Karen Veroy

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced-basis approximations - (Galerkin) projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem. In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation - a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin-earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equation illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation.

Certified Real-Time Solution of Parametrized Partial Differential Equations

Engineering analysis requires the prediction of (say, a single) selected “output” se relevant to ultimate component and system performance:* typical outputs include energies and forces, critical stresses or strains, flowrates or pressure drops, and various local and global measures of concentration, temperature, and flux. These outputs are functions of system parameters, or “inputs”, μ, that serve to identify a particular realization or configuration of the component or system: these inputs typically reflect geometry, properties, and boundary conditions and loads; we shall assume that μ is a P-vector (or P-tuple) of parameters in a prescribed closed input domain D ⊂ ℝp. The input-output relationship se(μ): D → ℝ thus encapsulates the behavior relevant to the desired engineering context.

Certified reduced basis methods for parametrized saddle point problems

We present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems. First, we develop new a posteriori error estimates that, unlike earlier approaches, provide upper bounds for the errors in the approximations of the primal variable and the Lagrange multiplier separately. The proposed method is an application of Brezzis theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods. Second, based on an analysis of Brezzis theory, we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost. Finally, we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and, compared to earlier approaches, computationally much more efficient. The method is applied to a Stokes flow problem in a two-dimensional channel with a parametrized rectangular obstacle. ...

REDUCED BASIS A POSTERIORI ERROR BOUNDS FOR THE STOKES EQUATIONS IN PARAMETRIZED DOMAINS: A PENALTY APPROACH

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method, built upon the penalty formulation for saddle point problems, provides error bounds not only for the velocity but also for the pressure approximation, while simultaneously admitting affine geometric variations with relative ease. The essential ingredients are: (i) dimension reduction through Galerkin projection onto a low-dimensional reduced basis space; (ii) stable, good approximation of the pressure through supremizer-enrichment of the velocity reduced basis space; (iii) optimal and numerically stable approximations identified through an efficient greedy sampling method; (iv) certainty, through rigorous a posteriori bounds for the errors in the reduced basis approximation; and (v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a (parametrized) rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter, and that the effects of the penalty parameter are relatively benign.

Analysis of Dispersive Waves Using the Wavelet Transform

Lamb waves have long been recognized to be a valuable tool in the nondestructive testing of plates and shells. The patterns of dispersion which characterize the propagation of Lamb waves through a material may be used to determine the thickness of a plate, the elastic properties of materials, as well as the condition and integrity of the sample. Unfortunately, the analysis of Lamb wave signals is complicated by the existence of at least two modes at any given frequency [1]. To efficiently determine the dispersive characteristics of a plate, it is therefore necessary to have a technique for analyzing broadband multimode signals. Methods for the time-frequency analysis of signals are especially useful in determining dispersion curves, since they reveal the time variation of each frequency component of the signal.

Time-frequency analysis of broadband dispersive waves using the wavelet transform

We explore the use of the Morlet wavelet transform in the time-frequency analysis of dispersive waves. The Morlet wavelet transform is a windowed Fourier transform wherein the window size varies with frequency. This allows for more effective analysis of the time-frequency characteristics of broadband signals. The Morlet wavelet transform coefficients evaluated at a particular center frequency indicate the group delay or time of arrival of a wave group whose Fourier spectrum is centered at the said frequency. The method is applied to both simulated and experimental data. In this study, multi-mode Lamb waves induced by laser generated ultrasound in a thin plate are considered.

Spectrotemporal analysis of guided-wave pulse-echo signals: Part 2. Numerical and experimental investigations

Part 2 of this paper seeks the numerical and experimental procedures for implementing the Morlet wavelet transform for the spectrotemporal analysis of multimode dispersive waves. The sensitivity due to the spectral parameters α and ω0, defining the bandwidth and center frequency of the mother wavelet, is studied through an error analysis of the simulation signals. The algorithm for determining the group delays is extended to multimode signals by introducing a thresholding technique. The ultimate goal of this study is to nondestructively detect and locate discontinuities in a thin plate. This is demonstrated by experimentally measuring the travel distance of a Lamb wave from a single waveform.

Spectrotemporal analysis of guided-wave pulse-echo signals: Part 1. Dispersive systems

In this paper, a wavelet-based time frequency analysis is presented to analyze guided-wave signals for rapid inspection of thin-walled structural members. The overall objective is to detect and locate discontinuities using a single broadband signal. Part 1 of this paper shows how the wavelet transform can be used to analyze a dispersive system. A straightforward procedure is developed to extract group delay information from the computed wavelet transform coefficients. The procedure is demonstrated by a simulation study for single-mode and simple dual-mode dispersion signals. Part 2 is an experimental study of multimode dispersion.

Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

We propose a certified reduced basis approach for the strong- and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.