Professor Nick Higham has been awarded the IMA Gold Medal 2020 by the Institute of Mathematics and its Applications. Institute Gold Medals are awarded every two years in recognition of outstanding contributions to mathematics and its applications.

This is the second such success for a member of the Department of Mathematics: Professor Fritz Ursell won the IMA Gold Medal 1994.

Dr Stefan Güttel will receive the 2021 James H. Wilkinson Prize in Numerical Analysis and Scientific Computing. This prestigious prize, established in 1979, is awarded every four years by the Society for Industrial and Applied Mathematics (SIAM). SIAM is the world’s largest professional association devoted to applied mathematics with over 14,000 individual members.

Dr Güttel’s research is in computational mathematics, and in particular, in efficient numerical algorithms for high-dimensional problems. The prize recognizes his contributions to the analysis, implementation, and application of rational and block Krylov methods. These methods have recently become very popular for the efficient solution of large eigenvalue problems, matrix equations, and in model order reduction.

Dr Güttel’s work is applicable to a wide range of problems in science and engineering. He maintains active collaborations with other Departments and companies such as Arup, Autotrader, AspenTech, N Brown Group, and Schlumberger-Doll Research. He also holds a fellowship with the Alan Turing Institute, the UK’s national institute for data science and artificial intelligence.

Dr Güttel will deliver a prize lecture at the 2021 SIAM Conference on Computational Science and Engineering (CSE21), to be held on a virtual platform in March 2021.

Sixteen publications by members of the NLA group feature among the top 40 most read articles in two SIAM journals, both leading venues for publications in numerical linear algebra.

Computing machines with floating-point capabilities started appearing in the late 1940s, and despite being regarded as an optional feature for decades to come, floating-point hardware equipped a number of the mainframe computers produced in the 1960s and 1970s. At the time, there was no consensus on what features a number system ought to have, and the arithmetics implemented by any two vendors could differ widely in terms of word sizes, floating-point representations, rounding behaviour, and accuracy of the operations implemented in hardware. This lack of standardization was a major hindrance to the portability of the source code, as it made it significantly harder to write and maintain architecture-independent code.

The way out of this predicament was the establishment of the IEEE 754-1985 standard, which dictated word sizes, representations, and accuracy requirements for two default floating-point formats: single and double precision. For about three decades, hardware vendors conformed with the standard, and the large majority of general-purpose processing units came equipped with IEEE-compliant arithmetic units. Motivated by the low-precision requirements of machine learning applications, however, in recent years hardware manufacturers have started exploring the realm of low-precision arithmetics, commercialising a variety of highly optimised hardware units that pursue efficiency at the expense of accuracy. The high-performance computing community has attempted to channel the extreme performance of this hardware and utilise it in more traditional scientific computing applications, where more stringent accuracy needs typically require the use of higher precision.

The broad range of different formats available poses a serious challenge to those trying to develop mixed-precision algorithms, since studying the numerical behaviour of an algorithm in different working precisions may require access to a number of high-end hardware platforms, including supercomputer-grade CPUs and GPUs. In order to alleviate the problem, several software packages for simulating low-precision floating-point arithmetics in software have been developed. Some of these are described in the table below. All these solutions execute each arithmetic operation in hardware, and use the software layer only to round the results to the desired number of significant bits of precision.

If the hardware and the software-simulated floating-point format are both IEEE compliant, rounding requires only standard mathematical library functions, but a straightforward implementation of the rounding formulae may potentially cause two kinds of issues. From a theoretical point of view, handling subnormals, underflow, and overflow demands special attention, and numerical errors can cause mathematically correct formulae to behave incorrectly when evaluated in finite arithmetic. In terms of performance, on the other hand, the algorithms obtained in this way are not necessarily efficient, as they build upon library functions that, being designed to handle a broad range of cases, might not be optimised for the specific needs of floating-point rounding functions.

In MATLAB, low-precision arithmetics can be simulated using the chop function, developed by Higham and Pranesh and discussed in a previous post. Mantas Mikaitis and I decided to port the functionalities of this MATLAB implementation to a lower level language, in order to obtain better performance. The scope of the project has progressively broadened, and we have recently released the first version of CPFloat, an open-source C library that offers a wide range of routines for rounding arrays of binary32 or binary64 numbers to lower precision. Our code exploits the bit-level representation of the underlying floating-point formats, and performs only low-level bit manipulation and integer arithmetic without relying on costly library calls.

The project is hosted in a GitHub repository where, along with the C code, we provide a MEX interface for MATLAB and Octave which is fully compatible with the chop function. The algorithms that underlie our implementation, as well as the design choices and the testing infrastructure of CPFloat are discussed in detail in a recent EPrint. In the numerical experiments shown there, CPFloat brings a considerable speedup (typically one order of magnitude or more) over existing alternatives in C, C++, and MATLAB. To the best of our knowledge, our library is currently the most efficient and complete software for experimenting with custom low-precision floating-point arithmetic available in any language.

Most general-purpose computing devices implement floating-point arithmetic defined in the IEEE 754-1985 standard (revised in 2008 and 2019) [1]. The standard provides various compulsory and recommended operations in order to prepare the hardware for as wide a space of applications as possible, as well as to assure reproducibility of numerical calculations across different devices. Hardware that is IEEE 754-compliant usually includes at least addition/subtraction, multiplication, division and square root operations. Each of these can be performed with various rounding modes, chosen by the user from round-to-nearest ties-to-even, round-toward-zero, round-toward-infinity, and round-toward-minus-infinity. Round-to-nearest provides the lowest error bound possible and thus is the default choice for most, but the other rounding modes find uses in various applications as well.

Given hardware with the five arithmetic operations and four rounding modes described above, how can we simulate other rounding modes efficiently when needed? Rounding modes that are not included in the IEEE 754 standard but are of interest in various research domains are, for example: round-to-odd [2], round-to-nearest ties-to-zero [3], stochastic rounding with 50/50 chance of rounding to either of the neighbouring values [4], or stochastic rounding with probabilities proportional to the distances between the exact value and the neighbouring floating-point values [5, 6]. The last of these is also used in fixed-point arithmetic: for example in machine learning [7] and ordinary differential equation (ODE) solvers [8].

In a recent preprint with Massimiliano Fasi [9] we address the problem of simulating the proportional variant of stochastic rounding of operations in precision , when there is no higher precision available in hardware (therefore is the working precision as well as the target precision for rounding). This rounding mode requires the round-off bits captured to be used in calculating the probabilities, but they are lost in IEEE 754-compliant operations of present hardware devices—an operation is performed and only the rounded result is returned in a register. A straightforward approach would be to perform operations in higher precision than and derive the probabilities from the high precision answer. However, this approach can result in low efficiency code because we would need to use software implementation of higher precision arithmetic operations; for example the MPFR package or other similar software packages developed for the purpose of computing in higher precisions than the working precision.

Our solution is to use error-free floating-point transformations [10] for addition/subtraction and multiplication, and similar transformations (yet not error free) for the division and square root, to compute the distances between the exact number and the two closest floating-point values. This can be done by having the original arguments and the rounded answer from the hardware operation, and knowing what rounding mode was used for producing it. Our algorithms presented in the paper cover two possible cases: when the rounding mode of the hardware operations is round-to-nearest ties-to-even (default) or a combination of the other three rounding modes. The former strategy results in slightly more complicated algorithms since round-to-nearest can round to both sides of the exact number, but the latter is slower due to costs associated with changing the rounding modes—this is true at least with Intel processors which we used for testing.

In simulating stochastically rounded binary64 (double precision) arithmetic operations on a computer equipped with IEEE 754-compliant operations in that precision, our algorithms achieved between 7.3 and 19 times higher throughput of arithmetic operations compared with the simulation that uses MPFR (Table 1), while not making any compromises in accuracy. In the preprint we also evaluate our algorithms in various applications where stochastic rounding is expected to bring advantages: summation of floating-point numbers and ODE solvers. In summary, our algorithms can provide significant speedup for programs that use stochastic rounding operations in working precision.

It is worth noting that the algorithms explored in [9] should not be used if higher precision than the target precision (the precision of which stochastic rounding we would like to simulate) is available in hardware. In such cases, the higher precision should be used if possible, and the high precision results from arithmetic operations should be rounded stochastically to the target precision using different algorithms: such as the ones used in the MATLAB chop [11] and CPFloat [12] packages (a separate blog post on CPFloat can be found here).

It is well established that mixed precision algorithms that factorize a matrix at a precision lower than the working precision can reduce the execution time and the energy consumption of parallel solvers for dense linear systems. Much less is known about the efficiency of mixed precision parallel algorithms for sparse linear systems. Existing work on the performance of mixed precision algorithms for solving sparse linear systems has shown that a speedup up to 2x can be expected, but these studies are limited to single core experiments. In this EPrint, Nick Higham, Françoise Tisseur, Craig Lucas and I evaluate the benefits of using single precision arithmetic in solving a double precision sparse linear systems on multi-core processors and GPUs, focusing on the key components of LU factorization and matrix–vector products.

We find that single precision sparse LU factorization is prone to a severe loss of performance due to the intrusion of subnormal numbers. This phenomenon of LU factorization generating subnormal numbers does not appear to have been observed before. In fact, the elements at the stage of Gaussian elimination are generated from the formula , where is a multiplier. If is a dense matrix of normalized floating-point numbers with norm of order 1, it is extremely unlikely that any of the will become subnormal. However, for sparse matrices we can identify a mechanism whereby fill-in cascades down a column and small multipliers combine multiplicatively. This mechanism is described, with an illustration for better understanding, in the EPrint, as well as strategies to automatically flush subnormals to zero, to avoid the performance penalties.

The figure below shows the speedup of single precision sparse LU factorization over double precision using the sparse direct solver MUMPS on 10 Intel Skylake cores. We used 36 sparse matrices from the SuiteSparse Matrix Collection, from which 21 matrices are of medium size with 700, 000 to 5, 000, 000 nonzero elements, and 15 larger matrices with 7,000,000 to 64,000,000 nonzeros. The orange bars represent the speeds achieved when subnormals are not flush to zero (FTZ) during the single precision LU factorization. For approximately 30% of the matrices, the orange bars are below the threshold of 1. This means instead of accelerating the computation, single precision slows it down compared with the double sparse LU decomposition. This has been corrected by flushing subnormals to zero as illustrated by the blue bars.

As for the overall performance, our experiments show that the anticipated speedup of 2x over a double precision LU factorization is obtained only for the very largest of our test problems. In the figure above, the matrices that exceed the speedup of 1.5x are exclusively of very large size. This result is not atypical, as two other sparse direct solvers considered in our work, PARDISO and SuperLU show a similar result. This contrasts with dense linear systems, where a 2x speedup is often achieved even for matrices of size as small as when single precision is used in place of double precision.

The speedups observed in our parallel experiments are clearly lower than the speedups reported in the existing works based on single core experiments. To understand this contrast, it is important to recall that sparse matrix factorization algorithms have a pre-processing step called reordering and analysis, before the actual numerical factorization. In most of the sparse direct solvers, the reordering and analysis step is serial or has a limited parallelism. Therefore, while the time spent in numerical factorization step decreases with increasing number of cores, the reordering and analysis time stagnates and can becomes a performance bottleneck. In addition, as the reordering and analysis step does not involve floating point computation, lowering the precision, from double to single, has a little impact on the speedup except in the case of very large matrices where the numerical factorization step remains dominant despite the use of multiple cores.

For iterative solvers, our results show that the performance gain in computing or applying incomplete factorization preconditioners in single precision is not appealing enough to justify the accuracy sacrifice, but we have observed a speedup of 1.5x from matrix–vector product kernels by using single precision. In future work, we will explore new approaches to integrate efficiently single precision matrix–vector product kernels and single precision preconditioners in double precision iterative solvers without accuracy loss.

Modern GPUs are equipped with mixed precision units called tensor cores that offer the capability of computing matrix–matrix products both at very high performance and with high accuracy. GPU tensor cores have been used to accelerate various numerical linear algebra algorithms. Among these, LU factorization is a natural candidate, since it can be written so as to heavily rely on matrix–matrix operations.

Haidar et al. were the first to propose an algorithm to compute the LU factorization of a general dense matrix exploiting GPU tensor cores, and obtained up to a 4x-5x speedup in the solution of various linear systems. Moreover, they also showed that the accuracy boost delivered by tensor cores could significantly improve the convergence of iterative methods preconditioned by the computed LU factors.

In a recent paper, Pierre Blanchard, Nick Higham, Florent Lopez, Srikara Pranesh, and I performed the rounding error analysis of an LU factorization algorithm exploiting what we called block FMAs—a generalization of tensor cores that also includes similar mixed precision units such as Google’s TPUs. We explained how block FMAs can be chained together to accumulate the result of intermediate computations in the higher precision (single precision for tensor cores), which allows for a significant accuracy boost.

However, these algorithms shared a common limitation: the matrix was stored in single, rather than half, precision. Indeed, if the matrix is stored in half precision, the accuracy boost is lost and the computed LU factors are essentially identical to those obtained by a standard LU factorization run entirely in half precision. This represents a lost opportunity, because storing the matrix in half precision not only halves the memory footprint, but also greatly reduces data movement, which brings a significant performance boost.

In a recent preprint, Florent Lopez and I propose a new LU factorization algorithm that is able to store the matrix in half precision without incurring such a significant accuracy loss. There are two main ingredients to this new algorithm. First, we switch from the standard right-looking scheme to a left-looking one, which, together with single precision buffers of controlled size, allows us to perform the update operations accurately. Second, we doubly partition the matrix so as to accurately perform the panel factorizations by exploiting tensor cores.

In our experiments, we factorize a range of test matrices using an NVIDIA V100 GPU. The two figures below summarize our main results by comparing three LU factorization algorithms: the standard algorithm (of Haidar et al. and Blanchard et al.) with the matrix stored in either single or half precision, and the new algorithm.

The figure on the left reports the backward error obtained by solving a linear system for a dense matrix with random entries. The figure on the right displays the performance in GigaFLOPS. Unlike the standard algorithm, the new one does not incur a significant loss of accuracy by storing the matrix in half precision. In doing so, it greatly reduces data movement which leads to significant speedup: the new algorithm can be up to twice faster than the standard one.

The need to compute matrix functions arises in many applications in science and engineering. Specialized methods exist for evaluating particular matrix functions, for example, the well-known scaling and squaring algorithm for the matrix exponential, Newton’s method for matrix sign function, and the inverse scaling and squaring method for the matrix logarithm. For some functions a specialized method is not available, in which case a general purpose algorithm is needed. The Schur-Parlett algorithm computes a general function of a matrix, with the function dependence restricted to the evaluation of on the diagonal blocks of the reordered and blocked Schur form. It evaluates on the nontrivial diagonal blocks via a Taylor series, so it requires the derivatives of and it also requires the Taylor series to have a sufficiently large radius of convergence. However, the derivatives are not always available or accurately computable.

In our recent preprint, Nick Higham and I present a version of the Schur-Parlett algorithm that requires only function values and uses higher precision arithmetic to evaluate on the diagonal blocks of order greater than 2 (if there are any) of the reordered and blocked Schur form. Our algorithm is inspired by Davies’s randomized approximate diagonalization method, but we show that the measure of error that underlies randomized approximate diagonalization makes it not a reliable numerical method for computing matrix functions. The key idea in our algorithm is to compute by diagonalization the function of a small random diagonal perturbation of each triangular block, where the perturbation ensures that diagonalization will succeed. Multiprecision algorithms have already been developed for the matrix exponential and the matrix logarithm, and those algorithms are tightly coupled to the functions in question, whereas here we place no restrictions on the function. And the algorithm we propose is not restricted only to a working precision of double since its framework is precision independent. We test our algorithm in double precision on a variety of test problems, and the results show that the algorithm generally behaves in a numerically stable fashion. We apply our algorithm to the matrix Mittag-Leffler function and show that it yields results of accuracy similar to, and in some cases much greater than, the state of the art algorithm for this function.

In the following example we employ the algorithm to compute in double precision the matrix Airy function, and compare the result with that produced by a spectral decomposition method (which breaks down if A is defective, and will be inaccurate if A is close to being defective).

We observe that the normwise relative difference between our algorithm and the spectral decomposition method is tiny. This is just one of the many matrix functions that no specialized algorithm is available for, and the MATLAB function, which implements the Schur-Parlett algorithm, is not able to compute. Our new algorithm greatly expands the class of readily computable matrix functions, and its MATLAB code is available here.

The RIKEN Center for Computational Science in Kobe, Japan, home to the supercomputer Fugaku. This machine, scheduled to become fully operational in 2021, heads the June 2020 edition of the TOP500 list and achieved an execution rate of 1.421 x 10^{18} (mixed-precision) floating-point operations per second (1.421 exaflops) on the HPL-AI Mixed-Precision Benchmark by solving a system of order 13,516,800.

Traditionally, the fastest supercomputers in the world are ranked according to their performance on HPL, a portable implementation of the High-Performance Computing Linpack Benchmark for distributed memory systems. This software package measures the rate of execution, expressed in binary64 floating-point operations per second, at which a computer solves a large dense linear system using LU factorization with row partial pivoting. This metric is an excellent predictor of the performance a machine will achieve on compute-bound tasks executed in binary64 arithmetic, but the test problem is not representative of typical artificial intelligence computations, where lower precision is typically considered satisfactory.

In an endeavour to consider both workloads at once, the Innovative Computing Laboratory (ICL) at the University of Tennesse, Knoxville, developed the HPL-AI Mixed-Precision Benchmark. The reference implementation of the benchmark solves a dense linear system of order to binary64 accuracy by complementing a low-precision direct solver with a high-precision refinement strategy. The code generates and in binary64, computes the LU factorization without pivoting of using only binary32 arithmetic, finds an approximate binary32 solution using forward and back substitution, and finally solves the linear system with GMRES in binary64 arithmetic, using as starting vector and the low-precision LU factors of as preconditioners.

In order to obtain the best possible rate of execution, the coefficient matrix must be dense and, as the benchmark is expected to be needed for , it should also be easy to generate at scale on a distributed memory environment. Furthermore, we require that have growth factor of order 1, so as to guarantee the stability of Gaussian elimination without pivoting, and be not too ill-conditioned, in order to ensure the convergence of GMRES. Currently, the reference implementation relies on a randomly generated matrix that is diagonally dominant by rows. This choice ensures the stability of LU factorization without pivoting, but the matrix thus constructed requires an amount of communication that depends on , and turns out to be extremely well conditioned, with an -norm condition number just above 4 for large .

In our EPrint Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting in LU Factorization, Nick Higham and I present a new parametric class of dense square matrices for which LU factorization without pivoting is stable. The parameters and that yield a specified -norm condition number can be computed efficiently, and once they have been chosen the matrix can be formed from an explicit formula in floating-point operations.

We also discuss several adaptations aimed at further improving the suitability of the generated matrix as a test problem for the HPL-AI benchmark, as our main goal is to provide a robust and efficient matrix generator for it. In particular, we explain how scaling the matrix can help transition from binary32 to binary16, in order to take advantage of the hardware accelerators for low precision such as the tensor cores that equip recent NVIDIA GPUs, and we discuss how the matrices can be tweaked to make the entries in the LU factors less predictable and slow down the convergence of GMRES.

For many years, the arithmetic operations available on most hardware were . More recently ( years ago) the FMA (fused multiply-add) operation also became prevalent in general purpose devices such as CPUs and GPUs. Software builds on top of these operations, for example compilers use a series of these and other hardware instructions to implement more complex functions and algorithms such as the exponential function, matrix multiply, or the algorithms for pseudo random number generation. These arithmetic operations have been standardized by the IEEE 754 floating-point standard since 1985 and most current systems are compliant with it.

Recently, because of the increasing adoption of machine learning, general purpose devices started to include inner product or matrix multiply-accumulate (MMA) operations in hardware. This is a generalization of a scalar FMA to vectors and matrices. Since it is performed in hardware, the expected speedup is achieved due to parallelism — instead of using a few FMA hardware units sequentially to multiply matrices as is the case in software, all of the elements of a matrix of some size are computed in parallel. Using the inner product and matrix multiply-accumulate operations in hardware, compute-bound applications that have high-intensity usage of them are sped up significantly.

Figure 1: Mixed-precision matrix muliply-accumulate operation on 4×4 matrices performed by the tensor cores.

NVIDIA GPUs are widely used for machine learning and other applications. In the latest TOP500 supercomputer list published in June 2020 [1], 112 computers are equipped with NVIDIA graphics cards. The main feature of the recent NVIDIA GPUs is hundreds of arithmetic units for performing the MMA operation — NVIDIA calls these tensor cores. Various applications outside machine learning are being explored in order to utilize very high arithmetic throughput that can be achieved when using MMAs in hardware [2]. Table 1 lists three recent NVIDIA architectures as well as other hardware devices with an MMA operation on chip. The NVIDIA V100 has the first version and the T4 has the second version of tensor cores; in these devices tensor cores work on matrices of size in mixed-precision of fp16 and fp32, as shown in Figure 1. The most recent revision of tensor cores in the A100 updated both the precisions available and the dimensions of input/output matrices. See the NVIDIA V100 and NVIDIA A100 whitepapers for more details.

Table 1: Processing units or architectures equipped with mixed-precision matrix multiply-accumulate accelerators. The notation refers to the matrix multiply-accumulate operation where and are matrices, and and have size and , respectively.

The MMA operation is not standardized by the IEEE 754, therefore various numerical features of tensor cores are not clear. Knowing such features as the support for subnormal numbers, order of operations, and rounding modes and normalization of significands in various parts of the MMA computation can be important, for example when doing error analysis of algorithms that utilize MMAs in order to derive error bounds [3]. As the application space utilizing tensor cores is growing beyond machine learning, understanding of the numerical behaviour of tensor cores will become increasingly useful. Even before such hardware is put to use in some applications, one might want to simulate the behaviour of tensor cores in order to develop numerical codes targeted to tensor cores, on a conventional hardware. Furthermore, there is also a question of differences in numerical behaviour between tensor cores and software MMA using the standard FMA hardware calls, for example differences that would appear when transitioning the existent software to use tensor cores.

Motivated by this, in our recent preprint [4] we investigate an experimental testing methodology of the tensor cores. The method consists of the following steps.

Identify a numerical feature that needs to be tested, for example, the rounding mode in the 5-operand adders that accumulate products in MMA.

Identify possible implementations, for example round to nearest or round toward zero.

Find floating-point inputs that would result in different outputs for each possible hardware behaviour in 2.

Observe outputs and make a conclusion based on expected outputs for each possible behaviour in 2.

Using this approach, we identified the following list of numerical features of the NVIDIA tensor cores.

Subnormal floating-point numbers are fully supported, both on the inputs and outputs.

5-operand adders accumulate 5 addends (4 products from and a value from ) starting from the largest in magnitude.

Round toward zero, rather than round to nearest (default in IEEE 754-compliant arithmetic), in the 5-operand adders is implemented.

Different normalization behaviour from the MMA implemented in software (tensor cores normalize the answer of the whole 5-element dot product at the end rather than after each addition of products).

Inner products without intermediate normalization are shown to be non-monotonic in rare cases (this result is more general than tensor cores, since to the best of our knowledge, most hardware implementations do not normalize on every addition due to lower hardware costs).

Our conclusion is that in the current version, the tensor cores on V100 and T4 (the A100 is not yet available to us) do not replicate the behaviour of the MMA implemented with IEEE 754 compliant FMA hardware operations. These numerical behaviours are expected in a hardware MMA optimized for reducing hardware cost and most likely is motivated by a fact that machine learning applications usually are claimed to not need all the IEEE 754 features and high precision in general. These results provide the parameters that can be used in rounding error analysis of tensor cores [3] which can be useful when developing numerical software.

Our CUDA code to test numerical features of tensor cores is available here.

[4] M. Fasi, N. J. Higham, M. Mikaitis, and S. Pranesh. Numerical Behavior of NVIDIA Tensor Cores. July 2020; revised October 2020. MIMS Eprint, published online.