Gradient Flow From A Random Walk in Hilbert Space

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).