High Performance Computing and Grid in Latvia: Status and Perspectives

Janis Rimshans and Bruno Martuzans

Institute of Mathematics and Computer Science, University of Latvia,

Riga, LV-1459, Latvia

BaltGrid, 2004, Vilnius, Lithuania

Introduction

• What? – HPC actual problems · charge and mass transfer in a non-linear media · ferroelectric/ferromagnetic and critical indices

• Why?– Multidimensional and complicated geometry problems need to be

supported by effective calculations

• How?– Numerical methods, Monte Carlo simulations– Parallel programming

• So what? – Grid connections to supercomputers

Charge transfer : IMCS University of Latvia (Latvia) & MCI Southern

Denmark University, supported by CIRIUS (Denmark)

Optically sensitive semiconductor plasma

GRx

J

t

n n

nnnn P

xJS

xTn

t

2

3

GRx

J

t

p p

pppp P

xJS

xTp

t

2

3

adeecc NNpnpnxx

ec ,

nnnn Tnxx

nJ

,

ppnp Tpxx

pJ

,

2

nnxnnen TnT

xnCS,

2

ppxpphp TpT

xpCS,

Charge transfer : IMCS University of Latvia (Latvia) & MCI Southern Denmark University, supported by CIRIUS (Denmark)

R.V.N. Melnik and J. Rimshans, Monotone schemes for time-dependentenergy balance models, ANZIAM J. 45 (E), C729-C743, 2004 (Proc. of 11thComputational Techniques and Applications Conference, CTAC-2003) R.V.N. Melnik and J.Rimshans, Numerical Analysis of Fast Transport in Optically Sensitive Semiconductors, Special Issue of DCDIS – 2003, DCDIS Series B, ISSN 1492-8760, Guelph, Ontario, Canada, p.1-6.

Shock waves: IMCS University of Latvia (Latvia) & DMR

National Science Foundation (USA)

Shocked solid conductors

)( T

Bnn n

Tk

qnD J

t

nn

J

nNq

d 0

t

vmq T

B.Martuzans. Yu. Skryl, M.M. Kuklja, Dynamic Response of the Electrone-Hole System in the shocked silicon. Latvian Journal of Physics and Technical Sciences. N4, pp. 56-68, 2003

Yu. Skryl, M.M.Kuklja, Numerical simulation of electron and hole diffusion in shocked silicon, AIP Conference Proceedings, 706(1), 267-270 (2004).

Convection-diffusion : IMCS University of Latvia (Latvia)

& SMS&EPCC University of Edinburgh, supported by Royal Society (UK)

Advective transport

x

CK

xx

Cv

t

C

210 v

K 0v

K

J.Rimshans and N.Smyth, Monotone exponential difference scheme for advection diffusion equation,Submitted for Numerical methods for partial differential equations, 2004.

Ferroelectric materials under alternate driving: ISSP&IMCS University of Latvia (Latvia)

Probability density: key entities

2

2 ),(),(

)(),(

P

tPtP

xP

H

PtP

Probability density of polarization

Variation of energy

Thermal noise strength is the most delicate counterpart of theory. Understanding of its dynamical origin is in progress

J. Hlinka and E. Klotins, Application of elastostatic Green function tensor technique to electrostriction in cubic, Hexagonal and Orthorombic crystals, J. Phys.: Condens. Matter 15 (2003) 5755-5764

Ferroelectric materials under alternate driving: ISSP&IMCS University of Latvia (Latvia)

-1

0

10

20000

40000

60000

0

1

2

-1

0

1

Probability density of polarization

P

t

Parameters of the model:

Quartic Landau-Ginzburg energy functional+periodic driving.Amplitude of driving voltage = A0/2Dimensionless frequency = 10^(-4)Diffusion constant (noise strength = 1/20Polarization = first moment of the instantaneous probability density

E. Klotins, Relaxation dynamics of metastable systems: application to polar medium, Physica A, 340 (2004) 196-200

Ferroelectric materials under alternate driving: IMCS University of Latvia (Latvia)

Spatially homogeneous case

0

300

0

sin1

P

ftAPPVf

Pt

f

310

309.0

05.0

1

1

A

V

J. Kaupužs, J. Rimshans, Polarization kinetics in ferroelectrics with regard to fluctuations, Cond-mat/0405124, 2004.

Critical exponents: IMCS University of Latvia (Latvia)

perturbation theory

xuxcxrdxTH 422/

4

kG

GDckr

kG ii

202

1 Dyson equation

4/5Predicted for 3D Ising model: Susceptibility exponent

Correlation length exponent 3/2Magnetization exponent d2

J.Kaupužs, Ann. Phys. (Leipzig) 10 (2001) 4, 299-331

Critical exponents: IMCS University of Latvia (Latvia)

J.Kaupužs, Proceedings of SPIE, vol. 5471, pp. 480-491, 2004; e-print cond-mat/0405197

L=384 L=410

Monte Carlo simulation results: magnetization exponent

ttL /256 0

L=41063.0 3/2

RG value

Main problems

• Convection – diffusion

• Advective transport

• Convection

• Monte Carlo simulations

• Data transmission

• Training

Advective transport

Ornstein-Uhlenbeck process

2

2

)(x

CK

x

Cx

t

C

Monotone condition

,/,12 2hKss

K

h2

2

1

Unconditionally monotone scheme

lil

iilii

i

lii

ii

l CCQCB

hCA

hC

111

11

1 11Usual schemes

11

11* ii

ii BA

hQ

1exp

exp

2/1

2/12/12/1

ii

iiii h

KA

1exp

1

2/12/12/1

ii

iii hKB

FTCS, Upwind, LaxWendrof

Advective transport

Ornstein-Uhlenbeck process

J.Rimshans and N.Smyth, Monotone exponential difference scheme for advection diffusion equation, Submitted for Numerical methods for partial differential equations, 2004.

Advective transport

Ornstein-Uhlenbeck process

J.Rimshans and N.Smyth, Monotone exponential difference scheme for advection diffusion equation, Submitted for Numerical methods for partial differential equations, 2004.

Convection-diffusion

Charge transfer

Implic

it

Half Im

plicit

Explicit m

ethods

t

nn

J

nNq

d 0

n

Tk

qnD

BnnJ

D

h2

2

1

dB

NDTk

qq )(

10

Ferroelectric materials under alternate driving: IMCS University of Latvia (Latvia)

Fokker-Planck equationLandau-Ginzburg hamiltonian

txPtxxP

cxPxPdxH ,,

242242

Langevin equation txP

H

t

P,

tSPckVfP

P

ftSPckVf

Pt

f

ik

ik

ikn

m

ni

k

rk

krk

rk

rkn

m

nr

k

nnn

n

n

nnnn

n

22

1

0,2

2

0

12

1

12 m, dimensions

J.Kaupužs, J.Rimshans, Polarization kinetics in ferroelectrics with regard to fluctuations, cond-mat/0405124, 2004.

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