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Optimization problems constrained by nonlinear partial differential equations have been the focus of intense research in scientific computing lately. Current methods for the parallel numerical solution of such problems involve sequential quadratic programming (SQP), with either reduced or full space approaches. In this paper we propose and investigate a class of parallel full space SQP Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithms. In LNKSz, a Lagrangian functional is formed and differentiated to obtain a Karush-Kuhn-Tucker (KKT) system of nonlinear equations. Inexact Newton method with line search is then applied. At each Newton iteration the linearized KKT system is solved with a Schwarz preconditioned Krylov subspace method. We apply LNKSz to the parallel numerical solution of some boundary control problems of two-dimensional incompressible Navier-Stokes equations. Numerical results are reported for different combinations of Reynolds number, mesh size and number of parallel processors. We also compare the application of LNKSz method to flow control problems against the application of NKSz method to flow simulation problems.

We address general approaches to the rational selection and validation of mathematical and computational models of tumor growth using methods of Bayesian inference. The model classes are derived from a general diffuse-interface, continuum mixture theory and focus on mass conservation of mixtures with up to four species. Synthetic data are generated using higher-order base models. We discuss general approaches to model calibration, validation, plausibility, and selection based on Bayesian-based methods, information theory, and maximum information entropy. We also address computational issues and provide numerical experiments based on Markov chain Monte Carlo algorithms and high performance computing implementations.

QUESO is a collection of statistical algorithms and programming constructs supporting research into the uncertainty quantification (UQ) of models and their predictions. It has been designed with three objectives: it should (a) be sufficiently abstract in order to handle a large spectrum of models, (b) be algorithmically extensible, allowing an easy insertion of new and improved algorithms, and (c) take advantage of parallel computing, in order to handle realistic models. Such objectives demand a combination of an object-oriented design with robust software engineering practices. QUESO is written in C++, uses MPI, and leverages libraries already available to the scientific community. We describe some UQ concepts, present QUESO, and list planned enhancements.

Abstract. We develop a class of V-cycle type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by nonlinear partial differential equations. Straightforward extensions of the one-level RAS to multilevel do not work due to the pollution effects of the coarse interpolation. We then introduce, in this paper, a pollution removing coarse-to-fine interpolation scheme for one of the components of the multi-component linear system, and show numerically that the combination of the new interpolation scheme with the RAS smoothed multigrid method provides an effective family of techniques for solving rather difficult PDE-constrained optimization problems. Numerical examples involving the boundary control of incompressible Navier-Stokes flows are presented in detail.Key words. Schwarz preconditioners, domain decomposition, multilevel methods, parallel computing, partial differential equations constrained optimization, inexact Newton, flow control.1. Introduction. There are two major families of Newton techniques for solving nonlinear optimization problems: reduced space methods, characterized by the partition of the problem into smaller ones at each Newton step, and full space ones. As computers become more powerful in processing speed and memory capacity, full space methods become more attractive, as exemplified by Lagrange-Newton-Krylov-Schur [3, 4] and one-level Lagrange-Newton-Krylov-Schwarz [25].A key element of any successful full space approach is the Jacobian preconditioner, which needs to be able to simultaneously reduce the condition number and provide good parallel scalability [20]. In this paper, we focus on fully coupled Schwarz type preconditioners in which all components of the linear system are treated equally, i.e., no variables are eliminated as in some Schur complement type approaches. Among Schwarz type preconditioners [29,30], the recently introduced restricted versions [6,9] seem to be more robust and computationally more efficient, especially for harder problems such as those indefinite, highly ill-conditioned, multi-components systems arising from PDE-constrained optimizations. The extension of the one-level restricted additive Schwarz method (RAS) to multilevel using the multigrid V-cycle idea and standard coarse to fine interpolations is straightforward, but may not work as expected due to the pollution effects of the interpolation. After many experiments with some flow control problems, we identified the source of the pollution at one of the three components of the Jacobian system, namely the Lagrange multiplier. Using a pollution removing interpolation scheme we have then been able to restore the robust and fast convergence of RAS. We only discuss linear versions of Schwarz methods even though nonlinear versions can also be obtained [8,13]. We refer to [2,19,28] for the analysis of some preconditionin...

In recent years, Bayesian model updating techniques based on measured data have been applied to many engineering and applied science problems. At the same time, parallel computational platforms are becoming increasingly more powerful and are being used more frequently by the engineering and scientific communities. Bayesian techniques usually require the evaluation of multi-dimensional integrals related to the posterior probability density function (PDF) of uncertain model parameters. The fact that such integrals cannot be computed analytically motivates the research of stochastic simulation methods for sampling posterior PDFs. One such algorithm is the adaptive multilevel stochastic simulation algorithm (AMSSA). In this paper we discuss the parallelization of AMSSA, formulating the necessary load balancing step as a binary integer programming problem. We present a variety of results showing the effectiveness of load balancing on the overall performance of AMSSA in a parallel computational environment.

In the present study, a general dynamic data-driven application system (DDDAS) is developed for real-time monitoring of damage in composite materials using methods and models that account for uncertainty in experimental data, model parameters, and in the selection of the model itself. The methodology involves (i) data data from uniaxial tensile experiments conducted on a composite material; (ii) continuum damage mechanics based material constitutive models; (iii) a Bayesian framework for uncertainty quantification, calibration, validation, and selection of models; and (iv) general Bayesian filtering, as well as Kalman and extended Kalman filters. A software infrastructure is developed and implemented in order to integrate the various parts of the DDDAS. The outcomes of computational analyses using the experimental data prove the feasibility of the Bayesian-based methods for model calibration, validation, and selection. Moreover, using such DDDAS infrastructure for real-time monitoring of the damage and degradation in materials results in results in an improved prediction of failure in the system. ‡ It is with great pleasure that we contribute this study in honor of our dear colleague, friend, and visionary in the field of numerical methods in engineering, Professor Ted Belytschko. We thank him for his numerous contributions to this field, for his statesmanship, and for his leadership over the many years of development of the subject of computational engineering.highly nonlinear material damage theories of the type used in contemporary fatigue analysis, fracture mechanics, and structural mechanics. These typically involve material parameters that exhibit uncertainties. In order to provide information for real-time monitoring of damage, the dynamically collected data of uniaxial tensile experiments conducted on composite materials [8] are taken into consideration. Thus, the system itself must be calibrated and validated, and the inherent uncertainties in data must be factored into a statistical analysis for the validation of the full system. A Bayesian framework is also developed for defining, updating, and quantifying uncertainties in the model, the experimental data, and the target quantities of interest. This paper is structured as follows. A summary of some physical models for damage that are considered for adoption along with the finite element solution procedure is presented in Section 2. This is followed in Section 3, the development of a corresponding DDDAS. In Section 4, the experimental results used in the statistical analyses are presented. Bayesian methods for model calibration, validation, and selection with quantification of uncertainties are outlined in Section 5. Section 6 describes the developed and implemented software infrastructure in order to integrate the acquired experimental data along with the finite element solution of the continuum damage mechanics model in order to calibrate the model and compute model plausibilities that guide the selection of the models themselves. This is accomplished...

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