CRYSTAL14: A program for the ab initio investigation of crystalline solids
Summary (3 min read)
1 Introduction
- Derivative-free optimization has experienced a renewed interest over the past decade that has encouraged a new wave of theory and algorithms.
- The authors explore benchmarking procedures for derivative-free optimization algorithms when there is a limited computational budget.
- ∗Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439.
- Users with expensive function evaluations are often interested in a convergence test that measures the decrease in function value.
- These performance profiles are useful to users who need to choose a solver that provides a given reduction in function value within a limit of µf function evaluations.
2 Benchmarking Derivative-Free Optimization Solvers
- Performance profiles, introduced by Dolan and Moré [5], have proved to be an important tool for benchmarking optimization solvers.
- Dolan and Moré define a benchmark in terms of a set P of benchmark problems, a set S of optimization solvers, and a convergence test T .
- Once these components of a benchmark are defined, performance profiles can be used to compare the performance of the solvers.
- In this section the authors first propose a convergence test for derivative-free optimization solvers and then examine the relevance of performance profiles for optimization problems with expensive function evaluations.
2.2 Data Profiles
- The authors can use performance profiles with the convergence test (2.2) to benchmark optimization solvers for problems with expensive function evaluations.
- Performance profiles compare different solvers, while data profiles display the raw data.
- The authors illustrate the differences between performance and data profiles with a synthetic case involving two solvers.
- Assume that solver S1 requires k1 simplex gradients to solve each of the first n1 problems, but fails to solve the remaining n2 problems.
- The limiting value of ρs(α) as α→∞ is the percentage of problems that can be solved with µf function evaluations.
3 Derivative-Free Optimization Solvers
- The selection of solvers S that the authors use to illustrate the benchmarking process was guided by a desire to examine the performance of a representative subset of derivative-free solvers, and thus they included both direct search and model-based algorithms.
- The authors note that some solvers were not tested because they require additional parameters outside the scope of this investigation, such as the requirement of bounds by imfil [8, 15].
- Other implementations of the Nelder-Mead method exist, but this code performs well and has a reasonable default for the size of the initial simplex.
- The NEWUOA code requires an initial starting point x0, a limit on the number of function evaluations, and the initial trust region radius.
- The authors effectively set all termination parameters to zero so that all codes terminate only when the limit on the number of function evaluations is exceeded.
4 Benchmark Problems
- The benchmark problems the authors have selected highlight some of the properties of derivativefree solvers as they face different classes of optimization problems.
- The problems in the benchmark set P are defined by a vector (kp, np,mp, sp) of integers.
- These functions are twice continuously differentiable on the level set associated with x0.
- The authors believe the noise in this type of simulation is better modeled by a function with both high-frequency and low-frequency oscillations.
- An advantage of the benchmark problems P is that a set of piecewise-smooth problems PPS can be easily derived by setting f(x) = m∑ k=1 |fk(x)|. (4.6) These problems are continuous, but the gradient does not exist when fk(x) = 0 and ∇fk(x) 6= 0 for some index k.
5 Computational Experiments
- The authors now present the results of computational experiments with the performance measures introduced in Section 2.
- This data is then processed to obtain a history vector hs ∈.
- For each problem, p ∈ P, fL was taken to be the best function value achieved by any solver within µf function evaluations, fL = mins∈S hs(xµf ).
- The authors present the data profiles for τ = 10−k with k ∈ {1, 3, 5, 7} because they are interested in the short-term behavior of the algorithms as the accuracy level changes.
- This accuracy level is mild compared to classical convergence tests based on the gradient.
5.1 Smooth Problems
- The data profiles in Figure 5.1 show that NEWUOA solves the largest percentage of problems for all sizes of the computational budget and levels of accuracy τ .
- With a budget of 10 simplex gradients and τ = 10−5, NEWUOA solves almost 35% of the problems, while both NMSMAX and APPSPACK solve roughly 10% of the problems.
- A benefit of the data profiles is that they can be useful for allocating a computational budget.
- Performance differences are also of interest in this case.
- Both plots in Figure 5.2 show that the performance difference between solvers decreases as the performance ratio increases.
5.2 Noisy Problems
- The authors now present the computational results for the noisy problems PN as defined in Section 4.
- An interesting difference between the data profiles for the smooth and noisy problems is that solver performances for large computational budgets tend to be closer than in the smooth case.
- For τ = 10−5, NEWUOA is the fastest solver on about 60% of the noisy problems, while it was the fastest solver on about 70% of the smooth problems.
- The performance differences between the solvers are about the same.
5.3 Piecewise-Smooth Problems
- The computational experiments for the piecewise-smooth problems PPS measure how the solvers perform in the presence of non-differentiable kinks.
- The data profiles for the piecewise-smooth problems, shown in Figure 5.5, show that these problems are more difficult to solve than the noisy problems PN and the smooth problems PS .
- Another interesting observation on the data profiles is that APPSPACK solves more problems than NMSMAX with τ = 10−5 for all sizes of the computational budget.
- The same behavior can be seen in the performance profile with τ = 10−1, but now the initial difference in performance is larger, more than 40%.
6 Concluding Remarks
- The authors interest in derivative-free methods is motivated in large part by the computationally expensive optimization problems that arise in DOE’s SciDAC initiative.
- This convergence test relies only on the function values obtained by the solver and caters to users with an interest in the short-term behavior of the solver.
- Performance of derivative-free solvers for larger problems is of interest, but this would require a different set of benchmark problems.
- The authors computational experiments used default input and algorithmic parameters, but the authors are aware that performance can change for other choices.
Acknowledgments
- The authors are grateful to Christine Shoemaker for useful conversations throughout the course of this work and to Josh Griffin and Lúıs Vicente for valuable comments on an earlier draft.
- Lastly, the authors are grateful for the developers of the freely available solvers they tested for providing us with source code and installation assistance: Genetha Gray and Tammy Kolda (APPSPACK[10]), Nick Higham (MDSMAX and NMSMAX [13]), Tim Kelley (nelder and imfil [15]), Michael Powell (NEWUOA [20] and UOBYQA [19]), and Ana Custódio and Lúıs Vicente (SID-PSM [4]).
Did you find this useful? Give us your feedback
Citations
1,108 citations
Cites background or methods from "CRYSTAL14: A program for the ab ini..."
...In Figure 2 we report the effect of the DIIS accelerator as compared with the Fock mixing + level-shifting scheme that was default in CRYSTAL14, benchmarked on our test set....
[...]
...In particular, the new developments and improvements with respect to the previous major release of the program, CRYSTAL14 (Dovesi et al., 2014), are discussed into detail....
[...]
...In particular, the new features of the code with respect to the previous major release, namely CRYSTAL14, include: (a) implementation of a DIIS scheme for SCF and CPHF/ KS convergence acceleration (now active by default); (b) fully automated implementation of the QHA for volume-dependent thermal properties; (c) input parameter-free implementation of Grimme’s DFT-D3 correction for weak dispersive interactions as well as a semiempirical composite method for molecular crystals; (d) calculation of elastic constants under pressure, directional elastic wave velocities, the piezo-optic tensor, and the piezoelectric tensor through an analytical CPHF/KS approach; (e) calculation of dynamic linear polarizabilities, SHG, and Pockels effect; (f ) self-consistent system-specific hybrid functionals; (g) improved scalability of the massively parallel version of the program, MPPCRYSTAL; (h) implementation of new tools for magnetic systems (evaluation of spin contamination, restricted open-shell HF, and noninteger spin locking); (i) atomic partition of the electron charge density according to the Hirshfeld-I scheme; (j) calculation of total and projected PDOS as well as its neutron-weighted counterpart (for simulation of INS); (k) calculation of X-ray diffraction spectra; and (l) evaluation of several electronic transport properties (electrical conductivity, Seebeck coefficient, electronic contribution to thermal conductivity, and transport across nanojunctions)....
[...]
...The CPHF/KS scheme was further extended to second-order in the perturbed wavefunction in CRYSTAL14, thus allowing for the calculation of static nonlinear properties (namely, second hyperpolarizabilities and n + 1 rule first hyperpolarizabilities) (Ferrero, Rérat, Orlando, Dovesi, & Bush, 2008; Orlando, Ferrero, Rérat, Kirtman, & Dovesi, 2009)....
[...]
...In the following of this section, we summarize the main developments made for these properties in the CRYSTAL17 version of the program with respect to CRYSTAL14....
[...]
1,042 citations
932 citations
785 citations
579 citations
References
81,985 citations
47,666 citations
47,477 citations
22,326 citations
17,531 citations
Related Papers (5)
Frequently Asked Questions (16)
Q2. Why is the XSF required to account for the effect of finite temperature?
Due to the fact that core and inner-valence electrons of atoms follow the movement of the respective nuclei, when ECD and related X-ray structure factors (XSF) are considered, it is mandatory to account for the effect of finite temperature, for instance by means of atomic harmonic Debye-Waller thermal factors.
Q3. How many monomers are used to calculate the optical properties of a polymer?
Convergence of the calculated optical properties to the infinite periodic polymer limit is essentially achieved for a chain length of 50 monomers.
Q4. What is the common use of geometry optimization in quantum chemical calculations?
Geometry optimization is mostly employed in quantum chemical calculations to obtain a nuclear configuration that is either• a stable structure of a given chemical species, or• can be used to estimate the transition configuration along a reaction path leading to determination of the rate for the corresponding process.
Q5. What is the acoustic wave velocities of a crystal?
The three acoustic wave velocities, also referred to as seismic velocities, can be labeled as quasi-longitudinal vp, slow quasi-transverse vs1 and fast quasi-transverse vs2, depending on their polarization with respect to the propagation direction.
Q6. What is the main challenge in extending the capabilities of crystal to handle large unit-cell systems?
Diminishing memory requirements has been a main challenging issue in extending the capabilities of Crystal to handle large unit-cell systems in the HPC context where the general trend implies reduced memory availability both in terms of GB/core and bandwidth.
Q7. What is the aim of using Eq. 19 as a starting point for their treatment?
The aim of using Eq. (19) as a startingpoint for their treatment here is top avoid explicit gradients of coefficients that would require the solution of an additional set of coupled–perturbed equations.
Q8. What is the reason why -SiO2 is widely used in the electronics industry?
12,66Due to its peculiar piezoelectric properties, α-SiO2 is another material that is widely utilized in the electronics industry.
Q9. How can the slope of frequency be related to the elastic constants of the monolayer?
By imposing equality10between the elastic strain energy of the monolayer and the corresponding vibrational energy of the nanotube, the slope of frequency versus 1/n for the latter can be related to the elastic constants of the monolayer.
Q10. Who is acknowledged for his invaluable contribution in the development of the code?
Victor R. Saunders is acknowledged for his invaluable contribution in the development of the code and in making it computationally efficient and numerically stable.
Q11. What is the effect of the MP2 approach on momentum space properties?
The recent development of an algorithm for computing CPs from the density matrix of the system, rather than from the crystalline orbitals, made possible the investigation of the effect of the adopted computational method on momentum space properties, even beyond the one-electron approximation, with the MP2 approach implemented in the Cryscor program.
Q12. What is the rate of increase relative to the corresponding static electronic property?
The rate of increase relative to the corresponding static electronic property is dictated primarily by the number of static fields used to characterize the process.
Q13. What are some of the new features of Crystal14 as regards electron densities?
Some of the new features of Crystal14 as regards electron densities are: i) the complete topological analysis of the ECD by means of the automated integration of the Topond package into Crystal14 (see Section VIII A); ii) the parallelization, with linear speed-up, of all the algorithms related to ECD and EMD; iii) calculation of anisotropic displacement parameters and DebyeWaller thermal factors for dynamical X-ray structure factors (see Section VIII B); iv) new algorithms for the analysis of the EMD (see Section VIII C).
Q14. How many MBytes is required for the grad step?
As an example, the 239 MBytes for the grad step in the case of the largest nanotube is about 1/10 of the memory commonly available in a single core standard machine.
Q15. What are the two parameters that determine the precision of coupled perturbed (hyper)polar?
For small bandgap polymers, the precision of coupled perturbed (hyper)polarizability calculations is mainly determined by two computational parameters, namely1.
Q16. What is the reason why the memory-occupation curve in the Figure is tending to an as?
This is the reason why the memory-occupation curve in the Figure appears as to be tending to an asymptotic value for large values of n, which represents the maximum amount of replicated data being stored to memory during an SCF+G calculation for MCM-41/X1.