1,721,164 research outputs found
Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems
No full textWe propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case. The method, which is based on an off-line-on-line decomposition procedure, allows at the on-line step large computational cost savings with respect to the "truth" approximation used for defining the reduced basis. An a posteriori error estimate is provided by mea ns of the goal-oriented analysis, thus associating an error bound to each optimal solution of the pa rametrized optimal control problem and answering to the demand for a reliable method. An adaptive procedure, led by the a posteriori error estimate, is considered for the generation of the reduced b asis space, which is set according to the optimal primal and dual solutions of the optimal control pr oblem at hand. © 2010 Society for Industrial and Applied Mathematics
Reduced basis method for parameterized elliptic advection-reaction problems
No full textIn this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal-dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one. Copyright 2010 by AMSS, Chinese Academy of Sciences
Reduced basis method and error estimation for parametrized optimal control problems with control constraints
No full textWe propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline-online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the "truth" one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems. © 2011 Springer Science+Business Media, LLC
Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS-LES modeling in a High Performance Computing framework
No full textIn this paper, we propose a semi-implicit approach for the time discretization of the Navier-Stokes equations with Variational Multiscale-Large Eddy Simulation turbulence modeling (VMS-LES). For the spatial approximation of the problem, we use the Finite Element method, while we employ the Backward Differentiation Formulas (BDF) for the time discretization. We treat the nonlinear terms arising in the variational formulation of the problem with a semi-implicit approach leading to a linear system associated to the fully discrete problem which needs to be assembled and solved only once at each discrete time instance. We solve this linear system by means of the GMRES method by employing a multigrid (ML) right preconditioner for the parallel setting. We validate the proposed fully discrete scheme towards the benchmark problem of the flow past a squared cylinder at high Reynolds number and we show the computational efficiency and scalability results of the solver in a High Performance Computing framework
A reduced 3D-0D fluid–structure interaction model of the aortic valve that includes leaflet curvature
Full text linkWe introduce an innovative lumped-parameter model of the aortic valve, designed to efficiently simulate the impact of valve dynamics on blood flow. Our reduced model includes the elastic effects associated with the leaflets’ curvature and the stress exchanged with the blood flow. The introduction of a lumped-parameter model based on momentum balance entails an easier calibration of the model parameters: Phenomenological-based models, on the other hand, typically have numerous parameters. This model is coupled to 3D Navier–Stokes equations describing the blood flow, where the moving valve leaflets are immersed in the fluid domain by a resistive method. A stabilized finite element method with a BDF time scheme is adopted for the discretization of the coupled problem, and the computational results show the suitability of the system in representing the leaflet motion, the blood flow in the ascending aorta, and the pressure jump across the leaflets. Both physiological and stenotic configurations are investigated, and we analyze the effects of different treatments for the leaflet velocity on the blood flow.PH-S
Accelerating Algebraic Multigrid Methods via Artificial Neural Networks
Full text linkWe present a novel deep learning-based algorithm to accelerate - through the
use of Artificial Neural Networks (ANNs) - the convergence of Algebraic
Multigrid (AMG) methods for the iterative solution of the linear systems of
equations stemming from finite element discretizations of Partial Differential
Equations (PDE). We show that ANNs can be successfully used to predict the
strong connection parameter that enters in the construction of the sequence of
increasingly smaller matrix problems standing at the basis of the AMG
algorithm, so as to maximize the corresponding convergence factor of the AMG
scheme. To demonstrate the practical capabilities of the proposed algorithm,
which we call AMG-ANN, we consider the iterative solution of the algebraic
system of equations stemming from finite element discretizations of
two-dimensional model problems. First, we consider an elliptic equation with a
highly heterogeneous diffusion coefficient and then a stationary Stokes
problem. We train (off-line) our ANN with a rich dataset and present an
in-depth analysis of the effects of tuning the strong threshold parameter on
the convergence factor of the resulting AMG iterative scheme
Isogeometric analysis for second order partial differential equations on surfaces
No full textWe consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds, specifically on surfaces in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by B-splines or NURBS. Our analysis addresses linear, nonlinear, time dependent, and eigenvalues problems involving the Laplace-Beltrami operator on surfaces. Moreover, we propose a priori error estimates under h-refinement in the general case of second order PDEs on the lower dimensional manifolds. We highlight the accuracy and efficiency of Isogeometric Analysis with respect to the exactness of the geometrical representations of the surfaces
Multipatch Isogeometric Analysis for electrophysiology: Simulation in a human heart
Full text linkIn the framework of cardiac electrophysiology for the human heart, we apply multipatch NURBS-based Isogeometric Analysis for the space discretization of the Monodomain model. Isogeometric Analysis (IGA) is a technique for the solution of Partial Differential Equations (PDEs) that facilitates encapsulating the exact representation of the computational geometry by using basis functions with high-order continuity. IGA features very small numerical dissipation and dispersion when compared to other methods for the solution of PDEs. The use of multiple patches allows to overcome the conventional limitations of single patch IGA, thanks to the gained flexibility in the design of the computational domain, especially when its representation is quite involved as in bioengineering applications. We propose two algorithms for the preprocessing of CAD models of complex surface and volumetric NURBS geometries with cavities, such as atria and ventricles: our purpose is to obtain geometrically and parametrically conforming NURBS multipatch models starting from CAD models. We employ those algorithms for the construction of an IGA realistic representation of a human heart. We apply IGA for the discretization of the Monodomain equation, which describes the evolution of the cardiac action potential in space and time at the tissue level. This PDE is coupled with suitable microscopic models to define the behavior at cellular scale: the Courtemanche-Ramirez-Nattel model for the atrial simulation, and the Luo-Rudy model for the ventricular one. Numerical simulations on realistic human atria and ventricle geometries are carried out, obtaining accurate and smooth excitation fronts by combining IGA with the multipatch approach for the geometrical representation of the computational domains, either surfaces for the atria or solids for the ventricles. (C) 2021 The Authors. Published by Elsevier B.V.CMC
Fast and robust parameter estimation with uncertainty quantification for the cardiac function
No full textBackground and objectives: Parameter estimation and uncertainty quantification are crucial in computational cardiology, as they enable the construction of digital twins that faithfully replicate the behavior of physical patients. Many model parameters regarding cardiac electromechanics and cardiovascular hemodynamics need to be robustly fitted by starting from a few, possibly non-invasive, noisy observations. Moreover, short execution times and a small amount of computational resources are required for the effective clinical translation. Methods: In the framework of Bayesian statistics, we combine Maximum a Posteriori estimation and Hamiltonian Monte Carlo to find an approximation of model parameters and their posterior distributions. Fast simulations and minimal memory requirements are achieved by using an accurate and geometry- specific Artificial Neural Network surrogate model for the cardiac function, matrix–free methods, automatic differentiation and automatic vectorization. Furthermore, we account for the surrogate modeling error and measurement error. Results: We perform three different in silico test cases, ranging from the ventricular function to the entire cardiocirculatory system, involving whole-heart mechanics, arterial and venous hemodynamics. By employing a single central processing unit on a standard laptop, we attain highly accurate estimations for all model parameters in short computational times. Furthermore, we obtain posterior distributions that contain the true values inside the 90% credibility regions. Conclusions: Many model parameters regarding the entire cardiovascular system can be fastly and robustly identified with minimal hardware requirements. This can be achieved when a small amount of non-invasive data is available and when high levels of signal-to-noise ratio are present in the quantities of interest. With these features, our approach meets the requirements for clinical exploitation, while being compliant with Green Computing practices
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