Leading-edge science and engineering depend on advanced computing for understanding, prediction, and control. In response to these needs, the field of computational science and engineering (CS&E) is evolving rapidly, to the point that it is now widely considered to be a new discipline by itself and a third pillar of the scientific enterprise, a peer alongside theory and physical experiment. CS&E is unique in that it enables progress in virtually all other disciplines by providing windows of discovery when traditional means of research reach their limits.
Because of its flexibility, computer simulation has become a universal tool. A simulation may serve as a virtual microscope that lets scientists observe the world of quantum physics much smaller than an atom, or it may be employed as a virtual telescope that allows us to explore how galaxies are forming in the universe. In this way, CS&E helps scientists to reach beyond our physical limitations in space and time. When physical experiments are too dangerous, too time-consuming, too expensive, or simply impossible, then advanced simulation techniques enable us to perform virtual experiments and to obtain scientific results otherwise beyond our capabilities.
Just as scientists employ computer simulations to improve our understanding of the world, engineers can use them to design better products. Virtual prototypes—simulated devices and processes—can increasingly replace their real counterparts to save time and to reduce cost, while simulation-based optimization techniques help engineers to develop better products. CS&E is used for research into innovative materials, to make vehicles safer and environmentally friendlier, to construct buildings that can withstand earthquakes and hurricanes, to improve energy efficiency everywhere, and to advance health care, among many other applications.
To begin CS&E research into a specific problem, researchers generally build a simulator that can correctly represent what has been experimentally observed. The creation of a successful simulator is often a very useful accomplishment in itself, since the work required for such a creation will typically lead to a much improved understanding of the physics (or chemistry, etc.) behind the observations. A working simulation will illuminate which effects are essential in the physics under study, which are essential for the observed behavior, and which can be neglected. In this phase, relatively simple scenarios and model geometries will often be sufficient to perform the necessary work. For example, it may be that a simple two-dimensional model will enable researchers to explore the essential mechanism under consideration. Up to this level, qualitative results are often sufficient.
However, as computational modeling continues to develop, CS&E research will aim to develop computational models that are sophisticated enough to be predictive. Good predictions will require quantitative accuracy, as well as the ability to address more complicated scenarios. This is often very difficult in practice. For example, methods that have been proven useful for computing the flow in rectangular driven cavities may still be very far from suitable in predicting the flow in an aneurysm in the human brain. Data structures and boundary conditions along with data input and output become much more challenging at this stage.
Once CS&E research has progressed to the level of being predictive, the final and ultimate step it must take to become indispensable in an engineering application is optimization. In general, an engineer wants to understand and make predictions about a device because he or she wants to improve the design; thus, he or she will want to use computing to compare different design choices and to decide how an engineering problem may be solved in the best possible way. Too often, practice in CS&E is still that a human varies the parameters, and predictive simulation is used to evaluate how well design criteria have been met. CS&E must enable the researcher to perform these optimization tasks seamlessly within a single model.
In addition to application domain knowledge, CS&E research requires expertise in advanced computing (elements of applied mathematics, scientific computing, and computer science). Additionally, many CS&E tasks involve complex data analysis to provide suitable input for the computations and visualizations required to help humans interpret the simulation outcome. Thus, many CS&E problems can be characterized by a pipeline that includes modeling techniques (mathematical and geometric), simulation techniques (discretizations, algorithms, data structures, software frameworks, and problem solving environments), and analysis techniques (data mining, data management, visualization, as well as error, sensitivity, stability, and uncertainty analyses).
This special issue on CS&E of the SIAM Journal on Scientific Computing is the result of a special call for papers that was announced at the SIAM Conference on Computational Science and Engineering held in Costa Mesa, California, February 19–23, 2007. Eighty-nine papers were submitted to the special issue and subjected to a review process as rigorous as the standard review procedures of the SIAM Journal on Scientific Computing. Three guest editors-in-chief headed an editorial board of 22 well-known researchers in CS&E. After a peer review process with two or more anonymous referees for each paper, 26 papers were eventually selected for publication and are now collected in this volume. The editors would herewith like to thank all contributors, including those whose papers do not appear in this volume. Several of these submissions may eventually appear as regular papers in SISC or other journals. As a result of our strict selectivity, we believe that this volume may be considered a milestone of case studies marking the current state of the art for research in CS&E by example.
As these papers clearly illustrate, CS&E research does not primarily seek to achieve new results for a given domain science but rather to provide new computational tools to researchers. Even though these tools are regularly developed to address challenging problems in specific domain sciences, they are generally transferable to other applications. For example, a method to simulate large ensembles of interacting particles may be useful to an astrophysicist studying galaxy formation as well as to a nanotechnology researcher exploring molecular dynamics. If the methods were not transferable to other applications, a collection of papers of such wide span of different applications into a single volume would be of dubious value. Ultimately, CS&E aims at developing a universal set of simulation methods and tools for the scientists and engineers of the future.
Nevertheless, high-level CS&E research differs from research in mathematics or computer science in that it is oriented at one or more realistic applications of a domain science or engineering discipline. Such problems often involve complicated three-dimensional geometries, multiple interacting scales, heterogeneities, anisotropies, and multiphysical or biological descriptions. Thus, they often thwart rigorous proofs of accuracy or efficiency, and so CS&E research must regularly address validation and verification by other means than traditional mathematical analysis.
Additionally, CS&E research cannot be limited to any single step of the simulation pipeline. A mathematical simulation algorithm is useful in CS&E only when it is also implemented on a real computer, and a new parallel processing paradigm becomes relevant for CS&E only when it is demonstrated to be applicable for a large-scale numerical computation. Of course, research for an individual piece of the CS&E pipeline, such as the development of a new finite element method, may also be very useful for progress in CS&E, but taken by itself such a development remains merely work in numerical analysis. In other words, CS&E is not just a new name for existing computational disciplines, nor—least of all—is it just mathematics in which the proofs are half finished. In fact, as opposed to other mathematical sciences, CS&E often achieves its progress through a clever combination of techniques and methods employed for the different stages of the CS&E pipeline. In such a case, the innovation of excellent research may consist in the creativity needed to synthesize a computational solution for a complex problem from the right building blocks.
Since this is a genuine volume on CS&E, it is impossible to classify the papers in it into single subtopics of CS&E. Just as genuine CS&E research must synthesize its solution from various building blocks, so each of these papers addresses several aspects of what we consider to be typical of CS&E research. Therefore, the assignment of a paper to any single such category is necessarily somewhat arbitrary. However, all papers of this special issue share a focus on the development of innovative computational methodology in the context of challenging applications, in the spirit of our rationale above.
Flow problems, when interpreted in a wide sense, form the largest class of applications that occurs in this special issue. This section includes papers on iterative solvers for flow problems, such as “Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations” by Persson and Peraire and turbulence modeling in “A Galerkin-Characteristic Method for Large-Eddy Simulation of Turbulent Flow and Heat Transfer” by El Amrani and Seaid. The flow applications in this volume span the scales, from climate modeling in “A New Approach to Atmospheric General Circulation Model: Global Cloud Resolving Model NICAM and Its Computational Performance” by Tomita, Goto, and Satoh to microfluidics in “Numerical Simulation of Particle Transport in a Drift Ratchet” by Brenk, Bungartz, Mehl, Muntean, Neckel, and Weinzierl. Also dealing with flow problems are “Hybrid Simulations of Reaction-Diffusion Systems in Porous Media” by Tartakovsky, Tartakovsky, Scheibe, and Meakin and “Output Functional Control for Nonlinear Equations Driven by Anisotropic Mesh Adaptation: The Navier–Stokes Equations” by Micheletti and Perotto. Reflecting a current trend in computational methods research, four papers in this volume deal with biomedical applications. They include “A Parallel Multilevel Technique for Solving the Bidomain Equation on a Human Heart with Purkinje Fibers and a Torso Model” by Washio, Okada, and Hisada; “Bridging Scales: A Three-Dimensional Electromechanical Finite Element Model of Skeletal Muscle” by Roehrle, Davidson, and Pullan; “Parallel Minimum $p$-Norm Solution of the Biomagnetic Inverse Problem for Realistic Signals Using Exact Hessian-Vector Products” by Bücker, Buecker, and Rupp; and “Long-Time Simulations on High Resolution Meshes to Model Calcium Waves in a Heart Cell” by Gobbert.
Adaptive methods in which the evolving solution and some aspect of the method (typically the discretization, but increasingly algorithmic parameters as well) interact so that the latter may be optimized, have been and clearly will remain a hot topic in CS&E research. Papers such as “PML Enhanced with a Self-Adaptive Goal-Oriented $hp$-Finite Element Method: Simulation of Through-Casing Borehole Resistivity Measurements” by Pardo, Demkowicz, Torres-Verdín, and Michler; “A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems” by Bangerth; and “Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation” by Deuflhard, Huisinga, Jahnke, and Wulkow illustrate such adaptive techniques in complex applications.
Reflecting a growth area whose techniques now have a home in a specialized SIAM journal, several papers are concerned with image processing, including “Adaptive Mesh Refinement for Nonparametric Image Registration” by Haber, Heldmann, and Modersitzki; “A Variational Shape Optimization Approach for Image Segmentation with the Mumford–Shah Functional” by Dogan, Morin, and Nochetto; and “Brain–Tumor Interaction Biophysical Models for Medical Image Registration” by Hogea, Davatzikos, and Biros.
Two papers deal with fast transforms, including “An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid” by Mehra and Kevlahan and “Fast Three-Dimensional Discrete Cosine Transform” by Lee, Chan, and Adjeroh. The paper by Narumi, Kameoka, Taiji, and Yasuoka presents the innovative use of multicore processors for molecular dynamics under the title “Accelerating Molecular Dynamics Simulations on PlayStation 3 Platform Using Virtual-GRAPE Programming Model.” The focus is on stochastic modeling and simulation methods in two papers: “Fast Monte Carlo Simulation Methods for Biological Reaction-Diffusion Systems in Solution and on Surfaces” by Kerr, Bartol, Kaminsky, Dittrich, Chang, Baden, Sejnowski, and Stiles; and “Towards a Statistical Theory of Texture Evolution in Polycrystals” by Barmak, Emelianenko, Golovaty, Kinderlehrer, and Ta'asan.
Finally, a group of five papers can be counted under the topic of optimization, identification and inverse problems (though these topics of course also play a role in the imaging papers above and others). The papers here include “Numerical Solution of an Inverse Problem of Imaging of Antipersonnel Land Mines by the Globally Convergent Convexification Algorithm” by Xin and Klibanov; “ORBIT: Optimization by Radial Basis Function Interpolation in Trust-Regions” by Wild, Regis, and Shoemaker; “Multigrid One-Shot Method for State Constrained Aerodynamic Shape Optimization” by Hazra; “Incremental Identification of Transport Coefficients in Convection-Diffusion Systems” by Karalashvili, Gross, Mhamdi, Reusken, and Marquardt; and “Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space” by Bui-Thanh, Willcox, and Ghattas.
To conclude, the guest editors-in-chief would like to thank again all contributors, associate editors, and the anonymous referees from industry, government laboratories, and universities across the globe for the work and enthusiasm they have put into this publication project. As SIAM seeks to chart a path on the research frontier of CS&E that is responsive to the interests of its members and illuminating to a vast community of developers and users not yet SIAM members, this collection helps to define CS&E, to transmit exciting developments, and above all to improve and focus our collective conversation about CS&E. What is the future role of CS&E in SIAM publications? This special issue complements a growing book series in leading to answers!
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