A generalized multi-resolution expansion for uncertainty propagation with application to cardiovascular modeling

Abstract

Computational models are used in a variety of fields to improve our understanding of complex physical phenomena. Recently, the realism of model predictions has been greatly enhanced by transitioning from deterministic to stochastic frameworks, where the effects of the intrinsic variability in parameters, loads, constitutive properties, model geometry and other quantities can be more naturally included. A general stochastic system may be characterized by a large number of arbitrarily distributed and correlated random inputs, and a limited support response with sharp gradients or event discontinuities. This motivates continued research into novel adaptive algorithms for uncertainty propagation, particularly those handling high dimensional, arbitrarily distributed random inputs and non-smooth stochastic responses. In this work, we generalize a previously proposed multi-resolution approach to uncertainty propagation to develop a method that improves computational efficiency, can handle arbitrarily distributed random inputs and non-smooth stochastic responses, and naturally facilitates adaptivity, i.e., the expansion coefficients encode information on solution refinement. Our approach relies on partitioning the stochastic space into elements that are subdivided along a single dimension, or, in other words, progressive refinements exhibiting a binary tree representation. We also show how these binary refinements are particularly effective in avoiding the exponential increase in the multi-resolution basis cardinality and significantly reduce the regression complexity for moderate to high dimensional random inputs. The performance of the approach is demonstrated through previously proposed uncertainty propagation benchmarks and stochastic multi-scale finite element simulations in cardiovascular flow.

Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels

“…In addition, several approaches for fluid-structure interaction (FSI) have been suggested to account for vessel wall deformability [13,20,27]. Recently, hemodynamic models have been used in the solution of complex problems in optimization [24,22] and uncertainty quantification [29,30,32,31,36].…”

Multi-fidelity classification using Gaussian processes: Accelerating the prediction of large-scale computational models

“…In this work, we present a novel multi-fidelity classifier using Gaussian process priors. While the multi-fidelity paradigm has been proposed for regression [16] and uncertainty quantification [41], this work represents one of the first attempts to formulate and implement a fully-Bayesian multi-fidelity classifier. Based on an autoregressive Gaussian process prior that enables the seamless integration of data with different levels of fidelity, our classifier outperforms the single-fidelity classifier both with synthetic data and in an application of cardiac electrophysiology.…”

Level set methods for stochastic discontinuity detection in nonlinear problems

“…One relies instead on a local measure of variance as a criterion for adaptivity. Other methods for stochastic discontinuous solutions that do not rely on explicit calculation of discontinuity locations include Padé approximation of discontinuous functions using rational functions [13], and multi-resolution analysis schemes based on multi-wavelet expansions that are robust to discontinuities due to hierarchical localization in stochastic space [15,16,17]. Iterative methods for computation of improved spectral expansions of non-smooth solutions to successively suppress the Gibbs phenomenon were introduced in [18].…”