Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> The Newton method to handle nonlinear terms with ROMs of dynamical systems <s> Abstract Methods for the approximation of large-scale dynamical systems will be surveyed. There are mainly two families namely, the SVD-based and Krylov-based approximation methods. The former family is based on the singular value decomposition and the second on moment matching. While the former has many desirable properties including an error bound, it cannot be applied to systems of high complexity. The strength of the latter on the other hand, is that it can be implemented iteratively and is thus appropriate for application to high complexity systems. An effort to combine the best attributes of these two families leads to a third class of approximation methods, which will be referred to as SVD/Krylov. Following a survey of these methods we will conclude with a new result concerning model reduction with preservation of passivity which is appropriate for application to large-scale circuits arising in VLSI chip performance verification. <s> BIB001 </s> Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> The Newton method to handle nonlinear terms with ROMs of dynamical systems <s> Lagrangian interpolation is a classical way to approximate general ::: functions by finite sums of well chosen, pre-defined, linearly ::: independent interpolating functions; it is much simpler to implement than ::: determining the best fits with respect to some Banach (or even Hilbert) ::: norms. In addition, only partial knowledge is required (here values on some ::: set of points). The problem of defining the best sample of points is ::: nevertheless rather complex and is in general open. In this paper we ::: propose a way to derive such sets of points. We do not claim that the ::: points resulting from the construction explained here are optimal in any ::: sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the ::: process is very general and simple to implement, and compared to situations ::: where the best behavior is known, it is relatively competitive. <s> BIB002 </s> Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> The Newton method to handle nonlinear terms with ROMs of dynamical systems <s> A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results. <s> BIB003 </s> Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> The Newton method to handle nonlinear terms with ROMs of dynamical systems <s> An approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. This method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow. The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. For linear problems, the desired transformation is induced by a special inner product, termed the "symmetry inner product", which is derived herein for several systems of physical interest. Connections are established between the proposed approach and other stability-preserving model reduction methods, giving the paper a review flavor. More specifically, it is shown that a discrete counterpart of this inner product is a weighted L 2 inner product obtained by solving a Lyapunov equation, first proposed by Rowley et al. and termed herein the "Lyapunov inner product". Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases. <s> BIB004
Reduced-order modeling is a general methodology to determine the principal information of a general high-dimensional problem and then reduce the problem, for example by projection. Reduction is generally possible when the M-Kolmogorov width into an admissible close set U of a Banach space V is rather small for a rather small integer M (the dimension of the approximate space). One of the main motivations to do that is to strongly reduce the computational cost for the numerical solution. Even if there are recent advances in nonlinear reduced order modeling, in particular with the empirical interpolation method (EIM) proposed by Maday et al. BIB002 , or discrete empirical interpolation method (DEIM) by Chaturantabut and Sorensen BIB003 , there are still some issues and open problems for nonlinear time-dependent problems. Dealing with general nonlinear terms and reduced-order modeling for dynamical systems may be a difficult task, because: • reduced-order models are expected to reproduce the stability of the system (for instance in the sense of Lyapunov, see BIB004 on this subject); • the local dynamics has to be reproduced, at least "at first order", involving a compatibility of the spectral properties between full and reduced systems; • the area visited by the trajectories into the state-space may be defined over a nonlinear manifold rather than in a linear subspace. Thus nonlinear dimensionality reduction methods would be better candidates for reduction. Balanced truncation strategy BIB001  for example is a trade-off in the reduction process to provide sufficient accuracy for controllability and observability of dynamical systems. However the theory mainly deals with linear time-invariant (LTI) systems. For time-dependent problems, one can adopt a greedy incremental strategy during time by adapting/enriching the low-dimensional subspace when the principal components are changing during time. But the price to pay is to online evaluate some (high-dimensional) nonlinear terms to control the error, what can be a penalizing factor of performance. If there is no other choice, parallel-in-time computing once again appears to be a complementary tool to keep global performance of the method.