COMMUNICATION New Directions in Numerical Computation Tobin A. Driscoll, Endre Süli, and Alex Townsend, Editors In August 2015 a distinguished collec- use of smooth func- tion of numerical analysts gathered at tions in two and higher Oxford to celebrate the sixtieth birth- dimensions was, until day of Nick Trefethen FRS and consider recently, limited to rela- the future of numerical analysis. Some tively simple domains. of the plenary speakers provided short Fundamental research essays for Notices. The full collection is on infinitely smooth online.1 two-dimensional inter- In a 1992 essay, “The Definition of polants may lead to Numerical Analysis”,2 Trefethen writes interesting new ap- of the field, “[O]ur central mission is proaches, yet the number of scientists to compute quantities that are typically working on them ap- uncomputable, from an analytical point Photo courtesy of Burchard Kaup. Nick Trefethen pears to be surprisingly of view, and to do it at lightning speed.” Jean-Paul Berrut small. These essays explore a few of the particulars of that mission. Bengt Fornberg Jean-Paul Berrut The Method—Not the Machine It used to be said that improvements in sci- A Puzzling Question about Numerical entific computing capabilities originate about Analysis equally from advances in algorithms and in hard- Why are so many academic mathematicians con- ware. During the last decade or two, the focus tent with piecewise smooth approximations to has shifted to building heroic-scale supercom- solutions to functional equations when those so- puting facilities. Maybe this is partly because lutions are known a priori to be smooth? Chebfun processing characteristics are quantifiable and impressively and beautifully demonstrates the easy to show in lists, with national and world records falling incessantly. However, the largest effectiveness of smooth one-dimensional approx- systems require inordinate amounts of power and imants. These may oscillate more than splines infrastructure, and they also become obsolete of comparable accuracy, but their convergence is very quickly. Yet algorithmic opportunities are as faster and automatically adjusts to the smooth- expansive as ever. ness of the underlying function. In contrast, the Endre Süli is professor of numerical analysis at the Uni- Tobin A. Driscoll is professor of mathematical sciences versity of Oxford. His email address is [email protected]. at the University of Delaware. His email address is ac.uk. [email protected]. Alex Townsend is assistant professor of mathematics at Cornell University. His email address is [email protected]. 1tobydriscoll.net/newdirections2015. Jean-Paul Berrut is professor of mathematics at the 2LN Trefethen, “The Definition of Numerical Analysis” Université de Fribourg. His email address is jean-paul. SIAM News, November 1992. [email protected]. For permission to reprint this article, please contact: Bengt Fornberg is professor of applied mathematics [email protected]. at the University of Colorado. His email address is DOI: http://dx.doi.org/10.1090/noti1363 [email protected]. 398 Notices of the AMS Volume 63, Number 4 Private industry when gadgets that we do not understand start perpetuates legacy doing things that we do not expect? algorithms to a This is not to say that data analysis will be surprising extent de- the only means of understanding. As soon as pre- spite the pressure dictability is established, scientists will be looking of economic incen- for explanations in terms of models and equations. tives. One example This suggests a reversal of the prevailing order of can be found in seis- inquiry, one that is already seen in mathematics. mic exploration for Conjectures are often formed based on computa- oil and gas. Finite dif- tional experiments, and possible steps in the proof ferences on regular are tested numerically before too much time is grids for modeling spent trying to prove them. Photo courtesy of Patrickof Campbell, Colorado. University wave propagation Bengt Fornberg were updated from second to fourth or- Nicholas J. Higham der in the 1980s. Now twentieth-order finite differences are widely used for exploration pro- Mixed Precision Computations duction work, but material interfaces are still For the last thirty years, most floating-point calcu- mostly treated only to first order. As a result, lations in scientific computing have been carried computations rely on extreme refinement to keep out in IEEE double-precision errors at a tolerable level. This approach has been arithmetic, which provides fine-tuned on massive systems while largely ignor- the elementary operations of ing the vast opportunities to exploit algorithmic addition, subtraction, multi- improvements. Much the same holds for geoscien- plication, and division at a tific simulations of the earth’s systems, such as relative accuracy of about climate and weather. 10−16. We are now seeing What matters in the long run is: the Method—not growing use of mixed pre- the Machine. cision, in which different floating point precisions are Anne Greenbaum combined in order to de- Photo courtesy of Pete Carr. liver a result of the required Nicholas J. Higham Return to the Days of Ptolemy accuracy at minimal cost. In the coming years we will learn more about what Single-precision arithmetic halves the costs of can and cannot be predicted through analysis of storing and transferring data and can take half the data, without nec- time of double-precision on the right hardware. essarily formulat- Quadruple-precision arithmetic is supported by ing a realistic some compilers, and arbitrary-precision arith- scientific model. metic is available in Fortran and C as well as Ptolemy was able to Python, Julia, and MATLAB™. accurately predict Multiple-precision is being used for iterative planetary motions refinement in linear algebra, rectification of algo- in this way. There rithmic instability, and checking of results. We can is great interest to- expect these practices to expand and new ones to day in predictions emerge as access to mixed precision becomes ever Anne Greenbaum of the stock mar- easier. ket, the weather, our purchasing preferences, and so on, mainly or entirely through the analysis of data. This is good news for the numerical linear al- Randy LeVeque gebra community, as these computations require algorithms such as the singular value decompo- Sharing the Code sition and rely on theoretical analysis connected A positive development in numerical analysis and with, for example, matrix completion problems. scientific computing is the increasing interest in It will be fascinating to see what new things archiving and sharing computer programs that are we learn about ourselves. Will there come a day when I trust Amazon’s book recommendations Nicholas J. Higham is Richardson Professor of Applied over my own instincts? What will be the effects Mathematics at The University of Manchester. His email address is [email protected]. Anne Greenbaum is professor of applied mathematics Randy LeVeque is professor of applied mathematics at the at the University of Washington. Her email address is University of Washington. His email address is rjl@amath. [email protected]. washington.edu. April 2016 Notices of the AMS 399 an integral part of research publica- tions. In mathematics it is unthinkable to publish a theorem without in- cluding a carefully written proof. Unfortunately we have not had the same expectation for code written to test algorithms. The “reproducible research” move- ment is attempting to address this. Journals and funding agencies are starting to require reproducibility, which is easier and more fun to accomplish than in the past due to Photo courtesy of Alan Perry. Randy LeVeque improvements in technology, includ- ing open-source repositories such as GitHub, virtualization and cloud comput- Photo courtesy of Peter Hudston. ing platforms, and browser-based notebooks for Attendees at the conference “New Directions exposition. in Numerical Computation,” held at Oxford Some people have been practicing reproducibil- University in August 2015 in honor of Nick ity for decades, and Nick Trefethen serves as a Trefethen’s sixtieth birthday. good example. He has always enjoyed polishing his code to make it accessible and informative in research papers and exposition as well as in high dimensionality, with the prototype of Darcy his software packages, from SCPACK to Chebfun. flow through a porous medium. A random field, To choose just one example from his oeuvre, try modeling for instance the permeability of an oil to imagine his Spectral Methods in Matlab without field, is naturally described by an infinite number the MATLAB™. It would still be a valuable book, of scalar random variables, giving rise to an infinite- but without the code to learn from and experiment dimensional problem. A very large finite number with, it would have had far less impact. We should of random variables may be needed to obtain a all take such pride and pleasure in sharing our good approximation. Some buzzwords for the cur- code! rent crop of numerical methods are (generalized) polynomial chaos, stochastic Galerkin, stochastic Ian H. Sloan collocation, sparse grids, multilevel Monte Carlo, quasi-Monte Carlo, and so on. High Dimensionality—A New Direction for Why is all of this interest coming now? Because Numerical Analysis only now is the computing power available to tackle realistic problems. Why will interest increase in High-dimensional problems will be an important the future? Because such problems are inherently part of numerical analysis in the future—perhaps hard and suffer from the famous “curse of di- not a new one, because they were mensionality”. Now we are tackling only simple initiated by Norbert Wiener in 1938. versions of such problems: for example, allowing The numerical analysis of high di- only random fields with “finite-dimensional noise”, mensionality was further advanced small variance, and long correlation lengths. As in the 1950s and 1960s through the computers become more powerful, some areas work of the number theorists Sobol, of numerical analysis may be effectively tamed, Hlawka, and Korobov, who (without but high-dimensional problems never fully so.