1
$\begingroup$

Suppose that I have the elliptic PDE

$\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, U_{x_1}(x_1,l_2)=0.$

I am trying to test out a basic inverse problem whereby I pick $n$ points in the domain, evaluate $U(x)$ at those points, and compute $Y_i = U(x_i)+ \epsilon_i$ where $\epsilon_i \sim N(0,\sigma^2)$ and $\epsilon_i$ are iid.

Now I pretend that I only have $Y_i, i=1,\dots,n$ at my disposal and I will attempt to recover $U(x_i)$ using the Bayesian approach.

Computing the density function of $\vec{Y}$ given $\vec{U(x)}$ (both of which are vectors of length $n$) is straightforward. But I am stuck on the issue of choosing a prior due to the following:

1) I am not sure how the maximum/minimum principle for elliptic pdes could be applied here since I do not know the values of $U$ on 2 sides of the rectangle. But running numerical simulations show that $U(x)$ is always within $[0,1]$. (Side question: could we apply some max/min principle of some sort to prove the bounds on $U$?)

2) In choosing a prior for the Bayes framework, it seems intuitive to me initially that the joint density of $U(x_i), i=1,\dots,n$ is the uniform density on $[0,1]^n$ because of point 1 above. So here's my major question: Why is it that if I choose the uniform density as a prior, the maximum a posteriori method performs very poorly, almost useless?

As an example, take $n=2$. The posterior distribution is then $\text{1}_{[0,1]^2}\exp\{-\frac{1}{2\sigma^2}((Y_1-U(x_1))^2+(Y_2-U(x_2))^2))\}$.

If we suppose that the true values are $U(x_1)= 0.4, U(x_2) = 0.2$ and that $\epsilon_1 = 1.1,\epsilon_2 = 0.2$ such that $Y_1 = 1.5$ and $Y_2 = 0.4$ then using the maximum a posteriori method, the estimates for $U(x_1)$ and $U(x_2)$ are $U(x_1)=1$ and $U(x_2) = 0.4$ which are not really close to the true values. Of course in this scenario, I used $\sigma^2 =2$ and surely lowering $\sigma$ would give better estimates but does this only imply that I should choose a more sophisticated prior?

I am asking because in this case, I have the actual PDE at hand but in reality, I don't so I would presume that a good first step would be choosing a uniform prior. (I'm new to inverse problems...)

Insights greatly appreciated.

$\endgroup$
2
  • $\begingroup$ What is the inverse problem here? What is the data and what is the unknown? As an inverse problems researcher, I'm quite confused... $\endgroup$ Aug 15, 2014 at 22:27
  • $\begingroup$ Hi Joonas, the inverse problem here is recovering $U(x)$ given knowledge of $Y$. But in order to verify the accuracy of the Bayesian framework, we will assume true values for $U(x)$, namely the solution of the PDE, compute $Y$ from the linear relation above, and use maximum a posteriori method to obtain estimates for $U(x)$, then compare the estimates with the true values $\endgroup$ Aug 15, 2014 at 22:46

1 Answer 1

0
$\begingroup$

Do I understand your problem correctly if I state you want to solve the following problem (assumption $A(x)=Id$) $ \hat U = \arg \min \{ \| U - Y \|^2_2 + \lambda \mathcal R( U) \} $ where the data $Y$ is given on a $N \times N $ grid and $\mathcal R(.)$ is a convex regularization functional? Or is your data given only at certain points on the grid?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.