Multilevel Quasi-Monte Carlo for Optimization under Uncertainty

09/29/2021
by   Philipp A. Guth, et al.
0

This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the CE method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and the single level quasi-Monte Carlo method.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/15/2023

Quasi continuous level Monte Carlo for random elliptic PDEs

This paper provides a framework in which multilevel Monte Carlo and cont...
research
11/07/2017

Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

This paper addresses optimization problems constrained by partial differ...
research
05/29/2020

MG/OPT and MLMC for Robust Optimization of PDEs

An algorithm is proposed to solve robust control problems constrained by...
research
09/06/2020

Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

We present a novel algorithmic approach and an error analysis leveraging...
research
06/25/2019

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

Practical structural engineering problems are often characterized by sig...
research
12/07/2022

Monte Carlo convergence rates for kth moments in Banach spaces

We formulate standard and multilevel Monte Carlo methods for the kth mom...
research
06/27/2022

Multilevel Quality Indicators (MQI): Methodology and Monte Carlo evidence

Background: Quality indicators are frequently used to assess the perform...

Please sign up or login with your details

Forgot password? Click here to reset