92854f0f5870bc3c5d2d356ad1bf24e3.ppt

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A Mesh-free Numerical Method for three-dimensional Nonlinear Schrödinger Equation Department of Computer Science and Information Systems Birkbeck, University of London Thomas C. L. Yue [email protected] com Feb 09, 2011 1

Overview • Physical motivation of the problem – Dimensionless Gross-Pitaevskii equation (GPE) • Introduction to Radial basis functions (RBF) – Global supported strictly positive definite radial basis functions – Compactly supported radial basis funtions – Kansa’s method (asymmetric collocation) • Meshfree solution of cubic Nonlinear Schrodinger Equation – Numerical experiments and validation 2

Physical Motivation 3

Physical Motivation History of Bose Einstein Condensation (BEC) [1, 2] • First predicted by Bose & Einstein (1924) • Experimentally observed in University of Colorado JILA lab (1995) What is BEC? [1, 2] • A phase of matter where all particles occupy the same quantum state • Occurs when diulated bosons (integer spin particles) gas are cooled to extremely low temperature (10 -9 K) • Individual particle wave functions behave as a single wave function 4

Physical Motivation 1. High temperature particle behaviour dominated 3. T=Tcrit Bose Einstein Condensate 2. Low temperature λd. B α T -0. 5 4. T=0 Giant Matter Wave Fig 1. A visual description of how a gas of bosonic-atoms behave at various temperatures (T). [1] 5

Experimental Results of BEC JILA (95’, Rb, 5, 000) ETH (02’, Rb, 300, 000)

Gross–Pitaevskii equation • Hartree–Fock approximation [1, 2] – The many-body wavefunction is written as productsof individual wave functions of each bosons [1, 2] • The Hamiltonian • The conserved quantities

Gross–Pitaevskii equation • At temperature T<

Gross–Pitaevskii equation • Rearranging the equation and defining the following constants • The dimensionless Gross–Pitaevskii equation Note: This is mathematical equivalent to the cubic Nonlinear Schrödinger Equation (NLS) 9

Existing numerical methods for Nonlinear Schrödinger Equation Existing numerical methods for NLS Spectral Methods – Pseudo-spectral method (Muruganandam et al) – Time splitting Fourier spectral approximation (Bao et al. ) – Split-step Fourier spectral method (Weideman) Mesh-based Methods – Galerkin spectral (Dion et al. ) – Finite Element (Carl Joachim, Berdal Haga) – Split-step finite difference method (Wang) 10

Existing numerical methods for Nonlinear Schrödinger Equation Existing numerical methods for NLS Spectral Methods – Pseudo-spectral method (Muruganandam et al) – Time splitting Fourier spectral approximation (Bao et al. ) – Split-step Fourier spectral method (Weideman) Mesh-based Methods – Galerkin spectral (Dion et al. ) – Finite Element (Carl Joachim, Berdal Haga) – Split-step finite difference method (Wang) Require mesh generation and re -meshing 11

Radial Basis Functions 12

Radial basis function • What is a radial basis function (RBF)? [4, 5] 13

RBF scattered data approximation • Given a set of data {x 1. . . x. N} and the corresponding known values {f(x 1). . f(x. N)}. Find the function f(x) that describes the data set. • Is the system guaranteed to be solvable? • Are the solutions unique? 14

RBF scattered data approximation Fig 2. Interpolation of f(x, y) with Gaussian RBF with c=1/3 and N=25. (left) shows the random generated data points, (mid) shows the centred at the collocation points, (right) shows the interpolated surface. 15

Background of radial basis functions • The system is solvable and unique provided the coefficient matrix is positive definite. [4, 5, 11] 16

Background of radial basis functions Globally supported strictly positive definite radial basis functions (GSRBF) • Leads to dense coefficient matrix • In many cases the coefficient matrix is ill-conditioned • For matrix inversion Schaback (2007) suggested – Singular Value Decomposition – Regularization techniques 17

Background of radial basis functions Compactly supported radial basis functions (CSRBF) • • Wu and Wendland introduced the compactly supported RBF (CSRBF) [4, 5] Leads to sparse coefficient matrix Reduce ill-conditioning of the resultant coefficient matrix The usage of CSRBF will be explored in 3 D NLS numerical experiment 18

Error Behaviour of RBF techniques • Trade off principle Schaback (1995) [5] Theorem: It is impossible to construct radial basis functions which guarantees good stability and small errors at the same time. • Driscoll and Fornberg (2002) observed the "Flat Limit” [6] c->∞ leads to highly ill-conditioned RBF interpolation matrix c->0 implies highly localized RBFs such that it fails to approximate data between collocation points 19

Error Behaviour of RBF techniques • Wright, Fornberg, Larsson (2004) [7] – With increasing shape parameter, interpolation error decreases sharply until the minimum numerical error is reached. – For any increasing shape parameter, interpolation error rapidly increases. The rapid decrease of interpolation error reaches a minimum. 20

Solving PDE with radial basis functions • Kansa (1990) proposed a direct approach to approximate the solution of PDE by • • where Ф represents any RBF and p(x) is basis polynomial of up to order m. Consider a linear PDE boundary value problem • where the linear operator L operates on the interior points Ω/∂Ω, the operator B specifies the boundary conditions for collocations on the boundaries ∂Ω. 21

Solving PDE with radial basis functions • Applying the RBF approximation the domain with Ni interior points in Ω/∂Ω and Nb boundary points on ∂Ω yields N equations • To remove the extra m degrees of freedom of the polynomial p(x) 22

Solving PDE with radial basis functions • Rewriting in matrix form • Note: The resultant PDE matrix is asymmetric. Hence Kansa method is also known as asymmetric collocation method. 23

Solving time-dependent PDE with θmethod and RBF • Some common methods for time-dependent PDE – θ-method – Runge-Kutta – Laplace Transform • θ-method – Based on the discretization of time-domain of the PDE. – The forward and backward time-step is weighted by (0≤θ≤ 1) • Consider the following time-dependent linear PDE problem 24

Solving time-dependent PDE with θmethod and RBF • constructing a time-domain mesh for M units, such that each time increment is denoted by tn=ndt, n=1. . M, dt=T/M. • Hence the approximated PDE problem becomes • Approximate spatial variables by radial basis functions (ie. Kansa method) 25

Meshfree Numerical Method for Nonlinear Schrödinger Equation 26

Mesh-free Numerical Method for Nonlinear Schrödinger Equation • Recall: The equation for modelling dynamics of Bose-Einstein condensate (time-dependent Gross–Pitaevskii equation) • The Gross–Pitaevskii equation is mathematical equivalent to the cubic Nonlinear Schrödinger equation. • The parameter q controls the interaction between particles – q>0 defocusing interaction – q<0 focusing interaction 27

Mesh-free Numerical Method for Nonlinear Schrödinger Equation • The full 3 D cubic Nonlinear Schrodinger equation (NLS) with initial and boundary conditions 28

Mesh-free Numerical Method for Nonlinear Schrödinger Equation • Key-steps for deriving the mesh-free method for NLS 1. 2. 3. 4. separate the original NLS into real r(x, t) and imaginary parts s(x, t) apply θ-method in time-domain linearize PDE using the approach in Dereli (2009) apply Kansa asymmetric collocation to spatial variables • Advantages of the proposed mathematical method 1. 2. 3. 4. entirely meshfree solves NLS in various dimensions d ≤ 3 flexible for selecting radial basis functions easy to implement (~200 lines of matlab code) 29

Derivation of the proposed method • Separating the original NLS with respect to real r(x, t) and imaginary parts s(x, t) yields a system of PDEs. • Applying θ-method in time-domain 30

Derivation of the proposed method • Using the approach by Dereli et al (2009) [8] the variables (r*, s*) are introduced to approximate the solutions sufficient close to (rn+1, sn+1) 31

Derivation of the proposed method • Defining an auxiliary variable • Rewrite the real and imaginary parts of NLS using the definition of (r*, s*) and α: (Real) (Imaginary) 32

Derivation of the proposed method • Apply the RBF approximation to the real part r(x, t) and imaginary part s(x, t) of the wavefunction Ψ (x, t) and its spatial derivatives 33

Derivation of the proposed method 34

Derivation of the proposed method 35

Derivation of the proposed method • Final matrix form results a system of 2 Nx 2 N equations • Solved via Singular Value Decomposition at each time-step to find RBF coefficients ζn+1 • Specific cases of θ-method – θ=0 explicit method – θ=0. 5 semi-implicit method – θ=1 implicit method 36

Implementation flow-chart start Set up physical geometries and potential function Compute initial conditions Kernel of the method while t

Numerical Experiments 38

Radial basis functions in this project Globally supported strictly positive definite radial basis function (GSRBF) Compactly supported radial basis function (CSRBF) for 3 D problem 39

1 D NLS numerical example • We consider a 1 D test case in Deconinck et al. (2001) to model the stability of Bose Einstein Condensates and Wang (2005). [11] 40

1 D NLS numerical example • Comparison of absolute error between split-step finite difference method (SSFD) in Weideman (1986) and split-step Fourier spectral (SSFS) in Wang (2005). [11] Table 1. Absolute error comparison of RBF-θ and earlier methods. The solution is computed using RBF= Gaussian, θ=0. 5, M=200, N=128, c=2. 5. Table 2. Maximum relative error and maximum RMS error of real and imaginary parts of the wavefunction at T=1 generated by different globally supported strictly positive definite RBFs with M=500, N=128. 41

Fig 6. Real and imaginary parts of the numerical solution and the corresponding relative error at T=1 computed by RBF=Gaussian, M=500, N=128, c=2. 5, θ=0. 5. 42

Fig 7. Particle density (top) and relative error (bottom) of numerical solution at T=1 with M=500, N=128, c=2. 5, θ=0. 5, RBF=Gaussian. 43

2 D NLS numerical experiment • Consider a 2 D defocusing interaction where q=1, k=1 44

2 D NLS numerical results Table 5. Maximum relative error, RMS error for different GSRBFs with M=2000, N=100, T=1. Table 6. Maximum relative error and RMS error of particle density at T=1 generated by different GSRBFs with M=2000, N=100. 45

Fig 10. Real and imaginary parts of numerical solutions and the corresponding relative error at time T=1 computed by M=2000, 46 N=100, c=0. 7, θ=1, RBF=Gaussian

Fig 11. Particle density (top) and relative error (bottom) of numerical solution at T=1 computed by M=2000, N=100, c=0. 7, θ=1, RBF=Gaussian 47

3 D NLS numerical experiment • Consider a 3 D focusing example where q=-1, k=2 48

3 D NLS numerical results Numerical results for all θ-methods and GSRBF combinations Table 7. Maximum relative error and RMS error of particle density at T=1 generated by various GSRBFs. 49

Fig 12. Real and imaginary parts of numerical solutions and the corresponding relative error at time T=1 computed by M=800, N=216, c=2. 0, θ=1, RBF=IMQ. 50

3 D NLS numerical results Numerical results for all θ-methods and GSRBF combinations Table 7. Maximum relative error and RMS error of particle density at T=1 generated by various GSRBFs. 51

3 D NLS numerical results Numerical results for all θ-methods and GSRBF combinations Can we speed up the simulation? ? ? Table 7. Maximum relative error and RMS error of particle density at T=1 generated by various GSRBFs, M=800, N=216. 52

Effects of shape parameter • Accuracy: error behaviour is consistent with observation Wright, Fornberg, Larsson (2004) • Computational time: 96% of the time is consumed by SVD Fig 13. Computational time for various shape parameters 53

Compactly supported radial basis functions (CSRBF) • Combined implicit method (θ=1) with CSRBF to overcome computationtime barrier • Matrix inversion is done via LU factorization • Reduced total simulation time by 85% compared to globally supported strictly positive radial basis functions Table 8. Illustration of maximum absolute error, maximum relative error and computation time for implicit RBF-θ method using various compactly supported radial basis functions. 54

Fig 15. Real and imaginary parts of numerical solutions and the corresponding relative error at time T=1 computed by M=800, N=216, c=6. 0, θ= 1, RBF=W 13(Wu 1, 3). 55

Table 9. Maximum absolute and relative error for various terminal time (T) generated using different RBFs with M=800, N=216, θ=1.

Summary of Results • Globally supported strictly positive definite RBFs (GSRBF) • Relative error of O(10 -4) -O(10 -3), RMS error O(10 -5)-O(10 -3) • Leads to dense matrices • Require sophisticated matrix inversion method (SVD) [10] • 96% of the time per iteration is consumed by matrix inversion • Compactly supported RBFs (CSRBF) • Offer same level of accuracy as GSRBF • Leads to sparse matrices • Can be solved by conventional methods such as LU factorization • Reduce the overall simulation time by 85% 57

Future Work • Shape parameter selection strategy • More sophisticated time integration scheme – For time dependent external potentials (Nistazakis et al) • On computational enhancements – Utilize more efficient data structures for large scale simulations – Explore parallelism using GPUs or High Performance Computing 58

Conclusion • Showed the physical motivation behind the BEC problem • Introduced the basics of RBFs – Classification – Asymmetric collocation for PDE • Proposed a new mesh-free method (RBF-θ) for cubic Nonlinear Schrödinger equation – θ-method in time – RBF approximation for spatial variables • Validated the RBF-θ method via numerical experiments – Relative error: O(10 -4) -O(10 -3) – RMS error: O(10 -5)-O(10 -3) 59

Thank you very much 60

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