To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally available.
So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot research-topics of practical relevance in numeric mathematics are.

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    quora.com/… – Beni Bogosel Sep 13 '15 at 11:40
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    scicomp.stackexchange.com – Thomas Klimpel Sep 13 '15 at 13:09
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    Wow, "should have lost its importance" is the exact opposite of the conclusion that I would have arrived at given that pretty much nothing is left untouched by computing! – Suvrit Sep 13 '15 at 13:44
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    Ironically, it is because of, rather than despite, increasing computational power that fast approximate algorithms are increasing in importance. In the early days, the boundary between computable and uncomputable was the interesting one. As computers became better, interest shifted to the boundary between P and NP. Nowadays, linear and sublinear-time algorithms are all the rage. The more powerful our computers, the more data we amass, and the less we can afford to carry out exact computations. – Timothy Chow Sep 13 '15 at 18:57
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    For a specific example where you might naively think that exact computation is used, consider integer linear programming. This is a purely combinatorial problem and "on paper" is most naturally solved using exact integer arithmetic. But in practice, all the best implementations (such as CPLEX) use floating-point computations, because exact integer arithmetic is much slower. – Timothy Chow Sep 13 '15 at 19:00
up vote 31 down vote accepted

No, research in numerical mathematics is still very relevant today.

One of the main challenges is big data: scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sparse matrices of dimensions 100k or 1M, for instance. Using traditional methods such as Gaussian elimination will take ages even on modern computers, and require way more RAM than they have. To obtain faster methods, one has to understand the structure of the problem and exploit, for instance, knowledge of the behaviour of similar smaller problems to construct better approximations. There is a lot of research on algorithms that can scale up to huge sizes and their mathematical properties. To mention just one example, designing randomized numerical algorithms and proving their effectiveness requires deep mathematical results.

For another extreme case, in some quantum chemistry applications, one has to compute approximate eigenpairs in a context in which a single vector of the dimension wouldn't fit in RAM using normal storage. $n=10^{30}$ isn't out of the question, for instance.

Another observation is that most problems aren't solved with infinite precision even on today's computers. High precision or exact rational arithmetic is a cause of major slowdowns of a very large factor on modern architectures, and doesn't really solve the issue: if you are using an unstable algorithm, things will go wrong even if you throw 500 digits of precision in it. For instance, the error in your measurements and input data will be amplified to a factor to which you have no more significant digits. People agree that the solution is designing more stable algorithms, not raising the precision. For instance, by exploiting symmetries and hidden structures in the data.

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    Especially the big-data issue seems to be a new challenge; are the research efforts for finding more stable algorithms focused on "classical" tasks like integration or solving differential equations or are there also relatively new problems to be addressed? – Manfred Weis Sep 13 '15 at 11:31
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    @ManfredWeis Both. An excellent example of a new big research field is compressed sensing. But people are also still doing research on challenging classical problems like the global convergence of the Newton method on polynomials. – Federico Poloni Sep 13 '15 at 11:50
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    A good way to get a feeling for what is going on could be browsing through the abstracts of a big conference in the field (for numerical linear algebra, see for instance this one, or checking out an issue of a review journal such as SIREV (siam.org/journals/sirev.php, unfortunately paywalled). – Federico Poloni Sep 13 '15 at 12:04
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    @ManfredWeis In "big data" computations, the focus is less on "high precision", because the computations are often done of very noisy data, and trying to obtain very high precision solutions is science fiction. However, the core ideas of "conditioning", "stability", "efficient numerical linear algebra" all are not only present but also crucial to the success of algorithms and software for dealing with large-scale data intensive tasks (e.g., in optimization, machine learning, data mining, etc.) – Suvrit Sep 13 '15 at 13:48
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    One minor quip: actually, 1M is not "big" these days :-) One has to have several 100M as dimension, or better yet, in the billions for it to be called "big" these days. – Suvrit Sep 13 '15 at 20:41

So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot research-topics of practical relevance in numeric mathematics are.

The question of practical relevance of numerical mathematics has very little to do with current hot research topics. Well established topics like cubic splines, basis splines, finite element methods, spectral methods, boundary integral methods, Krylov-subspace methods, sparse matrices, nested dissection, stiff ordinary differential equations, differential-algebraic equations, domain decomposition methods, sparse grids, FFT techniques, Monte Carlo methods, stochastic differential equations, Gaussian quadrature, Chebyshev methods, ..., ..., ... all remain relevant today.

This doesn't mean that the current hot research-topics are not of practical relevance, but they are the more challenging problems. For example, in 1998 R. Hiptmair finally managed to construct working multigrid methods for Maxwell's equations, which was challenging before due to a non-trivial kernel of the corresponding operator. In the following years, many similar methods were proposed. Unrelated to this, methods to efficiently deal with the indefinite systems arising from time harmonic Maxwell or wave equations (Helmholtz equation) have occurred only recently. However, there is a difference in that Hiptmair really was able to overcome all theoretical and practical difficulties due to better theory, while the stability of fast solvers for indefinite systems (time harmonic case) remains challenging in practice. An earlier case of hot research activity following a breakthrough was caused when Sethian developed level set methods and fast marching methods in 1988. (I know that naming Hiptmair or Sethian as the only inventors of the methods they promoted is historically inaccurate, but it nicely simplify things and gives precise dates.)

Most of the problems to which mathematics is applied nowadays cannot be solved symbolically; think of flow in an oil reservoir or air flow past a vehicle. These problems are modelled by complicated systems of nonlinear PDEs and have highly irregular boundaries, all of which make analytic solution impossible. Numerical approximations are the only reasonable approach, and they require billions, trillions (or more) arithmetic operations. If those operations were performed in "unlimited precision" (i.e., rational arithmetic) the size of the denominators required would be similarly large and it would quickly become impractical to store the solution. It would also be millions of times slower (compared to doing it in double precision floating point). Even in floating point, there are plenty of problems (e.g. turbulence modeling) that we cannot solve to acceptable accuracy in reasonable time on the largest supercomputers.

One other note: rounding errors are only one source of error in numerical computations; discretization error is frequently more significant. Even if you could use infinite precision for these computations, the stability of the numerical discretizations would continue to be an important area of research.

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