size
# Set initial value
u_0 = Expression('0', degree=1)
# Set boundary value
u_D = Expression('0', degree=1)
# Basic setup
mesh = UnitIntervalMesh(N_s)
V = FunctionSpace(mesh, 'P', 2)
def boundary(x, on_boundary):
return on_boundary
bc = DirichletBC(V, u_D, boundary)
# Define the potential
q = interpolate(Expression('0.1-0.05*sin(2*pi*(x[0]-0.25))', degree=1), V)
control = Control(q) # Want to differentiate with respect to q
# Define the variational problem (using backwards Euler)
u_next = Function(V)
u = interpolate(u_0,V)
v = TestFunction(V)
r = Expression('x[0] >= 0.215 && x[0] <= 0.315 ? (4*sin(4*pi*t)+0.001*pow(t,2)) : 0', \
degree=1, t=0)
F = ( (u_next - u)*v /dt + dot(q*grad(u_next), grad(v)) - r*v )*dx
# Start solution
for t in range(N_t):
# Update current time
t += dt
# At time n, we want r = r((n+1)dt)
r.t = t
# Compute solution
solve(F == 0, u_next, bc)
# Update u
u.assign(u_next)
# Now, I want the derivative of u(T) with respect to q
dudq = compute_gradient(u, control) # But obviously, this does not work.
```