global optimization algorithms for chemical applications

Introduction Algorithmic Details Applications and Pitfalls Related Other Projects Summary and Outlook 1/ 28

Global Optimization Algorithms for Chemical ApplicationsAlgorithms, Showcases and Other Projects

Johannes M. Dieterich

Institute for Physical ChemistryGeorg-August-University Gottingen

Germany

2012


Disclaimer

THIS IS A STRIPPED-DOWN VERSION.


The search space

General problem

The search space typically has various minima of different fitness. Only fewest (normally just one) are of optimal fitness. In many cases, the number of minima increases exponentially with

search space dimensionality. NPOs (NP optimization problems):

knapsack (packing problem) travelling salesman (shortest path problem)


Exponential scaling of search spaces: Should a chemist care?

The Lennard-Jones example

LJ clusters seem to exhibit exponential (exp(x2)) scaling of the numberof minima w.r.t. cluster size12

Clusters described by two-body forces are NP-hard optimizationproblems.3

1M. R. Hoare and J. McInnes, Faraday Disc. Chem. Soc., 32, 791 (1983).2LJ100 is considered to have 1 10

40 local minima.3L. T. Wille and J. Vennik, J. Phys. A: Math. Gen., 18, L419 (1985).


(Efficient) Exploration of the search space: Techniques

MC/MD-based methods Simulated annealing (SA)4: Crystal growth. Monte Carlo with minimizations (MCM)5: Basin hopping. Minimum hopping (MH)6: New MD-based algorithm.

Nature inspired algorithms

Genetic algorithms (GA)7: Darwinian/Lamarckian evolution. Evolutionary strategies (ES)8: Mutation-focused evolution. Particle-Swarm optimization (PSO)9: Model movements in a group.

4S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Science, 220, 671 (1983).5Z. Li and H A. Scheraga, Proc. Natl. Acad. Sci. USA, 84, 6611 (1987); D. J. Wales and H. A. Scheraga, Science, 285, 1368 (1999).6S. Goedecker, J. Chem. Phys., 120, 9911 (2004).7D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1989.8I. Rechenberg, Evolutionsstrategien: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution., 1973.9J. Kennedy and R. Eberhart, Proc. IEEE Int. Conf. Neural Networks, 1942 (1995).


Genetic Algorithms for Cluster Optimization

Status Highly optimized, system-specific implementations10

Typically only support for a few levels of theory Maximal support for binary systems11

Remaining Challenges

More efficient and general algorithms Extensible, futureproof framework Efficient and stable parallelization schemes Support for arbitrary levels of theory Support for arbitrary systems

arbitrary systems arbitrary degrees of freedom

10e.g. Hartke working group: similar yet different versions for TIP4P clusters, doped TIP4P clusters, TTM2F clusters, clathrates,...11see e.g. R. L. Johnston et al and R. Ferrando et al


Genetic algorithms in a nutshell

(Genetic) Operations

selection/mating crossover mutation (sanity check) (optional magic) local optimization

Maintaining/updating the population

diversity check add to the population

(a) selecting two parents







(b) crossover







(c) mutation







(d) collision (i.a.)

(e) dissociation (i.a.)







(g) opt. magic







(f) local opt.







(h) diversity check







(i) adding an individual


So, what is the magic?

Options

directed mutation: move a building block to a better position (requiressystem knowledge)12

explicit exchange: (randomly) exchange building blocks for a bettermixing in highly-mixed systems13

hybrid algorithms: couple GA/MCM for better search space explorationand exploitation14

GA/MCM

12Applications to water clusters by B. Hartke et al., currently in development for OGOLEM13See e.g. R. L. Johnston, Dalton Trans. 4193 (2003).14Some recent (mostly EA/SA based) implementations found in protein folding. DEM/SA/EP: M. Goldstein, E. Fredj and R. B. Gerber, J.

Comput. Chem., 32, 1785 (2011), SA/EA: Y. Sakae, T. Hiroyusa, M. Miki and Y. Okamoto, J. Comput. Chem., 32, 1353 (2011).


Differences in search space exploration

Exploration and exploitation differences

GA-based methods: good long-range exploration, good short-rangeexploitation (with minimizations)

MC/MD-based methods: good mid-range exploration and exploitation,good short-range exploitation (with minimizations)


Benchmarking genetic algorithms: Ackleys function

(a) full search space (2D) (b) fine structure (2D)

Function definition

f (~x) = 20 exp

0.21

n

ni=1

x2i

exp[1n

ni=1

cos(2xi)

]+20+e1 (1)

with the global minimum xi = 0.0 in the n-dimensional space.15

15D. H. Ackley, A connectionist machine for genetic hillclimbing, 1987; T. Back, Evolutionary algorithms in theory and practice, 1996.


Benchmarking genetic algorithms: Ackleys function II

(a) w/ local opt. (b) w/o local opt.

Conclusions The scaling is both w/ and w/o local optimization excellent for all tested

crossover operators (O(N1.1) and O(N1.0)). The benchmark function can be easily solved even for 1000 dimensions. Standard benchmark functions are too simple for GA-based global

optimization techniques.16

16J. M. Dieterich and B. Hartke, in preparation.


Hypersurfaces for cluster optimization

Requirements

exact description (to the sub-kcal regime) fast energy and gradient evaluations (since typically multiple hundreds of

thousands are needed) well-behaving (no discontinuities, distinct gradients)

(Partial) Solutions

specialized force fields: fast but potentially inaccurate ab initio methods: exact and (very) slow combinations of the two any other methods (semiempirics,...) depending on the system


OGOLEM methods and interfaces

Interfaces Tinker and OpenBabel:

standard force fields Amber, Gromacs and NAMD:

(custom) force fields for largerbiological systems

DFTB+:tight-binding methods

MNDO and MOPAC:(special) semiempirical methods

Orca:semiempirics, DFT and ab initio

CPMD and CP2K:plane wave DFT with periodicboundaries

MOLPRO:high(er)-level ab initio methods

Internal implementations

Standard LJ FF:employingLorentz-Berthelot mixingrules

High-level LJ FF:dedicated parameters foreach pair, arbitrary powerexpression

Gupta FF:two-body FF for metals

Amber FF:reparametrizable

UFF GGFF:

general Gaussian-basedtwo- and three-body FF


OGOLEM: Global optimization of chemical problems

Cluster structures arbitrary levels of theory arbitrary systems with arbitrary DoF first implementation of a N-phenotype X-over for N > 2 support for systems in environments (on surfaces, in pockets)

Parametrization (system-specific) potential parametrization against reference data currently focussed on energy-structure dependence for fitness

Molecular design

discrete optimization problem design of photo-switchable molecules17

tuning of substuents for tuning of excitation energies excitation energies using FOCI-AM1 (Mopac) or TDDFT (Orca)

17N. O. Carstensen, J. M. Dieterich and B. Hartke, Phys. Chem. Chem. Phys., 13 290 (2011).


Kanamycin A dimers

(a) (KA)2 (b) (KA)2Na+ (c) (KA)2K+

Methods Global optimization based on AMBER:GAFFv10 employing genotype

operators. Reoptimization using MOPAC:PM3. Final structure selection based on MOLPRO:DF-LMP2/cc-pVDZ. Structures seem to support the preference of sodium over potassium

also seen in experiment.18

18J. M. Dieterich, U. Gerstel, J.-M. Schroder, and B. Hartke, J. Mol. Model. doi: 10.1007/s00894-011-0983-x (2011).


Sodium-doped water clusters I: Techniques

Experimental requests

relatively exact description of (H2O)nNa (n


Sodium-doped water clusters II

(H2O)3Na

-200

0

200

400

600

800

1000

2800 3000 3200 3400 3600 3800 4000

rela

tive

IR tr

ansm

issi

on

wavenumber [cm-1]

(a) Type A

-600

-400

-200

0

200

400

600

800

1000

2800 3000 3200 3400 3600 3800 4000

rela

tive

IR tr

ansm

issi

on

wavenumber [cm-1]

(b) Type B

200

300

400

500

600

700

800

900

1000

2800 3000 3200 3400 3600 3800 4000

rela

tive

IR tr

ansm

issi

on

wavenumber [cm-1]

(c) Type C

allows for a reliable assignment of structural patterns may explain temperature dependence of experimental spectra19

level of theory becomes unfeasible for (H2O)X Na with X > 7

19R. Forck, J. M. Dieterich, R. A. Mata, T. Zeuch, Phys. Chem. Chem. Phys. accepted.


scaTTM3F: A modern TTM3-F

Project

global optimization of water clusters (bigger than 34 units) using TTM3-F as a backend Java2JNI2Fortran is nasty (SMP impossible)

Status purely Scala-based implementation (scaTTM3F) Aland (energy decomposition based globopt) niching directed mutation TBD


Water clusters: Some common TTM2-F/TTM3-F structures

(a) 3 (b) 4 (c) 5 (d) 6 (e) 7

(f) 8 (g) 9 (h) 10 (i) 11 (j) 12

(k) 13 (l) 14 (m) 15 (n) 16


Clever opt: Simplification rules!

V1

V2

V3

r1

r2

r3

R

R

The workflow

Vtot(r2) = V1(R r2) + V2(r2) + V3(R r2), R = 16.34A (2)

constrained BP86/def2-SVP optimizations (rotated planes for BF4) DF-MP2/cc-pVTZ single points GGFF[5] fitting Is this really reasonable?


Clever opt: Simplification rules! II

0.2

0.15

0.1

0.05

0

0.05

13 13.5 14 14.5 15 15.5 16 16.5 17 17.5

E [E

h]

r [a0]

GGFF[5] fit

(a) BF4 pot.

0.25

0.2

0.15

0.1

0.05

0

0.05

10 11 12 13 14 15 16 17 18

E [E

h]

r [a0]

GGFF[5] fit

(b) Cl pot.

E [k

Jm

ol

1 ]

r2 []

1500

1450

1400

1350

7 7.5 8 8.5 9 9.5 10

(c) (BF4)3

E [k

Jm

ol

1 ]

r2 []

1550

1500

1450

1400

1350

7 7.5 8 8.5 9 9.5 10

(d) (BF4)2ClE

[kJ

mol

1 ]

r2 []

1600

1550

1500

1450

1400

7 7.5 8 8.5 9 9.5 10

(e) (BF4)Cl2


Sn(III)/Ge(III) Hydrides

(a) GeH, BP (b) SnH, BP

Results20

BP86/def2-SVP Sn: SD-ECP optimizations and frequencies NBO analysis: Sn-H: 89.3% p-character, Ge-H: 84.8% lone pair: Sn: 79.4% s-character, Ge: 70.9% 2nd order perturbation energy NBO analysis: N lone pair donation into

metal p orbital (50 kcal/mol Sn, 35 kcal/mol Ge)

20S. Khan, P. P. Samuel, R. Michel, J. M. Dieterich, R. A. Mata, J.-P. Demers, A. Lange, H. Roesky, D. Stalke, Chem. Commun., in press.


Sn(III)/Ge(III) Hydrides: DFT-D?

(a) GeH, BP-D (b) SnH, BP-D3

What about DFT-D? local geometry around the metal similar to BP86 isopropyl geometry changes (methyl moiety points towards metal),

difference to experiment DF-MP2/cc-pVTZ favours the BP results (by 2.9 for Sn and 0.2 kcal/mol

for Ge)


Atomdroid: Mobile computational chemistry

User interface Molecular viewer Molecular builder Molecule manipulation/analysis PDB downloader Bluetooth sharing

Computational methods

UFF energy and gradient21

L-BFGS22, NEWUOA and Powell minimizations Monte Carlo Monte Carlo with Minimizations (MCM)23

21A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M. Skiff, J. Am. Chem. Soc., 114, 10024 (1992).22J. Nocedal, Math. of Comput., 35, 773 (1980) via RISO: an implementation of distributed belief networks in Java.23Z. Li and H A. Scheraga, Proc. Natl. Acad. Sci. USA, 84, 6611 (1987); D. J. Wales and H. A. Scheraga, Science, 285, 1368 (1999).


Benchmarks: Is it more than just a toy?

Prejudice

Mobile platforms are (and will always be) just for toys not for realapplications.

Hardware is not (and will never be) fast enough for real computations.

Benchmark systems

(c) MeOH (d) Coffein (e) Cryptand 222 (f) Neurokinin A (g) alpha RgIA

Hardware: PoV Mobii Tablet Tegra2 w/ Android 3.0 ROM


Benchmarks: Results

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100 120 140 160 180 200 220

time

[ms]

size [atoms]

energyenergy+gradient

System No. of atoms energy [ms] energy+gradient [ms]A 6 0.100.01 0.180.02B 24 0.860.01 1.480.02C 63 3.540.02 5.480.02D 155 17.400.09 26.470.07E 201 28.740.03 43.560.02


Summary and Outlook

general framework for global optimization exploitation of new hardware platforms: Android, CUDA (not in this talk) development of new parallelization schemes implementation and parametrization of classical force field expressions

as well as novel, flexible ones cooperations with experiment (Kanamycin A, doped water clusters) development of novel GA-based algorithms ...


AcknowledgementsThe group:Ricardo Mata group-leaderJoao Oliveira biosystemsJonas Feldt mobile computingThorsten Stolper opts and regionsMilica Andrejic QM/MM

Collaborations:Bernd Hartke water clustersSascha Frick hydration shells in reactionsThomas Zeuch cluster experimentsSvenja Fischmann internshipDietmar Stalke inorganic compounds IGuido Clever inorganic compounds IIMark Feyand inorganic compounds III

Bachelors:Sebastian Gerke (Gottingen), Maximilian Konrad (Gottingen), Sebastian Wille(Gottingen), Marvin Kammler (Gottingen)

IntroductionAlgorithmic DetailsApplications and PitfallsRelated Other ProjectsSummary and Outlook

0.0: 0.1: 0.2: 0.3: 0.4: 0.5: 0.6: 0.7: 0.8: 0.9: 0.10: 0.11: 0.12: 0.13: 0.14: 0.15: 0.16: 0.17: 0.18: 0.19: 0.20: 0.21: 0.22: 0.23: 0.24: 0.25: 0.26: 0.27: 0.28: 0.29: 0.30: 0.31: 0.32: 0.33: 0.34: 0.35: 0.36: 0.37: 0.38: 0.39: 0.40: 0.41: 0.42: 0.43: 0.44: 0.45: 0.46: 0.47: 0.48: 0.49: 0.50: 0.51: 0.52: 0.53: 0.54: 0.55: 0.56: 0.57: 0.58: 0.59: 0.60: 0.61: 0.62: anm0:

global optimization algorithms for chemical applications

Documents