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Plamen Fiziev BLTF, JINR, Dubna Parallel computations using Maple 17

Plamen Fiziev BLTF, JINR, Dubna Parallel computations using Maple 17. Mathematical Modeling and Computational Physics Dubna , 2013. The point of the talk is to present some of the achievements and technical problems in parallel calculations using last versions of Maple package.

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Plamen Fiziev BLTF, JINR, Dubna Parallel computations using Maple 17

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  1. PlamenFizievBLTF, JINR, DubnaParallel computations using Maple 17 Mathematical Modeling andComputational Physics Dubna, 2013

  2. The point of the talk is to present some of the achievements and technical problems in parallel calculations using last versions of Maple package. In particular, we intend to describe the simplicity of Parallel computations with Maple Linear Algebra Package DEtools Package MathematicalFunctionsPackage New methods for generalizedHeun’s functions THE HEUN PROJECT

  3. Some of the basic Maple 17 packages for doing Math and Computational Physics: • Linear Algebra Package • DEtoolsPackage • MathematicalFunctions Package (including Heun’s functions) • IntegrationToolsPackage • DifferentialGeometry Package • DiscreteTransforms Package • MultiSeries package • DynamicSystemsPackage • PDEtools Package • ScientificErrorAnalysis Package • Physics Package

  4. Supermicro 4U/tower SYS-7047A-TH 2 X CPU Intel Xeon E5-2670 Processor (20M Cache, 2.60 GHz, 8.00 GT/s Intel® QPI) 16 core (32 logical) Memory 128 GB DDR3-1600MHz ECC REG 2 X HDD SAS 300GB 15000RPM HDD SATA 2TB 7200RPM Enterprise • OS: • Scientific Linux release 5.4 x 64 • WARNING: using Scientific Linux instead of Windows seems to be criticalfor successful parallel computations on this and similar platforms using Maple 16/17. • Software: • Maple 17 • Fortran 90 • MPI

  5. Lenovo W520 Two Intel(R) Core (TM), i7-2670 QM CPU @ 2.20 GHz 4 core (8 logical) Memory 16 GB DDR3 SDRAM -1333MHz • OS: • Windows 7 x 64 • Software: • Maple 17.2

  6. Linear Algebra is already almost fully paralellezed using the threads technology - for a Neq X Neqrendom matrix A with(LinearAlgebra): Neq:=1200; A0 := RandomMatrix(Neq,Neq,datatype=float[8]): nrmA:=sqrt(add(add(A0[j,i]^2,i=1..Neq),j=1..Neq)): A:=A0/nrmA: EV:=Eigenvectors(A, output='list'): - Time=2 sec - Time=10 sec Eigenvalues: …………………………………………………………….

  7. Eigenvalues: Time versus Matrix size For such simple computations is CRITICAL the amount of RAMforonecore ! The same statement is true for the other Linear Algebra operations.

  8. Linear systems of ODEs with rendom coefficients and rendom initial conditions: • A0 := RandomMatrix(Neq,Neq,datatype=float[8]): • nrmA:=sqrt(add(add(A0[j,i]^2,i=1..Neq),j=1..Neq)):A:=A0/nrmA: • f:= (j) -> x[j](t):X:= Vector(Neq,f):df:=(j) -> diff(x[j](t),t):dX:=Vector(Neq,df): • F:=A.X: • dsys := seq(dX[m]=F[m],m=1..Neq): Neq := Number of equations parallelSyssolver:=proc() uses Grid; local me,numNodes,nrm,Xin,r,i,rply; global syss, NK, Neq, IC; me:= MyNode(); numNodes:= NumNodes(); Xin:=seq(x[i](0)=IC[i,me+1]/nrm,i=1..Neq); r:=syssolver(me+1); if me <> 0 then Send(0, r); else rply := r; for i from 1 to (numNodes-1) do r := Receive(i); rply := rply, r; end do; return [rply]; end if: end proc: IC0:=RandomMatrix(Neq,NK,datatype=float[8]);nrm:=sqrt(add(add(IC0[i,j]^2,i=1..Neq),j=1..NK)): IC:=IC0/nrm: syssolver:=proc(k) local ssys, nrm,Xin; global NK, Neq, IC; UseHardwareFloats:=true: nrm:=sqrt(add(IC[i,k]^2,i=1..Neq)); Xin:=seq(x[i](0)=IC[i,k],i=1..Neq); ssys:= dsolve({dsys,Xin},numeric);ssys(500): end proc: result:=Grid:-Launch(parallelSyssolver,imports=['syssolver','dsys','IC','NK','Neq'],numnodes=NK):

  9. An example for a parallelized procedure - Computing a Convex Hull from Maple help. ( => acceleration about 2 times) 500 ODEs => parallel 1.7 s sequential 23.5 s => 13.6 times (An almost maximal usage of all cores is reached via the inner Grid technology)

  10. Critical is the amount of RAM for one core !

  11. Examples of Linear ODEs with essential singular points 1. Euler’s example: Asymptotic Series Bi-section method Parallel Comp + Multi-section: M = # Cores+1

  12. Examples of Linear ODEs with essential singular points 2. Uniformed Neznamovet. al. system of ODEs for Dirac particle in Schwarzschild metric (arXiv: 1301.7595): Unknown functions: and

  13. 1. Regularization (no zeros in the denominator): 2. Iniformization(no roots): 3. Polar variables:

  14. The vector field on the torus {Φ, η} : • The relief of the vector field • Separatrix valley

  15. The phase portrait on the torus {Φ, η} : Node at => Unbounded states Saddle point at => Bounded states - Node at => Unbounded states

  16. Multi-section • M = # cores +1 • N times. • Practically • N ~ 10 • gives in a • short time • an accuracy • not worse • than

  17. After a finite number of multi-sections we collect several points (with good precision) on the separatrixand then we perform an interpolation,since we know the exact position of the singular point: η = 0 Φ = Thus we really reach the infinite point with respect to the variable: ρ

  18. Discrete spectrum of the bounded states of Dirac particle in Schwarzschild metric From for η asymptotic Solving by parallel computing the solutions of the corresponding ODEs one obtains: Then, according to Neznamov et.al. arXiv: 1301.7595, the spectral condition is

  19. Another formulation of the same problem A Schrödinger like form with quite complicated quasipotentialW(ρ,ε) which yields a second order ODE with five singular points – even not a Heun equation.

  20. The Heun Project http://tcpa.uni-sofia.bg/heun/heun_group.html Particular case: the Heun equation

  21. The confluent Heun equation: Huge amount of applications in all scientific areas: physics, astrophysics, astronomy, chemistry, biology, economics , e.t.c The only computer package able to operate with the Heun functions is still Maple. But there are a lot of problems, in spite of some improvements in Maple 17.2 after more then five years work. For example, problematic are thecalculations for big values of z : (due to existence of irregular singular point at z = ) Maple 10 -17 Maple 17.2 Conclusion: NEW CODES ARE NEEDED. The above example gives some idea what we really need.

  22. Thank you for your attention !

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