Related papers: Robust Optimization of PDEs with Random Coefficien…
In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel…
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range…
We present a Multi-Index Quasi-Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm…
In predictive modeling with simulation or machine learning, it is critical to accurately assess the quality of estimated values through output analysis. In recent decades output analysis has become enriched with methods that quantify the…
Computing risk measures of a financial portfolio comprising thousands of derivatives is a challenging problem because (a) it involves a nested expectation requiring multiple evaluations of the loss of the financial portfolio for different…
This paper presents multilevel hybrid transport (MLHT) methods for solving the neutral-particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multilevel Monte Carlo (MLMC)…
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly,…
Estimating risk measures such as large loss probabilities and Value-at-Risk is fundamental in financial risk management and often relies on computationally intensive nested Monte Carlo methods. While Multi-Level Monte Carlo (MLMC)…
Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic…
Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in…
This work describes a domain embedding technique between two non-matching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is…
In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of…
Optimizing highly complex cost/energy functions over discrete variables is at the heart of many open problems across different scientific disciplines and industries. A major obstacle is the emergence of many-body effects among certain…
The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty quantification in PDE models. It combines approximations at different levels of accuracy using a hierarchy of…
We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random…
We present a new Subset Simulation approach using Hamiltonian neural network-based Monte Carlo sampling for reliability analysis. The proposed strategy combines the superior sampling of the Hamiltonian Monte Carlo method with…
The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variables can be replaced by similar approximate random variables…
Multi-fidelity Monte Carlo (MFMC) is a variance reduction method that leverages a multi-fidelity ensemble of models of varying cost and accuracy levels. Constructing an MFMC estimator with optimal variance requires knowledge of the…
We consider the problem of estimating the probability of a large loss from a financial portfolio, where the future loss is expressed as a conditional expectation. Since the conditional expectation is intractable in most cases, one may…
We present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise…