be computed independently from each other). Although the method is deterministic it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.'
author:
- 'Houman Owhadi[^1]'
bibliography:
- 'RPS.bib'
title: |
    Multigrid with rough coefficients\
    and Multiresolution operator decomposition\
    from Hierarchical Information Games
---

Introduction {#secdt}
============

Scientific discovery as a decision theory problem
-------------------------------------------------

The process of scientific discovery is oftentimes based on intuition, trial and error and plain guesswork. This paper is motivated by the question of the existence of a rational decision framework that could be used to facilitate/guide this process, or turn it, to some degree, into an algorithm. In exploring this question, we will consider the problem of finding a method for solving (up to a pre-specified level of accuracy) PDEs with rough ($L^\infty$) coefficients as fast as possible with the following prototypical PDE (and its possible discretization over a fine mesh) as an example $$\label{eqn:scalar}
\begin{cases}
    -\diiv \big(a(x)  \nabla u(x)\big)=g(x) \quad  x \in \Omega;\, g \in L^2(\Omega),\text{ or }g \in H^{-1}(\Omega) \\
    u=0 \quad \text{on}\quad \partial 