@article {Simula.SC.74,
title = {Global C{^1} Maps on General Domains},
journal = {Mathematical Models and Methods in Applied Sciences (M3AS)},
volume = {19},
number = {5},
year = {2009},
note = {{\textcopyright} Copyright World Scientific Publishing Company},
pages = {803-832},
abstract = {In many contexts, there is a need to construct C{^1} maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C{^1}. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are dis- cussed and compared: the transfinite interpolation proposed by Gordon and Hall, the {\textquoteright}pseudo-harmonic{\textquoteright} extension proposed by Gordon and Wixom, a new generalization of the Gordon-Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean value extension proposed by Floater.},
doi = {10.1142/S0218202509003632},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}
@misc {Simula.SC.440,
title = {Real-Time Computation of CSF Flow},
howpublished = {Invited talk at a seminar for cerebrospinal fluid flow, University of Wisconsin, Madison, U.S.A., January 16th},
year = {2009},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist and Simone Deparis}
}
@misc {Simula.SC.438,
title = {Real-Time Flow Simulation},
howpublished = {Invited talk at The Chiari Institute, New York, U.S.A., January 14th},
year = {2009},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist and Simone Deparis}
}
@misc {Simula.SC.622,
title = {The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case},
howpublished = {Invited talk at the reduced basis mini-symposium at ICOSAHOM 09, Trondheim, Norway, June 22-26},
year = {2009},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}
@misc {Simula.SC.624,
title = {The Spectral Element Method Used to Assess the Quality of a Deformed Mesh},
howpublished = {contributed talk at ICOSAHOM 09, Trondheim, Norway, June 22-26},
year = {2009},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}
@inbook {Simula.SC.26,
title = {The Reduced Basis Element Method for Fluid Flows},
booktitle = {Analysis and Simulation of Fluid Dynamics},
series = {Advances in Mathematical Fluid Mechanics},
year = {2007},
pages = {129-154},
publisher = {Birkh{\"a}user},
organization = {Birkh{\"a}user},
chapter = {8},
address = {Basel},
abstract = {The reduced basis element approximation is a discretization method for solving partial differential equations that has inherited features from the domain decomposition method and the reduced basis approximation paradigm in a similar way as the spectral element method has inherited features from domain decomposition methods and spectral approximations. We present here a review of the method directed to the application of fluid flow simulations in hierarchical geometries. We present the rational and the basics of the method together with details on the implementation. We illustrate also the rapid convergence with numerical results.},
isbn = {978-3-7643-7741-0},
author = {Alf Emil L{\o}vgren and Einar M. R{\o}nquist and Yvon Maday},
editor = {C. Calgaro and J.-F. Coulombel and T. Goudon}
}
@misc {Simula.SC.130,
title = {Reduced Basis Modeling of Complex Flow Systems},
howpublished = {Invited talk at the Modelling and Scientific Computing group at EPFL, Switzerland, November, 2007. Also presented at the workshop on Modelling and Computation of Biomedical Processes at CBC, Simula, June 7-14},
year = {2007},
author = {Alf Emil L{\o}vgren and Einar M. R{\o}nquist and Yvon Maday}
}
@article {Maday.2006.1,
title = {The Parareal-in-Time Algorithm: Basics, Stability Analysis and More},
journal = {journal},
year = {2006},
abstract = {The parareal-in-time algorithm allows to take benefit of a parallel architecture of large scale computing resources in order to decrease the restitution time for the numerical simulation of time dependent problems. The method can be presented as a predictor corrector scheme where the predictor is based on a coarse grain and inexpensive simulation, solved in a serial manner and the expensive corrector can be spread on different processors. Like for domain decomposition techniques, the algorithm is based on the decomposition of the global simulation time interval into slabs, each slab being dedicated to a processor. After reminding the basics of the approximation, we discuss the stability of the algorithm for a system of autonomous differential equations. The stability function for the algorithm is derived and analyzed, based on various choices of schemes in time for the coarse and the fine propagator. We then present some complementary analysis that provides the frame for the definition of the cheap coarse simulation. Finally, numerical results for the viscous Burger{\textquoteright}s equation are presented.},
author = {Yvon Maday and Einar M. R{\o}nquist and Gunnar Andreas Staff}
}
@inproceedings {Lovgren.2006.4,
title = {A Reduced Basis Element Method for Complex Flow Systems},
journal = {ECCOMAS CFD 2006, European Conference on Computational Fluid Dynamics},
year = {2006},
publisher = {TU Delft},
type = {Conference},
abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations within domains belonging to a certain class. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into smaller blocks that are topologically similar to a few reference shapes (or generic computational parts). Associated with each reference shape are precomputed solutions corresponding to the same governing partial differential equation, and similar boundary conditions, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to the computational domain is then taken to be a linear combination of the precomputed solutions, mapped from the reference shapes for the different blocks to the actual domain. The variation of the geometry induces non-affine parameter dependence, and we apply the empirical interpolation technique to achieve an offline/online decoupling of the reduced basis procedure. Some results for incompressible flow systems have already been presented, and the focus here will be to further improve the offline/online decoupling of problems with non-affine parameter dependence. To this end we use the empirical interpolation method to approximate the parameter depen- dent operators. We also present a generalized transfinite interpolation method intended to produce global C1 mappings from the reference shapes to each corresponding block of the computational domain.},
isbn = {90-9020970-0},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist},
editor = {P. Wesseling and E. O{\~n}ate and J. P{\'e}riaux}
}
@article {Lovgren.2006.1,
title = {A Reduced Basis Element Method for the Steady Stokes Problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
volume = {40},
number = {3},
year = {2006},
pages = {529-552},
abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual computational part. We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition, to a posteriori error estimation, and to {\textquotedblleft}gluing{\textquotedblright} the local solutions together in the multidomain case.},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}
@article {Lovgren.2006.3,
title = {A Reduced Basis Element Method for the Steady Stokes Problem: Application to Hierarchical Flow Systems},
journal = {Model. Identif. Control},
volume = {27},
number = {2},
year = {2006},
pages = {79-94},
abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations, and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of sub-domains that are similar to a few reference domains or generic computational parts. Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to the mapping of the velocity fields, to satisfying the inf-sup condition, and to gluing the local solutions together in the multi domain case.We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}
@inproceedings {Lovgren.2005.2,
title = {A Reduced Basis Element Method for the Steady Stokes Problem: Application to Hierarchical Flow Systems},
journal = {SIMS 2005, 46th Conference on Simulation and Modeling},
year = {2005},
publisher = {SIMS 2005 and Tapir Academic Press},
type = {1},
address = {Trondheim, Norway},
abstract = {The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations, and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of sub-domains that are similar to a few reference domains or generic computational parts. Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to the mapping of the velocity fields, to satisfying the inf-sup condition, and to gluing the local solutions together in the multi domain case.We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.},
author = {Alf Emil L{\o}vgren and Yvon Maday and Einar M. R{\o}nquist}
}