Parallel, adaptive multigrid methods for parabolic PDEs and - PowerPoint PPT Presentation

Parallel, adaptive multigrid methods for parabolic PDEs and applications Feng Wei Yang Department of Mathematics University of Sussex F.W.Yang@sussex.ac.uk 25 January 2016 Feng Wei Yang Seminar at Warwick 25 January 2016 1 / 46

Objectives To solve complex non-linear parabolic systems by applying: 2 nd order central Finite Difference Method (FDM) 2 nd order Backward Differentiation Formula (BDF2) Nonlinear multigrid method with Full Approximation Scheme (FAS) Adaptive Mesh Refinement (AMR) Adaptive time-stepping Parallel techniques Feng Wei Yang Seminar at Warwick 25 January 2016 2 / 46

Outline Multigrid methods Linear multigrid Nonlinear multigrid Thin film models from Gaskell et al. and Validation Tumour modelling and a model from Wise et al. Optimal control for whole cell tracking Feng Wei Yang Seminar at Warwick 25 January 2016 3 / 46

Jacobi/Gauss-Seidel iterative methods Well-known methods Require diagonally-dominant matrices Typically have complexity of O ( n 2 ) for general sparse matrices ... Smoothing property High frequency of error Low frequency of error S.H. Lui Numerical Analysis of Partial Differential Equations , 2011 Feng Wei Yang Seminar at Warwick 25 January 2016 4 / 46

Convergence of a typical Jacobi iterative method source: nkl.cc.u-tokyo.ac.jp Feng Wei Yang Seminar at Warwick 25 January 2016 5 / 46

Multigrid V-cycle Finest grid Grid level 4 Grid level 3 Grid level 2 Coarsest grid Grid level 1 y x Feng Wei Yang Seminar at Warwick 25 January 2016 6 / 46

Linear multigrid A linear problem: Au = b , (1) exact error can be obtained as E = u − v , (2) residual can be calculated as: r = b − Av . (3) Error equation: AE = A ( u − v ) = Au − Av (4) = b − Av = r . Feng Wei Yang Seminar at Warwick 25 January 2016 7 / 46

Linear multigrid Feng Wei Yang Seminar at Warwick 25 January 2016 8 / 46

Nonlinear multigrid The Error Equation (4) does not exist in a nonlinear case Full Approximate Scheme (FAS) For problem on coarser grids, a modified RHS is included Feng Wei Yang Seminar at Warwick 25 January 2016 9 / 46

Nonlinear multigrid Feng Wei Yang Seminar at Warwick 25 January 2016 10 / 46

A nonlinear point-wise smoother Let’s consider our nonlinear problem: A ( v ) = f . It can be rewritten as: F ( v ) = 0 . Then the Newton-like nonlinear point-wise smoother at a particular grid point ( i , j ) ∈ Ω can be the following: F ( v ) v ℓ +1 , t +1 = v ℓ, t +1 − . i , j i , j F ′ ( v ℓ, t +1 ) i , j Feng Wei Yang Seminar at Warwick 25 January 2016 11 / 46

Domain decomposition and guard cells 16 17 12 13 14 15 10 8 9 11 6 7 4 5 2 3 1 boundary Feng Wei Yang Seminar at Warwick 25 January 2016 12 / 46

Droplet spreading model ∂ h � � �� � � �� h 3 h 3 ∂ p ∂ p ∂ ∂ x − B o + ∂ ∂ t = ǫ sin α ∂ x 3 ∂ y 3 ∂ y p = −△ ( h ) − Π( h ) + B o h cos α with Neumann boundary conditions: ∂ n h = 0 ∂ n p = 0 on ∂ Ω Gaskell et al. Int. J. Numer. Meth. Fluids , 45:1161-1186, 2004 Feng Wei Yang Seminar at Warwick 25 January 2016 13 / 46

Our solver Cell-centred 2 nd order finite difference method PARAMESH library for mesh generation and AMR Fully implicit BDF2 method with adaptive time-stepping MLAT variation of FAS multigrid at each time-step Newton-block 2 × 2 Red-Black (weighted) Gauss-Seidel smoother Full weighting restriction and bilinear interpolation Parallelism achieved using MPI Feng Wei Yang Seminar at Warwick 25 January 2016 14 / 46

Newton-block smoother Update at a grid point ( i , j ): − 1 � F h i , j ( h , p ) � h ℓ +1 , t +1 � h ℓ, t +1 ∂ F h ∂ F h   � � � ∂ h t +1 ∂ p t +1 i , j i , j = − p ℓ +1 , t +1 p ℓ, t +1  ∂ F p ∂ F p  F p i , j ( h , p ) ∂ h t +1 ∂ p t +1 i , j i , j i , j i , j Feng Wei Yang Seminar at Warwick 25 January 2016 15 / 46

Validation 32x32 64x64 5 128x128 256x256 512x512 1024x1024 4 3 h 0 (t) 2 1 0 0 0.2 0.4 0.6 0.8 1 t − 5 x 10 Results from Gaskell et al. on the left and our results on the right. Feng Wei Yang Seminar at Warwick 25 January 2016 16 / 46

Multigrid linear complexity 2 10 CPU time required Line with slope of 1 Average CPU time per time step (seconds). 1 10 0 10 4 5 6 7 10 10 10 10 No. grid points on the finest grid. A log-log plot demonstrating the linear complexity of multigrid. Feng Wei Yang Seminar at Warwick 25 January 2016 17 / 46

Multigrid performance Results from Gaskell et al. on the left and our results on the right. Feng Wei Yang Seminar at Warwick 25 January 2016 18 / 46

AMR AMR with initial condition on the left and final solution on the right. Feng Wei Yang Seminar at Warwick 25 January 2016 19 / 46

Adaptive time-stepping − 7 x 10 11 adaptive time − stepping 1024x1024 10 9 8 7 Time step size 6 5 4 3 2 1 0 0.2 0.4 0.6 0.8 1 Time − 5 x 10 Evolution of δ t during T = [0 , 1 × 10 − 5 ]. Feng Wei Yang Seminar at Warwick 25 January 2016 20 / 46

Adaptive multigrid solver Cases No. leaf nodes Uniform 1024 2 1,048,576 AMR 168,480 Cases No. time CPU time steps (seconds) Fixed δ t 1000 16721.3 ATS 45 574.4 Feng Wei Yang Seminar at Warwick 25 January 2016 21 / 46

Animation Feng Wei Yang Seminar at Warwick 25 January 2016 22 / 46

Parallel efficiency vs. Actual computational time 100% 500 AMR 32x32x32 − 256x256x256 AMR 32x32x32 − 256x256x256 AMR 64x64x64 − 256x256x256 AMR 64x64x64 − 256x256x256 450 90% The computational time (seconds) 400 80% 70% 350 Parallel efficiency 300 60% 250 50% 200 40% 150 30% 100 20% 50 10% 0 0% 0 20 40 60 80 1 4 16 64 Number of cores Number of cores Feng Wei Yang Seminar at Warwick 25 January 2016 23 / 46

Tumour modelling - avascular tumour growth Starts with a small cluster of cells Nutrient supply through diffusion Internal adhesion force Three layers of cells: Proliferative cells Dormant cells Dead cells (necrosis) Volume loss in necrotic core source: www.bioinfo.de Feng Wei Yang Seminar at Warwick 25 January 2016 24 / 46

Tumour modelling - vascular tumour growth TAF chemical factor Inducing blood vessel (angiogenesis) Exponential growth rate Develop secondaries through metastasis source: www.maths.dundee.ac.uk Feng Wei Yang Seminar at Warwick 25 January 2016 25 / 46

Tumour model from Wise et al. ∂ t φ T = M ∇ · ( φ T ∇ µ ) + S T ( φ T , φ D , n ) − ∇ · ( u S φ T ) f ′ ( φ T ) − ǫ 2 ∆ φ T = µ ∂ t φ D = M ∇ · ( φ D ∇ µ ) + S D ( φ T , φ D , n ) − ∇ · ( u S φ D ) − ( ∇ p − γ = ǫ µ ∇ φ T ) u S ∇ · u S = S T ( φ T , φ D , n ) 0 = ∆ n + T C ( φ T , n ) − n ( φ T − φ D ) with mixed boundary conditions: µ = p = 0 n = 1 ∂ n φ T = ∂ n φ D = 0 on ∂ Ω , S. M. Wise, J. S. Lowengrub, V. Cristini, Math. Comput. Modelling , 53: 1-20, 2011. Feng Wei Yang Seminar at Warwick 25 January 2016 26 / 46

Tumour model from Wise et al. ∂ t φ T = M ∇ · ( φ T ∇ µ ) + S T ( φ T , φ D , n ) − ∇ · ( u S φ T ) f ′ ( φ T ) − ǫ 2 ∆ φ T = µ ∂ t φ D = M ∇ · ( φ D ∇ µ ) + S D ( φ T , φ D , n ) − ∇ · ( u S φ D ) − ( ∇ p − γ [ u S = ǫ µ ∇ φ T )] S T ( φ T , φ D , n ) − ∇ · ( γ − ∆ p = ǫ µ ∇ φ T ) 0 = ∆ n + T C ( φ T , n ) − n ( φ T − φ D ) with mixed boundary conditions: µ = p = 0 n = 1 ∂ n φ T = ∂ n φ D = 0 on ∂ Ω , S. M. Wise, J. S. Lowengrub, V. Cristini, Math. Comput. Modelling , 53: 1-20, 2011. Feng Wei Yang Seminar at Warwick 25 January 2016 27 / 46

Validation t=100 t=200 Validation between results of Wise et al. and ours. Feng Wei Yang Seminar at Warwick 25 January 2016 28 / 46

2 nd order convergence rate For variable φ T Levels Time steps Infinity norm Ratio Two norm Ratio 5(128 2 ) 1250 - - - - 6(256 2 ) 9 . 118 × 10 − 2 7 . 836 × 10 − 3 2500 - - 7(512 2 ) 1 . 322 × 10 − 2 1 . 131 × 10 − 3 5000 6 . 90 6 . 93 8(1024 2 ) 2 . 579 × 10 − 3 2 . 367 × 10 − 4 10000 5 . 13 4 . 78 9(2048 2 ) 6 . 415 × 10 − 4 5 . 833 × 10 − 5 20000 4 . 02 4 . 06 2 nd order convergence rate seen from the model of tumour growth. Feng Wei Yang Seminar at Warwick 25 January 2016 29 / 46

3-D results t=50 t=100 t=150 t=200 Feng Wei Yang Seminar at Warwick 25 January 2016 30 / 46

3-D results Cutting through t=200 x plane Cutting through Cutting through y plane z plane Feng Wei Yang Seminar at Warwick 25 January 2016 31 / 46

Model objectives To track the morphology of cells and reconstruct their movements: V. Peschetola et al. Cytoskeleton , 2013 Feng Wei Yang Seminar at Warwick 25 January 2016 32 / 46