Use Of Double Precision Research Articles

Open to Evolve Embodied Intelligence

For the goal of automatically evolving Embodied Intelligence (EI), we investigate an open software architecture inspired by the high surface area to volume ratio of animal lungs, which aims to avoid information theoretic limits on long term evolution experiments (LTEE) encountered with monolithic genetic programming trees. Instead individuals are teams composed of 1023 trees whose inputs and outputs are linked by a low entropy loss branching data (air) pathway. Most trees are shallow and software engineering’s failed disruption propagation (FDP) is observed in the small fraction of deep trees. After initial search, most improvements are at intermediate depths and performance is still rising even after 100 000 generations.Despite the use of double precision for the bifurcating data interconnect, some information loss is seen, particularly in early generations. The static optimisation benchmark, appears to encourage early convergence, which locks the population into possibly sub-optimal phenotypes. Later thousands of small improvements, sometimes in large bloated ensemble members, appear to compensate for early overfitting.Using tournament fitness selection and subtree crossover, we target pure nested side-effect free floating point functions, which are known to have low FDP, and high fidelity data paths, in the hope of generating code which is not too robust so as to prevent on going improvement. However, we again find genetic changes deep within trees are silent. For single precision, we find a maximum evolvability sweet spot with trees of depth 10 to 100. Accordingly, we suggest to evolve very large very complex programs needed for Embodied Intelligence, an open structure with a high surface area permitting most mutation sites to be within 10-100 levels of the organism’s environment, and many better placed test oracles to monitor the impact of mutated code, will be needed.

Performance impact of precision reduction in sparse linear systems solvers.

It is well established that reduced precision arithmetic can be exploited to accelerate the solution of dense linear systems. Typical examples are mixed precision algorithms that reduce the execution time and the energy consumption of parallel solvers for dense linear systems by factorizing a matrix at a precision lower than the working precision. Much less is known about the efficiency of reduced precision in parallel solvers for sparse linear systems, and existing work focuses on single core experiments. We evaluate the benefits of using single precision arithmetic in solving a double precision sparse linear system using multiple cores. We consider both direct methods and iterative methods and we focus on using single precision for the key components of LU factorization and matrix–vector products. Our results show that the anticipated speedup of 2 over a double precision LU factorization is obtained only for the very largest of our test problems. We point out two key factors underlying the poor speedup. First, we find that single precision sparse LU factorization is prone to a severe loss of performance due to the intrusion of subnormal numbers. We identify a mechanism that allows cascading fill-ins to generate subnormal numbers and show that automatically flushing subnormals to zero avoids the performance penalties. The second factor is the lack of parallelism in the analysis and reordering phases of the solvers and the absence of floating-point arithmetic in these phases. For iterative solvers, we find that for the majority of the matrices computing or applying incomplete factorization preconditioners in single precision provides at best modest performance benefits compared with the use of double precision. We also find that using single precision for the matrix–vector product kernels provides an average speedup of 1.5 over double precision kernels. In both cases some form of refinement is needed to raise the single precision results to double precision accuracy, which will reduce performance gains.

Static Block-Recursive Cholesky Algorithm for a Distributed Memory Cluster

This paper investigates the block-recursive parallel algorithm for Cholesky decomposition for a super-computer with distributed memory. For ease of scaling, the number of cores is a degree of the number two. We investigate the resistance of the algorithm to the accumulation of computational errors and the scaling efficiency of the static block-recursive algorithm.Note that the problem does not have an exact solution in rational numbers (for example, LU-decompo-sition has an exact solution), so it is necessary to use approximate calculations and there are no other approaches than calculations with some accuracy. We have not been able to find systematic studies on the accumulation of error in the Cholesky algorithm, so we conduct such studies in this paper. elements are integers with a fixed number of binary digits and a uniform distribution.We have shown that for dense matrices, starting with matrix size 64, the use of double precision (standard floating-point arithmetic and 64-bit machine word) does not allow to obtain an error less than 1, even in simple cases. As the size of the coefficients increases, the error only increases, so the use of such arithmetic for matrices of even larger size loses its meaning. To solve this problem, we used BigDecimal arithmetic for large matrices. It allows you to programmatically specify the accuracy, which is specified as the number of decimal places. First, we determine the required number of characters for BigDecimal so that the calculation error of this matrix does not exceed one, and then we experiment with different numbers of cores for such a matrix. We give recommendations for dense matrices, the elements of which were given randomly with a uniform distribution. For such matrices, based on their size and the number of decimal places in their elements, we recommend the choice of accuracy for machine arithmetic and the number of cores for calculations.Manuscript received 09.06.2020

On the risks of using double precision in numerical simulations of spatio-temporal chaos

Due to the butterfly-effect, computer-generated chaotic simulations often deviate exponentially from the true solution, so that it is very hard to obtain a reliable simulation of chaos in a long-duration time. In this paper, a new strategy of the so-called Clean Numerical Simulation (CNS) in physical space is proposed for spatio-temporal chaos, which is computationally much more efficient than its predecessor (in spectral space). The strategy of the CNS is to reduce both of the truncation and round-off errors to a specified level by implementing high-order algorithms in multiple-precision arithmetic (with sufficient significant digits for all variables and parameters) so as to guarantee that numerical noise is below such a critical level in a temporal interval t∈[0,Tc] that corresponding numerical simulation remains reliable over the whole interval. Without loss of generality, the complex Ginzburg-Landau equation (CGLE) is used to illustrate its validity. As a result, a reliable long-duration numerical simulation of the CGLE is achieved in the whole spatial domain over a long interval of time t∈[0,3000], which is used as a reliable benchmark solution to investigate the influence of numerical noise by comparing it with the corresponding ones given by the 4th-order Runge-Kutta method in double precision (RKwD). Our results demonstrate that the use of double precision in simulations of chaos might lead to huge errors in the prediction of spatio-temporal trajectories and in statistics, not only quantitatively but also qualitatively, particularly in a long interval of time.

Long duration SPH simulations of sloshing in tanks with a low fill ratio and high stretching

Achieving stable and accurate simulations of long duration sloshing with low fill ratios using smooth particle hydrodynamics (SPH) is a challenging problem. Its solution requires a judicious choice of SPH formulation to minimize the effect of errors introduced into the simulation by boundary conditions, dissipation terms or computer arithmetic. We assess the difficulties and common pitfalls of such simulations and propose a SPH method to deal with them effectively. The formulation combines an efficient use of double precision, wall boundary conditions using ghost particles, and a δ-SPH scheme for stability with minimal energy dissipation. The proposed SPH formulation accurately matches the experimental data, both in terms of surface elevations and forces on the tank, of the study of tuned liquid dampers by Reed et al. 1998, later reproduced by ESA/ESTEC, over a wide range of frequencies.

Closed-form solution for consolidation of three-layer soil with a vertical drain system

A closed-form solution for consolidation of three-layered soil with a vertical drain system is proposed in this paper. The solution is precise but has been proven to have some convergence problems. The use of double precision in the calculation program was found to be insufficient to guarantee convergence of the results. After symbolic operation was adopted in Matlab, the convergence was improved. Further study showed that the convergence problem still exists in cases when the calculation parameters for the different layers are significantly different. Hence, a new method is proposed to enlarge the controlling precision for the overall average degree of consolidation of the entire thickness of the soil and add controlling precision for the overall average degree of consolidation of each soil layer. Using this new method, the convergence problem is solved. An example case is presented of calculations for a three-layer embankment with a vertical drain system in Yueqing Bay in China. The pore pressure calculated along the depth of the embankment agrees well with in situ measurements.

A program for accurate solutions of two-electron atoms

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform. Program summary Title of program: corr2el Catalogue identifier: ADUX Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUX Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PC Operating systems or monitors under which the program has been tested: Sun Solaris 9, Linux Programming language used: ANSI C Memory required to execute with typical data: 3 Mwords or more No. bits in a word: 32 No. processors used: arbitrary Has the code been vectorized or parallelized: parallelized Number of lines in distributed program, including test data, etc.:5885 Number of bytes in distributed program, including test data, etc.: 26 199 Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences. Method of solution: An iterative eigenvalue-solver that requires only the action of the Hamiltonian on a trial function is applied. The two-electron wave function is expanded in a sum of partial waves. The finite difference method is then applied to approximate the derivatives of the pair functions. The total action of the Hamiltonian on the partial waves, including correlation effects, is computed using highly optimized routines. Restriction on the complexity of the problem: The Hamiltonian employed here does not take relativistic or finite nuclear mass effects into account. The amount of computing time may become unreasonable for excited states far above the ground state. The use of double precision puts a limit on the accuracy obtainable. Typical running time: This ranges from half a minute (to obtain 10 significant figures for the s-limit of the Helium ground state) to perhaps a day for advanced examples depending on the level of parallelization. Unusual features of the program: The implicitly restarted Arnoldi method used to obtain the eigenvalues is implemented by using the ARPACK/PARPACK program library [ http://www.netlib.org/arpack]. This package also depends on the standard numerical libraries BLAS and LAPACK [ http://www.netlib.org/lapack]. Good performance is obtained by using Sun's optimized performance libraries [ http://www.sun.com]. By using a 64-bit environment (Ultra SPARC III and Solaris 9), memory limitations are non-problematic. Shared memory is used in the parallel version. Fast communication between the nodes is made over shared memory using Sun's implementation of MPI.

Derivation of closed analytical expressions for Rosen-Morse Franck-Condon factors

The expressions for Rosen–Morse Franck–Condon factors derived previously yield a double sum with alternating terms. For higher values of the quantum number the numerical calculation of the Franck–Condon factors by electronic computers using these expressions leads to numerical overflow inspite of the use of double-precision (32 digits) arithmetic. High values for the quantum number in the final ground state of the Rosen–Morse potential occur in molecular nonradiative rate calculations. Furthermore, the expressions show a lack of clearness with respect to the parameters of the potential. For out-of-plane modes exact closed form expressions and exact recurrence relations are derived. Asymptotic expressions for the matrix elements are calculated. Exact closed form expressions for matrix elements with quantum numbers which correspond to regions close to the dissociation barrier are given.

ITERATIVE SOLUTION OF LARGE THREE-DIMENSIONAL BEM ELASTOSTATIC ANALYSES USING THE GMRES TECHNIQUE

The GMRES (Generalized Minimal RESidual) iterative method is receiving increased attention as a solver for the large dense and unstructured matrices generated by boundary element elastostatic analyses. Existing published results are predominantly for two-dimensional problems, of only medium size. When these methods are applied to large three-dimensional problems, which actually do require efficient iterative methods for practical solution, they fail. This failure is exacerbated by the use of the pre-conditioning otherwise desirable in such problems. The cause of the failure is identified as being in the orthogonalization process, and is demonstrated by the divergence of the ‘true’ residual, and the residual calculated during the GMRES algorithm. It is shown that double precision arithmetic is required for only the small fraction of the work comprising the orthogonalization process, and exploitation of this largely removes the penalties associated with the use of double precision. Additionally, it is shown that full re-orthogonalization can be employed to overcome the lack of convergence, extending the applicability of the GMRES to significantly larger problems. The approach is demonstrated by solving three-dimensional problems comprising ∼4000 and ∼5000 equations. © 1997 John Wiley & Sons, Ltd.

Improvement of complex arithmetic by use of double elements

It is shown that use of double precision in complex multiplication and division can significantly reduce rounding errors in these operations. Use of double precision accumulation of inner products in complex multiplication gives very close to the best possible error bound. Use of the same in complex division reduces error bounds, but a double precision divide is necessary to get close to the best possible bound.

Special features of two-dimensional finite elements used in computed thin-walled shell designs

1. In the majority of cases, unsatisfactory results obtained in computing thin-walled shell design using two-dimensional finite elements are caused by either errors of approximation, or numerical instability of the algorithm. 2. It is expedient to use methods of solution that are stable against rounding errors to investigate thin-walled shell designs having desired functions with steep gradients, and also in using an irregular finite-element grid. 3. Use of double precision leads to a reduction in the numerical error in all cases. In this case, the conditioning number may serve as a criterion for selection of the corresponding method of solution and use of double precision.

Counting Schemes for Nuclear Data Acquisition on Small Computers

In using a small computer for accumulating a pulse height spectrum, it is often the case that the word length is too small to store the counts on a one-word-per-channel basis. The use of double precision to avoid this problem is wasteful of storage space. One way to avoid this waste is to store the differences between counts in two adjacent channels rather than the counts themselves. This can be achieved by adding 1 for each new count to the channel corresponding to that count and subtracting 1 from the next channel. The original spectrum can be recovered by summing the difference counts from one end of the storage array. Overflows can still occur when the original count distribution is increasing or decreasing rapidly (e.g., in the peaks of the spectrum), but these will, in general, not be too frequent and can be given special treatment as they occur. One method of doing this is to tag one bit of the word in the count array corresponding to the channel which has overflowed and using the remainder of the word to give the location of a double word which contains the full value of the count. Another method to avoid the use of double precision involves the use of a stochastic counting scheme for accumulating the spectrum.