Results 1  10
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23
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Cited by 99 (12 self)
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Reduced basis method for finite volume approximations of parametrized linear evolution equations
 M2AN, Math. Model. Numer. Anal
"... The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RBmethods have so far mainly been applied to finite element sch ..."
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Cited by 66 (24 self)
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The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RBmethods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for general evolution problems and the derivation of rigorous aposteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized. This is the basis for a rapid online computation in case of multiplesimulation requests. We introduce a new offline basisgeneration algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convectiondiffusion problem demonstrate the efficient applicability of the approach. 1
Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation
 SIAM J. Sci. Comput
"... Abstract. We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtain ..."
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Cited by 34 (17 self)
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Abstract. We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEIgreedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronised way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space–compression or even space–dimensionality reduction. We perform empirical investigations of the error convergence and run–times. In all cases we obtain a good run–time acceleration.
A reduced basis method for evolution schemes with parameterdependent explicit operators
 ETNA, Electron. Trans. Numer. Anal
"... Abstract. During the last decades, reduced basis (RB) methods have been developed to a wide methodology for model reduction of problems that are governed by parametrized partial differential equations (P 2 DEs). In particular equations of elliptic and parabolic type for linear, low polynomial or mon ..."
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Cited by 20 (9 self)
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Abstract. During the last decades, reduced basis (RB) methods have been developed to a wide methodology for model reduction of problems that are governed by parametrized partial differential equations (P 2 DEs). In particular equations of elliptic and parabolic type for linear, low polynomial or monotonic nonlinearities have been treated successfully by RB methods using finite element schemes. Due to the characteristic offlineonline decomposition, the reduced models often become suitable for a multiquery or realtime setting, where simulation results, such as fieldvariables or output estimates, can be approximated reliably and rapidly for varying parameters. In the current study, we address a certain class of timedependent evolution schemes with explicit discretization operators that are arbitrarily parameter dependent. We extend the RBmethodology to these cases by applying the empirical interpolation method to localized discretization operators. The main technical ingredients are: (i) generation of a collateral reduced basis modelling the effects of the discretization operator under parameter variations in the offlinephase and (ii) an online simulation scheme based on a numerical subgrid and localized evaluations of the evolution operator. We formulate an aposteriori error estimator for quantification of the resulting reduced simulation error. Numerical experiments on a parametrized convection problem, discretized with a finite volume scheme, demonstrate the applicability of the model reduction technique. We obtain a parametrized reduced model, which enables parameter variation with fast simulation response. We quantify the computational gain with respect to the nonreduced model and investigate the error convergence.
Numerical solution of parametrized NavierStokes equations by reduced basis methods
 Numer. Methods Partial Differential Equations
"... empirical interpolation. We apply the reduced basis method to solve NavierStokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized nonlinear transport term in the reduced basis framework, including the case of nonaffine parametric dependence tha ..."
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Cited by 19 (11 self)
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empirical interpolation. We apply the reduced basis method to solve NavierStokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized nonlinear transport term in the reduced basis framework, including the case of nonaffine parametric dependence that is treated by an empirical interpolation method. This method features (i) a rapid global convergence owing to the property of the Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in the parameter space, and (ii) the offline/online computational procedures which decouple the generation and projection stages of the approximation process. This method is well suited for the repeated and rapid evaluations required
REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATION OF EVOLUTION EQUATIONS ON PARAMETRIZED GEOMETRIES
"... parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on timeconsuming parameterstudies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such P2DEs. Reduc ..."
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Cited by 10 (7 self)
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parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on timeconsuming parameterstudies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such P2DEs. Reduced Basis (RB) methods are a means to achieve this goal. These methods have gained popularity over the last few years for model reduction of finite element approximations of elliptic and instationary parabolic equations. We present a RB method for parabolic problems with general geometry parameterization and finite volume (FV) approximations. After a mapping on a reference domain, the parabolic equation leads to a convectiondiffusionreaction equation with anisotropic diffusion tensor. Suitable FV schemes with gradient reconstruction allow to discretize such problems. A model reduction of the resulting numerical scheme can be obtained by an RB technique. We present experimental results, that demonstrate the applicability of the RB method, in particular the computational acceleration. Key words. Reduced basis methods, model reduction, geometry transformation, heat equation AMS subject classifications. 76M12, 76R50, 35K05
Reduced basis approximation and error bounds for potential flows
"... in parametrized geometries ..."
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A Reduced Basis Hybrid Method for the Coupling of Parametrized Domains Represented by Fluidic Networks
"... In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent. The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar bl ..."
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Cited by 6 (2 self)
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In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent. The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled. The RBHM is built upon the reduced basis element method (RBEM) and it takes advantage from both the reduced basis methods (RB) and the domain decomposition method. We move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, representative solutions, corresponding to the same governing partial differential equations, are computed for different values of some parameters of interest, representing, for example, the deformation of the blocks. A generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration. The desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions. The geometrical parametrization of the domain, by transfinite mapping, induces nonaffine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a sub– sequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem
SPACEADAPTIVE REDUCED BASIS SIMULATION FOR TIMEDEPENDENT PROBLEMS
"... Abstract. We address the task of model reduction of parametrized evolution equations. Detailed simulations of such partial differential equations are frequently expensive to compute due to the space resolution of the discretization and not suitable for multiquerysettings, i.e. multiple simulation ..."
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Cited by 3 (1 self)
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Abstract. We address the task of model reduction of parametrized evolution equations. Detailed simulations of such partial differential equations are frequently expensive to compute due to the space resolution of the discretization and not suitable for multiquerysettings, i.e. multiple simulation requests with varying parameters. Reduced basis (RB) methods are increasingly popular methods to solve such parametrized problems. The currently existing RBmethods for timedependent problems use identical dimensionality N of the reduced model for all timesteps. This may be suboptimal, as different solution structures may require different dimensionalities N at different times, or a prescribed error tolerance should be obtained. In the current presentation, we extend a recently introduced RBscheme, in order to adaptively choose N in time. This adjustment of the model dimension is based on a posteriori error estimators, which can be computed rapidly during the online simulation. We provide experimental insights based on two advectiondiffusion problems. We demonstrate, that the trialanderror process of the Nfixed approach for obtaining a desired model accuracy, can be circumvented by the Nadaptive approach. In examples, where the solution complexity is changing over time, the Nadaptive approach yields an overall gain in computation time. 1
Reduced Basis Method for Nonlinear Explicit Finite Volume Approximations of Hyperbolic Evolution Equations
"... We address the task of model reduction for parametrized scalar hyperbolic or convection dominated parabolic evolution equations. These are problems which are characterized by a parameter vectorµ ∈ P from some set of possible parameters P ⊂ Rp, and the evolution problem is to determine u(t,µ) ∈ L∞(Ω ..."
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We address the task of model reduction for parametrized scalar hyperbolic or convection dominated parabolic evolution equations. These are problems which are characterized by a parameter vectorµ ∈ P from some set of possible parameters P ⊂ Rp, and the evolution problem is to determine u(t,µ) ∈ L∞(Ω) ∩ L2(Ω) on an open bounded domain Ω ⊂ Rd and finite time interval t ∈ [0, T], T> 0 such that ∂tu(µ) + L(t,µ)(u(t,µ)) = 0, u(0,µ) = u0(µ), and suitable boundary conditions are satisfied. Here u0(µ) ∈ L∞(Ω) ∩ L2(Ω) are the parameterdependent initial values, L(t,µ) is the parameter dependent spatial differential operator. We assume a discretization with explicit finite volume schemes and first order time discretization, which yields discrete solutions ukH(µ) ∈ WH, k = 0,...,K in the Hdimensional finite volume space WH ⊂ L∞(Ω) ∩ L2(Ω) approximating u(tk,µ) at the time instants 0 = t0 < t1 <... < tK = T. Such detailed simulations are frequently expensive to compute due to the space resolution and not suitable for use in multiquery settings, i.e. multiple simulation requests with varying parameters µ. Reduced Basis Methods are increasingly popular methods to solve such parametrized problems, aiming at a problemdependent simulation scheme, that approximates the detailed solutions ukH(µ) by efficiently computed reduced solutions ukN ∈ WN. Here WN ⊂ L2(Ω) is an Ndimensional reduced basis space with suitable reduced basis ΦN which is generated in a problem specific way based on snapshots of detailed solutions for suitably chosen time instants ki and