@article {2017, title = {Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts}, journal = {Biomechanics and Modeling in Mechanobiology}, volume = {16}, year = {2017}, pages = {1373-1399}, doi = {10.1007/s10237-017-0893-7}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851\&doi=10.1007\%2fs10237-017-0893-7\&partnerID=40\&md5=c388f20bd5de14187bad9ed7d9affbd0}, author = {F. Ballarin and Elena Faggiano and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Sonia Ippolito and Roberto Scrofani} } @article {2017, title = {Reduced Basis Methods for Uncertainty Quantification}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {5}, year = {2017}, month = {08/2017}, pages = {869}, type = {reviewed}, chapter = {813}, abstract = {
In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babu{\v s}ka, F. Nobile, and R. Tempone,\ SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.
Read More:\ http://epubs.siam.org/doi/abs/10.1137/151004550
This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.
Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.
This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.
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}, keywords = {reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis}, issn = {978-3-319-02089-1}, doi = {10.1007/978-3-319-02090-7}, author = {Alfio Quarteroni and Gianluigi Rozza} } @article {2014, title = {A weighted empirical interpolation method: A priori convergence analysis and applications}, number = {ESAIM: Mathematical Modelling and Numerical Analysis;volume 48; issue 4; pages 943-953}, year = {2014}, publisher = {EDP Sciences}, abstract = {We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Lo{\`e}ve expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.}, doi = {10.1051/m2an/2013128}, url = {http://urania.sissa.it/xmlui/handle/1963/35021}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {KoshakjiQuarteroniRozza2013, title = {Free Form Deformation Techniques Applied to 3D Shape Optimization Problems}, journal = {Communications in Applied and Industrial Mathematics}, year = {2013}, abstract = {The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation.}, doi = {10.1685/journal.caim.452}, author = {Anwar Koshakji and Alfio Quarteroni and Gianluigi Rozza} } @article {NegriRozzaManzoniQuarteroni2013, title = {Reduced basis method for parametrized elliptic optimal control problems}, journal = {SIAM Journal on Scientific Computing}, volume = {35}, number = {5}, year = {2013}, pages = {A2316{\textendash}A2340}, abstract = {We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.}, doi = {10.1137/120894737}, author = {Federico Negri and Gianluigi Rozza and Andrea Manzoni and Alfio Quarteroni} } @article {2013, title = {A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics}, number = {SISSA preprint;15/2013/MATE}, year = {2013}, institution = {SISSA}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {2013, title = {A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows}, number = {SISSA preprint;14/2013/MATE}, year = {2013}, institution = {SISSA}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {ChenQuarteroniRozza2013, title = {Stochastic optimal robin boundary control problems of advection-dominated elliptic equations}, journal = {SIAM Journal on Numerical Analysis}, volume = {51}, number = {5}, year = {2013}, pages = {2700{\textendash}2722}, abstract = {In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.}, doi = {10.1137/120884158}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {ChenQuarteroniRozza2013a, title = {A weighted reduced basis method for elliptic partial differential equations with random input data}, journal = {SIAM Journal on Numerical Analysis}, volume = {51}, number = {6}, year = {2013}, pages = {3163{\textendash}3185}, abstract = {In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.}, doi = {10.1137/130905253}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2012, title = {Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty}, journal = {Mathematical Modelling and Numerical Analysis, in press, 2012-13}, number = {SISSA;42/2012/M}, year = {2012}, publisher = {Cambridge University Press}, abstract = {We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework.}, keywords = {shape optimization}, url = {http://hdl.handle.net/1963/6337}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @inbook {2012, title = {Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs}, booktitle = {Springer, Indam Series, Vol. 4, 2012}, number = {SISSA Preprint;44/2012/M}, year = {2012}, publisher = {Springer}, organization = {Springer}, abstract = {The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates.}, keywords = {solution manifold}, url = {http://hdl.handle.net/1963/6340}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {10671, title = {Simulation-based uncertainty quantification of human arterial network hemodynamics}, journal = {International Journal Numerical Methods Biomedical Engineering}, number = {SISSA preprint;}, year = {2012}, publisher = {Wiley}, abstract = {This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time.}, keywords = {uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} }