CPU and GPU parallel Kramers-Klein calculationsB.I.Iaparov, A.N. Karmatsky, A.S. Moskvin

Ural Federal University, Yekaterinburg

1D Brownian motion of particle in an external potential field

• Reaction kinetics• Ion channel kinetics• Josephson tunneling • Etc.

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Langevin equation

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Kramers-Klein equation𝑚�̈�+𝑚𝛾 �̇�+𝑚𝑓 ′ (𝑥 )=𝑚 Γ (𝑡 )

Kramers-Klein equation

• Parabolic equation in the v-direction• Hyperbolic equation in the x-direction

𝜕𝜌𝜕𝑡 =−𝑣 𝜕 𝜌𝜕 𝑥 + 𝑓 ′ (𝑥 ) 𝜕 𝜌

𝜕𝑣 +𝛾 𝜕 (𝑣 𝜌 )𝜕 𝑣 +

𝛾𝑘𝐵𝑇𝑚

𝜕2 𝜌𝜕 𝑣2

Numerical method(Araujo et al.,2015)

Rectangular domain:

Uniform grid:

Numerical method(Araujo et al.,2015)

Accuracy , conditionally stable(Araujo et al.,2015))

Types of boundary conditions

• Dirichlet:

Types of boundary conditions

• Dirichlet: • Absorbing:

Types of boundary conditions

• Dirichlet: • Absorbing:

• Reflecting:

Implementation details• PC: 4x Intel Core i5-4460 3.2 GHz, NVIDIA GTX 750• Uran supercomputer(NVIDIA Tesla M2050) from Institute of

Mathematics and Mechanics UrB RAS.• CPU program: C++(g++ -O3)• GPU: CUDA + g++ for host code

Parallelization

• CPU: boundaries in series, interior in parallel• GPU: the whole domain in one kernel except Reflecting BCs• Interior: Stencil pattern• Boundaries: with Dirichlet and Absorbing BCs also Stencil• Reflecting BCs:

Parallelization(Reflecting BCs)• Reflecting:

The first kernel calculates interior+half of boundaries(say with v > 0), the second one calculates the second half of boundaries(with v < 0)

Comparison with analytic solution in case of parabolic potential(Risken Frank,1996)

𝜀(𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝑛𝑜𝑟𝑚)=8.3𝑒−4

Performance resultsDirichlet Absorbing Reflecting

1x CPU 592.6 s 601.2 s 600.8 s

4x CPU 218.3 s (2.7x) 226.74 s (2.7x) 220 s(2.7x)

1x GPU(GTX 750) 114.7 s (5.2x) 116.6 s(5.2x) 122 s(4.9x)

1x GPU(Tesla M2050)

39.8 s (14.9x) 40 s(15x) 43.9 s(13.7x)

Conclusions

• We parallelized a scheme from(Araujo et al.,2015) on CPU and GPU • We showed that both CPU and GPU parallel computing provide a

signifcant gain in the solving time of the Kramers-Klein equation. Even home PC's GPU can solve the equation for an acceptable time.• Calculations on different grids have shown that duration of solving

depends linearly on

Thank you for attention!