Rapid Reliable Solution
of the Parametrized Partial Differential Equations
of Continuum Mechanics and Transport



Methodology: Reduced Basis Methods and A Posteriori Error Estimators

We include here selected recent theses from our group relevant to reduced basis approximation and a posteriori error estimation.

L Dedè, Adaptive and Reduced Basis Methods for Optimal Control Problems in Environmental Applications, PhD Thesis, MOX-Dipartimento di Matematica, Politecnico di Milano, Italy, June 2008.full text

DBP Huynh, Reduced Basis Approximation and Application to Fracture Problems, PhD Thesis, Singapore-MIT Alliance, National University of Singapore, October 2007.full text
This thesis focuses on the development of reduced basis methods for calculation of Stress Intensity Factors (SIF) in linear elasticity. Key ingredients include an expanded linear (non-coercive) formulation for quadratic outputs such as the Energy Release Rate (ERR); associated reduced basis approximations, (Natural-Norm as well as Successive Constraint Method) inf-sup lower bounds, and a posteriori error bounds; sensitivity methods for accurate extraction of the ERR and subsequently SIF; and finally (symbolic and numerical) mapping techniques for efficient and automatic treatment of crack geometry. Mode I, Mode II, and mixed Mode I-II cracks are considered, as well as many-query applications such as fatigue-induced crack growth and design against fracture. (For a related but simpler — and also less rigorous — approach to SIFs, see Arch Comput Methods Eng 2008 doi: 10.1007/s11831-008-9019-9.)

S Sen, Reduced Basis Approximation and A Posteriori Error Estimation for Non-Coercive Elliptic Problems: Application to Acoustics, PhD Thesis, Massachusetts Institute of Technology, June 2007. full text
This thesis focuses on the development of reduced basis methods for problems in acoustics with particular emphasis on acoustic waveguide applications. The key ingredients are complex-valued field descriptions; quadratric ``intensity" outputs such as the Transmission Coefficient; (Successive Constraint Method) inf-sup lower bounds and a posteriori error bounds for non-coercive non-Hermitian operators; and appropriate waveguide radiation boundary conditions. The methodology is applied to the characterization and design of simple acoustic filters.

GSH Pau, Reduced Basis Method for Quantum Models of Crystalline Solids, PhD Thesis, Massachusetts Institute of Technology, June 2007. full text
This thesis focuses on reduced basis methods for multiple-eigenvalue/eigenfunction problems with application to quantum mechanics (electronic structure) calculations. The key ingredients are appropriate spaces for accurate multiple-eigenvalue/eigenfunction approximation; asymptotic a posteriori error bounds for multiple eigenvalues/eigenfunctions; and finally treatment of (non-polynomial) nonlinear eigenproblems by the Empirical Interpolation Method. Two applications/model systems, corresponding to two rather different reduced basis contexts, are considered: electronic ground state calculations based on Density Functional Theory; and spectral property calculations based on band structure.

A Tan Yong Kwang, Reduced Basis Methods for 2nd Order Wave Equation: Application to One Dimensional Seismic Problem, Masters Thesis, Singapore-MIT Alliance, National University of Singapore, 2006.full text
This thesis focuses on RB approximation and a posteriori error estimation for the (Newmark-in-time discretization of the) second order wave equation. Numerical tests are performed for a simple problem in one space dimension. For sufficiently smooth initial conditions the RB approximation converges rapidly. The a posteriori error bounds for the usual (kinetic plus potential) energy are stable with respect to the truth FE discretization parameters and also the dimension of the RB approximation space.

G Rozza, Shape Design by Optimal Flow Control and Reduced Basis Techniques: Applications to Bypass Configurations in Haemodynamics, PhD Thesis, École Polytechnique Fédérale de Lausanne, November 2005.full text
This (reduced basis component of this) thesis focuses on reduced basis methods for the incompressible Stokes and Navier-Stokes equations, with particular emphasis on (i) stable "supremizer-enhanced" velocity spaces to ensure discrete div-stability, (ii) adjoint-based treatment of pressure and velocity output functionals, and (iii) empirical interpolation methods for non-affine/curvilinear geometries. The methods are applied to several moderate Reynolds number flow problems, including a many-parameter model for an arterial stenosis bypass device.

NC Nguyen, Reduced-Basis Approximation and A Posteriori Error Bounds for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis, PhD Thesis, Singapore-MIT Alliance, National University of Singapore, July 2005.full text
This thesis focuses on methods for (i) noncoercive problems — the Helmholtz equation of acoustics and elasticity, (ii) problems non-affine in the parameter, and (iii) problems (highly) non-linear in the state variable; for the latter two classes of problems, the "empirical interpolation method" of Paper 6 and Paper 13 above is developed and applied. The thesis also considers the systematic incorporation of reduced basis methods into robust inverse problem formulations (see Paper 10 above); examples include nondestructive evaluation for cracks and material damage in elastic solids, and acoustic inverse scattering for obstacle identification.

MA Grepl, Reduced-basis Approximations and A Posteriori Error Estimation for Parabolic Partial Differential Equations, PhD Thesis, Massachusetts Institute of Technology, May 2005.full text
This thesis focuses on methods for parabolic partial differential equations, including linear heat equations (as described in Paper 8 above), the convection-diffusion equation, and problems non-affine in the parameter and/or non-linear in the state variable (see Papers 6 and 13 above). Emphasis is on effective adjoint approximation, rigorous a posteriori error estimation, and optimal sampling procedures. Examples include nondestructive evaluation of concrete delamination cracks, identification of passive scalar sources, and reaction-diffusion (spontaneous ignition) systems.

K Veroy, Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications, PhD Thesis, Massachusetts Institute of Technology, June 2003.full text

DV Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations, PhD Thesis, Massachusetts Institute of Technology, February 2003.full text
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