When we think about progress in HPC, most of us use hardware speed, as reported in listings like the Top500, as our yardstick. But, is that the whole story – or even its most important component? HPC hardware and the attendant systems software and tools suites are certainly necessary for progress. But to harness HPC for practical problem solving, we also need the right math, as expressed in our solvers and applications algorithms. Hardware is tangible and visible while math is seen through the mind’s eye – and is easily overlooked. Lately, there hasn’t been much public discussion of HPC’s math. Where has it gone? Has it matured to the point of invisibility – or is it still a vibrant and dynamic part of HPC? Let’s take a look.

“Unglamorous but Critical”

From the early days of HPC, math was clearly seen as a vital element. In December of 1982, the Report of the Panel on Large Scale Computing in Science and Engineering, also known as the Lax Report, was published. One of its recommendations called for (emphasis mine):

Increased research in computational mathematics, software, and algorithms necessary to the effective and efficient use of supercomputer systems

Twenty years later, in July of 2003, the Department of Energy (DOE)’s Office of Science published: A Science-Based Case for Large-Scale Simulation, also known as the SCaLeS Report (Volume 1, Volume 2). Among other things, it reiterated the critical role of solvers in HPC (emphasis mine):

Like the engine hidden beneath the hood of a car, the solver is an unglamorous but critical component of a scientific code, upon which the function of the whole critically depends. As an engine needs to be properly matched to avoid overheating and failure when the vehicle’s performance requirements are pushed, so a solver appropriate to the simulation at hand is required as the computational burden gets heavier with new physics or as the distance across the data structures increases with enhanced resolution.

Solvers & Speedup

When improvements in HPC hardware performance are discussed, mention is often made of Moore’s Law and the desire to keep pace with it. Perhaps less well known is the observation that algorithm speedups have historically matched hardware speedups due to Moore’s Law. For example, consider this excerpt from the SCaLeS Report:

The choice of appropriate mathematical tools can make or break a simulation code. For example, over a four-decade period of our brief simulation era, algorithms alone have brought a speed increase of a factor of more than a million to computing the electrostatic potential induced by a charge distribution, typical of a computational kernel found in a wide variety of problems in the sciences. The improvement resulting from this algorithmic speedup is comparable to that resulting from the hardware speedup due to Moore’s Law over the same length of time (see Figure 13).

Figure 13.

Top: A table of the scaling of memory and processing requirements for the solution of the electrostatic potential equation on a uniform cubic grid of n × n × n cells.

Bottom: The relative gains of some solution algorithms for this problem and Moore’s Law for improvement of processing rates over the same period (illustrated for the case where n = 64).

Algorithms yield a factor comparable to that of the hardware, and the gains typically can be combined (that is, multiplied together). The algorithmic gains become more important than the hardware gains for larger problems. If adaptivity is exploited in the discretization, algorithms may do better still, though combining all of the gains becomes more subtle in this case.

Time to Solution

So, if hardware gains and algorithmic gains could be “multiplied together,” what would that imply? If we are currently targeting a 1,000 fold increase in hardware speed over the present decade and if algorithmic gains keep pace, then in ten years we’ll have improved our problem solving capability by a factor of 1,000,000. Thus we’d be able to solve today’s problems in one millionth of their current solution time or use today’s time to solution to tackle problems a million times harder. Sounds pretty impressive. Is the necessary math on track to make this happen?

Obviously, things aren’t as simplistic as I’ve made them out to be. To get the multiplicative effect, algorithms and hardware architectures should be independent of one another. But in real HPC life, algorithms and hardware architectures interact. Fast algorithms are usually “complicated” and complicated algorithms are best implemented on “simple” uncomplicated architectures. Historically, when new, more complicated, hardware architectures are introduced we revert to simpler and slower solvers. Consequently, the optimistic estimates of improvement in time to solution may not materialize. In fact, time to solution could go up. This effect can go largely unnoticed by the general community because simpler solvers can require lots of mathematical operations and faster architectures spit out more operations per second. Thus in this situation, applications codes can run “fast” but produce solutions slowly.

As we move toward extreme scale HPC hardware, the interaction of algorithms and hardware architectures is becoming more important than ever. Last year, DOE’s Office of Advanced Scientific Computing Research (ASCR) published a Report on the Workshop on Extreme-Scale Solvers: Transition to Future Architectures. In it, the following observation is made (emphasis mine):

The needs of extreme-scale science are expected to drive a hundredfold increase in computational capabilities by mid-decade and a factor of 1,000 increase within ten years. These 100 PF (and larger) supercomputers will change the way scientific discoveries are made; moreover, the technology developed for those systems will provide desktop performance on par with the fastest systems from just a few years ago. Since numerical solvers are at the heart of the codes that enable these discoveries, the development of efficient, robust, high-performance, portable solvers has a tremendous impact on harnessing these computers to achieve new science. But future architectures present major challenges to the research and development of such solvers. These architectural challenges include extreme parallelism, data placement and movement, resilience, and heterogeneity.

Solver Dominance

The extreme-scale solver report goes on to address the issue of solver dominance:

Increasing the efficiency of numerical solvers will significantly improve the ability of computational scientists to make scientific discoveries, because such solvers account for so much of the computation underlying scientific applications.

This figure, taken from the extreme-scale solver report, shows that for a typical application as processor count and problem size increase, the time spent in the application’s solver (blue), relative to the time spent in the rest of the application’s code (pink), grows and soon dominates the total execution time.

What’s to be done about this – especially as we anticipate the move to exascale architectures?

ExaMath

In an attempt to find some answers, ASCR has formed an Exascale Mathematics Working Group (EMWG) “for the purpose of identifying mathematics and algorithms research opportunities that will enable scientific applications to harness the potential of exascale computing.”

At ASCR’s request, the EMWG has organized a DOE Workshop on Applied Mathematics Research for Exascale Computing (ExaMath13). ExaMath13 is taking place on 21-22 August and encompasses 40 presentations, selected on the basis of two-page position papers submitted to the EMWG a few months ago. About 2/3s of the presenters are from the DOE Labs with the rest coming from universities. The seventy five submitted position papers from which the 40 presentations were selected may be found at the EMWG website. They make interesting reading and reinforce one’s optimism about the applied math community’s commitment to meeting the challenges posed by exascale architectures.

As the ExaMath problem is complex, it’s not surprising that most of the position papers deal with intricate mathematics. However, a few also address the bigger picture. To mention just one of those, Ulrich Ruede’s paper, entitled: New Mathematics for Exascale Computational Science?, summarizes the challenges faced by the applied math community particularly well:

I believe that the advent of exascale forces mathematics to address the performance abyss that widens increasingly between existing math theory and the practical use of HPC systems. Tweaking codes is not enough – we must turn back and analyze where we have not yet thought deeply enough, developing a new interdisciplinary algorithm and performance engineering methodology. Beyond this, exascale opens fascinating new opportunities in fundamental research that go far beyond just increasing the mesh resolution.

So, it looks like HPC’s math is back in the foreground. There are lots of bright folks in the applied math community. Let’s see what they come up with to address the difficulties posed by ExaMath.