Finite element methods for Maxwell's equations.

This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

The spatially and temporally variable parameters and inputs to complex groundwater models typically result in long runtimes which hinder comprehensive calibration, sensitivity, and uncertainty analysis. Surrogate modeling aims to provide a simpler, and hence faster, model which emulates the specified output of a more complex model in function of its inputs and parameters. In this review paper, we summarize surrogate modeling techniques in three categories: data-driven, projection, and hierarchical-based approaches. Data-driven surrogates approximate a groundwater model through an empirical model that captures the input-output mapping of the original model. Projection-based models reduce the dimensionality of the parameter space by projecting the governing equations onto a basis of orthonormal vectors. In hierarchical or multifidelity methods the surrogate is created by simplifying the representation of the physical system, such as by ignoring certain processes, or reducing the numerical resolution. In discussing the application to groundwater modeling of these methods, we note several imbalances in the existing literature: a large body of work on data-driven approaches seemingly ignores major drawbacks to the methods; only a fraction of the literature focuses on creating surrogates to reproduce outputs of fully distributed groundwater models, despite these being ubiquitous in practice; and a number of the more advanced surrogate modeling methods are yet to be fully applied in a groundwater modeling context.

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A review of surrogate models and their application to groundwater modeling

Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

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Certified reduced basis approximation for parametrized partial differential equations and applications

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

We propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic Maxwell's equations. The truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous Galerkin approximation of the linear partial differential equation. The formulation allows the use of different approximation spaces for solving the primal and the dual truth approximation problems to respect the characteristics of both problem types, leading to an overall reduction in the off-line computational effort. The main features of the method are the following: (i) rapid convergence on the entire representative set of parameters, (ii) rigorous a posteriori error estimators for the output, and (iii) a parameter independent off-line phase and a computationally very efficient on-line phase to enable the rapid solution of many-query problems arising in control, optimization, and design. The versatility and performance of this approach is shown through a numerical experiment, illustrating the modeling of material variations and problems with resonant behavior.

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Certified Reduced Basis Methods and Output Bounds for the Harmonic Maxwell's Equations

In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In (Huynh et al., C. R. Acad. Sci. Paris Ser. I Math. 345 (2007) 473-478), the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on- line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound is obtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.

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Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell"s problem

A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations

We present a reduced basis element method (RBEM) for the time-harmonic Maxwell’s equation. The RBEM is a Reduced Basis Method (RBM) with parameters describing the geometry of the computational domain, coupled with a domain decomposition method. The basic idea is the following. First, we decompose the computational domain into a series of subdomains, each of which is deformed from some reference domain. Then, we associate with each reference domain precomputed solutions to the same governing partial differential equation, but with different choices of deformations. Finally, one seeks the approximation on a new domain as a linear combination of the corresponding precomputed solutions on each subdomain. Unlike the work on RBEM for thermal fin and fluid flow problems, we do not need a mortar type method to “glue” the various local functions. This “gluing” is done “automatically” thanks to the use of a discontinuous Galerkin method. We present the rationale for the method together with numerical results showing exponential convergence for the simulation of a metallic pipe with both ends open.

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/A%20seamless%20reduced%20basis%20element%20method%20for%202D%20Maxwell's%20Problem.pdf

A seamless reduced basis element method for 2D Maxwell"s problem: An introduction

In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who prefer a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.

Yanlai Chen

Papers

Certified Reduced Basis Methods and Output Bounds for the Harmonic Maxwell's Equations

Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell"s problem

A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations

A seamless reduced basis element method for 2D Maxwell"s problem: An introduction

Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework