Reduced Models and Parallel Computing for Uncertainty Quantification in Cardiovascular Mathematics Open Access

Abstract

Computational fluid dynamics (CFD) has been progressively adopted in the last decade for studying the role of blood flow in the development of arterial diseases. While computational (in silico) investigations - compared to more traditional in vitro and in vivo studies - are generally more flexible and cost-effective, the adoption of CFD for computer-aided clinical trials and surgical planning is still an open challenge. The computational time to accurately and reliably solve mathematical models can be too long for the fast-paced clinical environment - especially in emergency scenarios, and quantifying the reliability of the results comes at an even higher computational cost. Moreover, the in silico analysis of large numbers of patients calls for significant computational resources. Hospitals and healthcare institutions are expected to outsource numerical simulations, which, however, raises concerns about privacy, data protection, and efficiency in terms of cost and performance. In such an articulated and complex scenario, this work addresses the challenges described above by (i) introducing a novel reduced model that guarantees levels of accuracy comparable to those achieved by high-fidelity 3D models, roughly at the same computational cost as the inexpensive yet inaccurate 1D models, by combining the Finite Element Method to describe the main stream dynamics with Spectral Methods to retrieve the transverse components; (ii) designing a new method for uncertainty quantification in large-scale networks that greatly enhances parallelism by performing uncertainty quantification at the subsystem level, and propagating uncertainty information encoded as polynomial chaos coefficients via overlapping domain decomposition techniques; (iii) providing an objective criterion to measure the performance of different parallel architectures based on the user’s priorities in terms of budget and tolerance to delay, and reducing the execution time by choosing a task-worker mapping strategy ahead of simulation time, and optimizing the amount of overlap in the domain decomposition phase.

Table of Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Hierarchical Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

  2.1 Introduction and Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

  2.2 HiMod in Cylindrical Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.2.1 The geometric setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.2.2 The reference basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

  2.3 Scalar Advection-Diffusion-Reaction Problems. . . . . . . . . . . . . . . . . . . . . . 17

    2.3.1 Numerical Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

  2.4 The Navier-Stokes equations in cylindrical coordinates . . . . . . . . . . . . . . . 28

    2.4.1 The HiMod formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    2.4.2 The inf-sup condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    2.4.3 Pole Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.4.4 Steady case: Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    2.4.5 Unsteady case: Womersley flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    2.4.6 Choice of the size of the modal space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

  2.5 Numerical Tests in Axisymmetric and Non-Axisymmetric Domains . . . . . 47

    2.5.1 Axisymmetric models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    2.5.2 Non-axisymmetric models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    2.5.3 Patient-specific geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

  2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Network Uncertainty Quantification via Domain Decomposition. . . . . . . . . . 62

  3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

  3.2 The DDUQ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    3.2.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    3.2.2 Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    3.2.3 Network solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    3.2.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

  3.3 Numerical test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    3.3.1 1D heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    3.3.2 2D nonlinear heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

  3.4 DDUQ acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    3.4.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    3.4.2 Geometric Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    3.4.3 Full Approximation Scheme (FAS) for non-linear problems. . . . . . . . 100

    3.4.4 Algebraic Multigrid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

  3.5 Multigrid Methods for DDUQ Network Problems in Matrix Form. . . . . . 103

    3.5.1 p-Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    3.5.2 h-Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    3.5.3 Preliminary results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

  3.6 h-Multigrid Methods for UQ in Networks . . . . . . . . . . . . . . . . . . . . . . . . . 125

    3.6.1 Prolongation and restriction operators . . . . . . . . . . . . . . . . . . . . . . . . 125

    3.6.2 Smoother and coarse-grid operators . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    3.6.3 The definition of the residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

  3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4 Reduced-order models for uncertainty quantification in the 

 cardiovascular network via DDUQ.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

  4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

  4.2 The Transversally-Enriched Pipe-Element Method . . . . . . . . . . . . . . . . 137

    4.2.1 Pipe discretization strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    4.2.2 Transversally enriched approximation . . . . . . . . . . . . . . . . . . . . . . . . 140

  4.3 Uncertainty quantification on blood flow problems . . . . . . . . . . . . . . . . 142

    4.3.1 Blood flow problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

    4.3.2 Geometrical decomposition of the vasculature. . . . . . . . . . . . . . . . . . 145

    4.3.3 Formulation by subdomain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

    4.3.4 The DDUQ-TEPEM algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

  4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

    4.4.1 Test setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    4.4.2 Scalability tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    4.4.3 Towards realistic geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

  4.5 DDUQ for unsteady problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

    4.5.1 Reduced 1D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    4.5.2 DDUQ formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

    4.5.3 Numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

  4.6 Final remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5 Platform and algorithm effects on computational fluid dynamics . . . . . . . 187

  5.1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

    5.1.1 The numerical problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

    5.1.2 Domain decomposition techniques for the solution of Partial

       Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    5.1.3 Packages used by the numerical solver . . . . . . . . . . . . . . . . . . . . . . . . 195

  5.2 CFD Experiences on clouds, grids and on-premise resources. . . . . . . . . 196

    5.2.1 Heterogeneous Target Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

    5.2.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

    5.2.3 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

  5.3 Adaptive mapping of parallel components on physical resources. . . . . 209

    5.3.1 Test case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

    5.3.2 Offine mesh partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

    5.3.3 Evaluation procedure and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

  5.4 Experimental optimization of parallel 3D overlapping domain 

    decomposition schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

    5.4.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

  5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

  7.1 Bottom-Up basis functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

  7.2 HiMod coeffcients for the Advection-Diffusion-Reaction Equations. . 231

  7.3 HiMod coeffcients for the Navier-Stokes equations. . . . . . . . . . . . . . . . 232

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Applied Mathematics Engineering, Biomedical Computer Science
Keyword	Parallel computing Uncertainty quantification Finite elements Spectral methods Cylindrical coordinates Overlapping domain decomposition Networks Performance analysis 3D patient-specific geometries Large-scale applications Cloud computing Model reduction Applied computing in medicine Multigrid methods Scalability Platform heterogeneity Cardiovascular mathematics Hemodynamics Multilevel methods Computational fluid dynamics
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Ruthotto, Lars, Emory University Blanco, Pablo J., Scientific Computing National Laboratory of Brazil Nagy, James, Emory University

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