An adaptive POD approximation method for the control of advection-diffusion equations

We present an algorithm for the approximation of a finite horizon optimal control problem for advection-diffusion equations. The method is based on the coupling between an adaptive POD representation of the solution and a Dynamic Programming approximation scheme for the corresponding evolutive Hamilton-Jacobi equation. We discuss several features regarding the adaptivity of the method, the role of error estimate indicators to choose a time subdivision of the problem and the computation of the basis functions. Some test problems are presented to illustrate the method.Comment: 17 pages, 18 figure

Model order reduction approaches for infinite horizon optimal control problems via the HJB equation

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods

An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations

We discuss model reduction for a particular class of quadratic-bilinear (QB) descriptor systems. The main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QBODEs [Benner et al. '16] to a class of descriptor systems in an efficient and reliable way. Recently, it has been shown in the case of linear or bilinear systems that a direct extension of interpolation-based model reduction techniques to descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we aim at transforming the considered QB descriptor system into an equivalent QBODE system by means of projectors for which standard model reduction techniques for QBODEs can be employed, including aforementioned interpolation scheme. Subsequently, we discuss related computational issues, thus resulting in a modified algorithm that allows us to construct \emph{near}--optimal reduced-order systems without explicitly computing the projectors used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier-Stokes equations

Deep Bilevel Learning

We present a novel regularization approach to train neural networks that enjoys better generalization and test error than standard stochastic gradient descent. Our approach is based on the principles of cross-validation, where a validation set is used to limit the model overfitting. We formulate such principles as a bilevel optimization problem. This formulation allows us to define the optimization of a cost on the validation set subject to another optimization on the training set. The overfitting is controlled by introducing weights on each mini-batch in the training set and by choosing their values so that they minimize the error on the validation set. In practice, these weights define mini-batch learning rates in a gradient descent update equation that favor gradients with better generalization capabilities. Because of its simplicity, this approach can be integrated with other regularization methods and training schemes. We evaluate extensively our proposed algorithm on several neural network architectures and datasets, and find that it consistently improves the generalization of the model, especially when labels are noisy.Comment: ECCV 201

Stabilization by sparse controls for a class of semilinear parabolic equations

Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P. The second author was supported by the Austrian Science Fund (FWF) under grant SFB F32 (SFB “Mathematical Optimization and Applications in Biomedical Sciences”) and by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program

A weighted reduced basis method for parabolic PDEs with random data

This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate according to a probability measure in the set-up of the reduced space. The error of stochastic finite element solutions is usually measured in a root mean square sense regarding their dependence on the stochastic input parameters. An orthogonal projection of a snapshot set onto a corresponding POD basis defines an optimum reduced approximation in terms of a Monte Carlo discretization of the root mean square error. The errors of a weighted POD-greedy Galerkin solution are compared against an orthogonal projection of the underlying snapshots onto a POD basis for a numerical example involving thermal conduction. In particular, it is assessed whether a weighted POD-greedy solutions is able to come significantly closer to the optimum than a non-weighted equivalent. Additionally, the performance of a weighted POD-greedy Galerkin solution is considered with respect to the mean absolute error of an adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure

Order reduction approaches for the algebraic Riccati equation and the LQR problem

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices

Spatially dispersed corporate headquarters: a historical analysis of their prevalence, antecedents, and consequences

Our study, which complements recent works challenging the traditional conceptualization of the CHQ as a single organizational unit, has a dual purpose. First, in descriptive terms, we set out to explore the prevalence of spatially dispersed CHQs in a historical context. Second, we aim to shed additional light on the CHQ’s spatial design by exploring internal antecedents and potential consequences. Building on arguments from information-processing theory, we propose that the strategic complexity facing the CHQ (affecting its information-processing demands) is associated with the likelihood of a spatially dispersed CHQ (affecting its information-processing capacity). In line with our dual purpose, we conduct a historical study drawing on survey and archival data covering 156 public firms domiciled in four countries (Germany, the Netherlands, the UK, and the US) in the late 1990s. Our results provide empirical support for the hypothesized associations between strategic complexity and the CHQ’s spatial design. Moreover, although we find no empirical support for the expected contingency effects, the results suggest that a spatially dispersed CHQ can have negative effects on CHQ and firm performance. Overall, our theoretical arguments and empirical results advance our knowledge about complex CHQ configurations

Why do firms launch corporate change programs? A contingency perspective on strategic change

We study strategic change as a visible and substantive action by examining the circumstances under which firms launch corporate change programs. Drawing on prior literature and corroborated by insights from interviews with executives, we propose a contingency perspective on the launch of corporate change programs (i.e. that different types of programs are launched under different circumstances). To do so, we combine arguments for three general motives for launching a corporate change program with two distinct types of corporate change programs. More specifically, we argue that firms are more likely to launch growth-oriented programs when the market situation is buoyant, when they have prior experience, and when they are underperforming. Furthermore, we argue that firms are more likely to launch efficiency-oriented programs when there is a new CEO, when they are underperforming, and when they are facing high levels of organizational complexity. To test our hypotheses regarding the motives for launching programs, we conducted a large-scale empirical study. Using hand-collected data for the European financial services and insurance industry over a ten-year period, we found support for our predictions. We discuss the implications of these findings for strategic change research

Ice-algal carbon supports harp and ringed seal diets in the European Arctic: evidence from fatty acid and stable isotope markers

Sea-ice declines in the European Arctic have led to substantial changes in marine food webs. To better understand the biological implications of these changes, we quantified the contributions of ice-associated and pelagic carbon sources to the diets of Arctic harp and ringed seals using compound-specific stable isotope ratios of fatty acids in specific primary producer biomarkers derived from sea-ice algae and phytoplankton. Comparison of fatty acid patterns between these 2 seal species indicated clear dietary separation, while the compound-specific stable isotope ratios of the same fatty acids showed partial overlap. These findings suggest that harp and ringed seals target different prey sources, yet their prey rely on ice and pelagic primary production in similar ways. From Bayesian stable isotope mixing models, we estimated that relative contributions of sympagic and pelagic carbon in seal blubber was an average of 69% and 31% for harp seals, and 72% and 28% for ringed seals, respectively. The similarity in the Bayesian estimations also indicates overlapping carbon sourcing by these 2 species. Our findings demonstrate that the seasonal ice-associated carbon pathway contributes substantially to the diets of both harp and ringed seals