Advances concerning multiscale methods and uncertainty quantification - PowerPoint PPT Presentation

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE P. Bastian, C. Engwer, J. Fahlke, M. Geveler, D. living knowledge Göddeke, O. Iliev, O. Ippisch, R. Milk, J. Mohring, WWU Münster January 25th, 2016 S. Müthing, M. Ohlberger, D. Ribbrock, S. Turek

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 2 /21 Outline ◮ Abstraction of Multiscale Methods ◮ Implementation + Results ◮ MLMC living knowledge ◮ Integration with MLMC Framework WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 3 /21 EXA-DUNE: "Flexible PDE Solvers, Numerical Methods, and Applications" ◮ 7 groups from 5 german universities ◮ Aim: develop new numerical, algorithmic and computational techniques to enable exa-scale computing for PDEs on heterogeneous massively parallel architectures living knowledge ◮ DUNE and FEAST (TU Dortmund, hardware-oriented numerics) WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 4 /21 Problem Statement Assume we are looking for u ǫ ∈ U solving R ǫ [ u ǫ ]( v ) = 0 ∀ v ∈ U where R ǫ : U �→ U ′ Example: living knowledge U = H 1 0 (Ω) , R ǫ [ u ] = b ǫ [ u ] − I WWU Münster � � b ǫ [ u ]( v ) = A ǫ ∇ u ∇ v , I ( v ) = fv Ω Ω � x : Ω �→ R d × d � A ǫ ( x ) = A ǫ , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 5 /21 Scales ◮ Idea: Make use of possible scale-separation ◮ Macroscopic scale represented by coarse-scale space U H ⊂ U on coarse partition T H of Ω living knowledge ◮ Microscopic scale represented WWU Münster by fine-scale space U h ⊂ U on fine partition T h of Ω ◮ Spaces should be nested U H ⊂ U h ⊂ U , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 6 /21 Abstract Method ◮ Let π U H : U �→ U H denote a projection onto the coarse space with U f , h = { u h ∈ U h : π U h ( u h ) = 0 } ◮ Compute and approximate discrete solution living knowledge u ǫ µ, h = u H + u f , h ∈ U H ⊕ U f , h satisfying WWU Münster R ǫ µ [ u H + u f , h ]( v H ) = 0 ∀ v H ∈ U H R ǫ µ [ u H + u f , h ]( v f , h ) = 0 ∀ v f , h ∈ U f , h ◮ Concrete choices for U H , U h , π U H and further localization of fine scale equation yield various multiscale methods. , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 7 /21 Pure MPI Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 7 /21 Pure MPI Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 7 /21 Pure MPI Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 8 /21 Strong Scaling Setup ◮ CG-FEM Discretizations via DUNE-gdt with DUNE-pdelab backend ◮ YASPGRID (cubes) from DUNE-grid on coarse and fine scale ◮ 64 3 cubes on coarse scale with 8 3 fine cells each, 134.217.728 total living knowledge ◮ Hardware: SuperMUC Phase 2, 256 to 8192 MPI ranks WWU Münster ◮ coarse system: DUNE-istl BiCGStab with ILUT preconditioning ◮ UMFPACK direct sparse solve for local problems , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 9 /21 Testcase Setup Standard diffusion problem in Ω = [ 0 , 1 ] 3 , with boundary conditions u ( x ) = 0 on ∂ Ω \ x 3 ∈ { 0 , 1 } and Neumann-zero elsewhere. With diffusion A ǫ and source f ǫ given as: 2 π x 1  � � ��  2 2 + cos 0 0 ǫ 1 + 1 A ǫ ( x 1 , x 2 , x 3 ) := 1 2 π x 1 � � 0 2 cos 0 8 π 2   ǫ living knowledge 0 0 0 f ǫ ( x ) := − ∇ · ( A ǫ ( x ) ∇ v ǫ ( x )) WWU Münster v ǫ ( x 1 , x 2 , x 3 ) := sin ( 2 π x 1 ) sin ( 2 π x 2 ) 2 π x 1 � � + ǫ 2 cos ( 2 π x 1 ) cos ( 2 π x 2 ) sin ǫ , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 10 /21 Analytical Solution living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 11 /21 2 6 Overall Coarse solve 2 5 Local assembly + solve Coarse assembly 2 4 Ideal Speedup 2 3 living knowledge 2 2 WWU Münster 2 1 2 0 2 8 2 9 2 10 2 11 2 12 2 13 # Cores , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 12 /21 Problem: Coarse solve ◮ Very few degrees of freedom per MPI-rank. Strong Scaling ends with only 32 cells per rank. living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 12 /21 Problem: Coarse solve ◮ Very few degrees of freedom per MPI-rank. Strong Scaling ends with only 32 cells per rank. ◮ Decrease number of ranks while increasing amount of work per rank! living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 12 /21 Problem: Coarse solve ◮ Very few degrees of freedom per MPI-rank. Strong Scaling ends with only 32 cells per rank. ◮ Decrease number of ranks while increasing amount of work per rank! living knowledge ◮ Simple for local problems: they’re already embarrassingly WWU Münster parallel , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 13 /21 Hybrid MPI/Shared Memory Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 13 /21 Hybrid MPI/Shared Memory Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 13 /21 Hybrid MPI/Shared Memory Implementation living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 14 /21 Hybrid Strong Scaling Setup ◮ 8 3 cubes on coarse scale with 32 3 fine cells each, 16.777.216 total ◮ Hardware: CHEOPS (RRZK Cologne) living knowledge ◮ 16 to 128 MPI-ranks, one hexacore CPU per rank: 96 to 768 WWU Münster cores used , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 15 /21 2 4 Overall Coarse solve 2 3 Local assembly + solve Coarse assembly Ideal 2 2 Speedup 2 1 living knowledge WWU Münster 2 0 2 − 1 2 7 2 8 2 9 # Cores , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 16 /21 UQ and MC ◮ Consider ground water flow through some real world domain living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 16 /21 UQ and MC ◮ Consider ground water flow through some real world domain ◮ Problem: full permeability field not accessible living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)

W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE 16 /21 UQ and MC ◮ Consider ground water flow through some real world domain ◮ Problem: full permeability field not accessible ◮ (With some assumptions) Possible to determine its many parameters from a number of measurements living knowledge WWU Münster , , René Milk (rene.milk@wwu.de)