# A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data

## TL;DR

This paper introduces a fully adaptive multilevel stochastic collocation method for elliptic PDEs with random data, combining adaptive spatial mesh refinement with anisotropic sparse grid stochastic discretization to optimize computational efficiency.

## Contribution

It presents a novel hierarchical adaptive strategy that optimizes both spatial and stochastic discretizations based on sample-dependent hierarchies, with rigorous analysis and optimal error control.

## Key findings

- Significant reduction in computational complexity compared to single-level methods.
- Demonstrated reliability of error control through numerical examples.
- Effective combination of adaptive spatial and stochastic discretizations.

## Abstract

We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.