Max Gunzburger - Böcker

This volume is the proceedings of the Workshop on Optimal Design and Control that was held in Blacksburg, Virginia, April 8-9, 1994. The workshop was spon­ sored by the Air Force Office of Scientific Research through the Air Force Center for Optimal Design and Control (CODAC) at Virginia Tech. The workshop was a gathering of engineers and mathematicians actively in­ volved in innovative research in control and optimization, with emphasis placed on problems governed by partial differential equations. The interdisciplinary nature of the workshop and the wide range of subdisciplines represented by the partici­ pants enabled an exchange of valuable information and also led to significant dis­ cussions about multidisciplinary optimization issues. One of the goals of the work­ shop was to include laboratory, industrial, and academic researchers so that anal­ yses, algorithms, implementations, and applications could all be well-represented in the talks; this interdisciplinary nature is reflected in these proceedings. An overriding impression that can be gleaned from the papers in this volume is the complexity of problems addressed by not only those authors engaged in appli­ cations, but also by those engaged in algorithmic development and even mathemat­ ical analyses. Thus, in many instances, systematic approaches using fully nonlin­ ear constraint equations are routinely used to solve control and optimization prob­ lems, in some cases replacing ad-hoc or empirically based procedures.

This volume is the proceedings of the Workshop on Optimal Design and Control that was held in Blacksburg, Virginia, April 8-9, 1994. The workshop was spon­ sored by the Air Force Office of Scientific Research through the Air Force Center for Optimal Design and Control (CODAC) at Virginia Tech. The workshop was a gathering of engineers and mathematicians actively in­ volved in innovative research in control and optimization, with emphasis placed on problems governed by partial differential equations. The interdisciplinary nature of the workshop and the wide range of subdisciplines represented by the partici­ pants enabled an exchange of valuable information and also led to significant dis­ cussions about multidisciplinary optimization issues. One of the goals of the work­ shop was to include laboratory, industrial, and academic researchers so that anal­ yses, algorithms, implementations, and applications could all be well-represented in the talks; this interdisciplinary nature is reflected in these proceedings. An overriding impression that can be gleaned from the papers in this volume is the complexity of problems addressed by not only those authors engaged in appli­ cations, but also by those engaged in algorithmic development and even mathemat­ ical analyses. Thus, in many instances, systematic approaches using fully nonlin­ ear constraint equations are routinely used to solve control and optimization prob­ lems, in some cases replacing ad-hoc or empirically based procedures.

The book presents the state of the art of nonlocal modeling and discretization and novel analyses of a class of nonstandard nonlocal models. These models have recently become a viable alternative to classical partial differential equations when the latter are unable to capture effects such as discontinuities and multiscale behavior in a system of interest. Because of their integral nature, nonlocal operators allow for the relaxation of regularity requirements on the solution and for capturing multiscale effects and thus have been successfully used in scientific and engineering applications such as diffusion processes, fracture mechanics, heterogeneous material response, subsurface transport, turbulence, and image processing, to name a few. Although the use of nonstandard models is novel, the book provides extensive background and a thorough analysis and description of their discretization methods, enabling it to serve as a gentle and practical introduction to nonlocal modeling for readers who are not familiar with nonlocality.

This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs.

This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs.

This volume is focused on the review of recent algorithmic and mathematical advances and the development of new research directions for Mathematical Model Approximations via RAMSES (Reduced order models, Approximation theory, Machine learning, Surrogates, Emulators, Simulators) in the setting of parametrized partial differential equations also with sparse and noisy data in high-dimensional parameter spaces. The book is a valuable resource for researchers, as well as masters and Ph.D students.