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Advances concerning multiscale methods and uncertainty quantification - - PowerPoint PPT Presentation

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W ESTFLISCHE W ILHELMS -U NIVERSITT M NSTER Advances concerning multiscale methods and uncertainty quantification in E XA -D UNE P. Bastian, C. Engwer, J. Fahlke, M. Geveler, D. living knowledge Gddeke, O. Iliev, O. Ippisch, R. Milk, J.

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Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE

  • P. Bastian, C. Engwer, J. Fahlke, M. Geveler, D.

Göddeke, O. Iliev, O. Ippisch, R. Milk, J. Mohring,

  • S. Müthing, M. Ohlberger, D. Ribbrock, S. Turek

January 25th, 2016

living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 2 /21

Outline

◮ Abstraction of Multiscale Methods ◮ Implementation + Results ◮ MLMC ◮ Integration with MLMC Framework

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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

◮ DUNE and FEAST (TU Dortmund, hardware-oriented numerics)

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 4 /21

Problem Statement

Assume we are looking for uǫ ∈ U solving Rǫ[uǫ](v) = 0 ∀v ∈ U where Rǫ : U → U

′

Example: U = H1

0(Ω),

Rǫ[u] = bǫ[u] − I bǫ[u](v) =

  • Ω

Aǫ∇u∇v, I(v) =

  • Ω

fv Aǫ(x) = A x

ǫ

  • : Ω → Rd×d

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 5 /21

Scales

◮ Idea: Make use of possible

scale-separation

◮ Macroscopic scale represented

by coarse-scale space UH ⊂ U

  • n coarse partition TH of Ω

◮ Microscopic scale represented

by fine-scale space Uh ⊂ U on fine partition Th of Ω

◮ Spaces should be nested

UH ⊂ Uh ⊂ U

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 6 /21

Abstract Method

◮ Let πUH : U → UH denote a projection onto the coarse space

with Uf,h = {uh ∈ Uh : πUh(uh) = 0}

◮ Compute and approximate discrete solution

uǫ

µ,h = uH + uf,h ∈UH ⊕ Uf,h satisfying

Rǫ

µ[uH + uf,h](vH) = 0

∀vH ∈ UH Rǫ

µ[uH + uf,h](vf,h) = 0

∀vf,h ∈ Uf,h

◮ Concrete choices for UH, Uh, πUH and further localization of fine

scale equation yield various multiscale methods.

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 7 /21

Pure MPI Implementation

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 7 /21

Pure MPI Implementation

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 7 /21

Pure MPI Implementation

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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 ◮ 643 cubes on coarse scale with 83 fine cells each, 134.217.728

total

◮ Hardware: SuperMUC Phase 2, 256 to 8192 MPI ranks ◮ coarse system: DUNE-istl BiCGStab with ILUT preconditioning ◮ UMFPACK direct sparse solve for local problems

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 9 /21

Testcase Setup

Standard diffusion problem in Ω = [0, 1]3, with boundary conditions u(x) = 0 on ∂Ω\x3 ∈ {0, 1} and Neumann-zero

  • elsewhere. With diffusion Aǫ and source f ǫ given as:

Aǫ(x1, x2, x3) := 1

8π2

  2

  • 2 + cos
  • 2π x1

ǫ

  • 1 + 1

2 cos

  • 2π x1

ǫ

 f ǫ(x) := − ∇ · (Aǫ (x) ∇vǫ(x)) vǫ(x1, x2, x3) := sin(2πx1) sin(2πx2) + ǫ

2 cos(2πx1) cos(2πx2) sin

  • 2π x1

ǫ

  • ,

, René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 10 /21

Analytical Solution

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 11 /21

28 29 210 211 212 213

# Cores

20 21 22 23 24 25 26

Speedup Overall Coarse solve Local assembly + solve Coarse assembly Ideal

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 12 /21

Problem: Coarse solve

◮ Very few degrees of freedom per MPI-rank. Strong Scaling ends

with only 32 cells per rank.

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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!

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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!

◮ Simple for local problems: they’re already embarrassingly

parallel

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 13 /21

Hybrid MPI/Shared Memory Implementation

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 13 /21

Hybrid MPI/Shared Memory Implementation

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 13 /21

Hybrid MPI/Shared Memory Implementation

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 14 /21

Hybrid Strong Scaling Setup

◮ 83 cubes on coarse scale with 323 fine cells each, 16.777.216

total

◮ Hardware: CHEOPS (RRZK Cologne) ◮ 16 to 128 MPI-ranks, one hexacore CPU per rank: 96 to 768

cores used

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 15 /21

27 28 29

# Cores

2−1 20 21 22 23 24

Speedup Overall Coarse solve Local assembly + solve Coarse assembly Ideal

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 16 /21

UQ and MC

◮ Consider ground water flow through some real world domain

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 16 /21

UQ and MC

◮ Consider ground water flow through some real world domain ◮ Problem: full permeability field not accessible

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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

◮ (Aggregate) Quantities of interest Q(ω) (ex. total flow trough

domain) can only be characterised by stochastical distributions

, , René Milk (rene.milk@wwu.de)

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living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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

◮ (Aggregate) Quantities of interest Q(ω) (ex. total flow trough

domain) can only be characterised by stochastical distributions

◮ Conventionally this uncertainty is quantified by means of

Monte Carlo methods

, , René Milk (rene.milk@wwu.de)

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SLIDE 27

living knowledge WWU Münster

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 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

◮ (Aggregate) Quantities of interest Q(ω) (ex. total flow trough

domain) can only be characterised by stochastical distributions

◮ Conventionally this uncertainty is quantified by means of

Monte Carlo methods

◮ But: Notoriously bad convergence

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 17 /21

MLMC Ideas

The main idea behind Multi Level Monte Carlo Method is to split Q(ω) = Q0(ω) + Q1(ω) − Q0(ω) + · · · + QL(ω) − QL−1(ω) + Q(ω) − QL(ω) with L + 1 increasingly accurate (+ time intensive) solvers. We reduce its variance accordingly, with many quick evaluations for the coarse solution and few slower solves for the differences. These solves are independent of one another → trivial parallelization. MLMC can greatly improve order of convergence compared with MC.

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 18 /21

MLMC Implementation

◮ DUNE module with very little requirements on pluggable

solvers

◮ Optimizes selection of slow/fast solves to minimize time to

solution

◮ Distributes work across groups of MPI-Ranks ◮ Tries to mimimize the communication overhead for estimating

mean times and variances

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 19 /21

MLMC Testcase setup

◮ stationary single phase flow through a unit cube ◮ random permeability field ◮ Boundary condition: constant pressure difference between left

and right end faces

◮ Aggregate quantity Q(ω) sought is the total flow through the

cube

◮ up to 1024 CPUs of the ITWM Cluster in Kaiserslautern

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 20 /21

Scaling of a 2-level MsFEM/FEM and comparison with Standard MC

500 1000 2 4 6 8 number of processors speedup ideal actual 0.1 0.2 0.3 2000 4000 6000 8000 10000 12000 14000 correlation length computation time [s] MC MLMC

, , René Milk (rene.milk@wwu.de)

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER

Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE 21 /21

Thank you for your attention.

Get the code: https://github.com/wwu-numerik/dune-multiscale https://github.com/wwu-numerik/dune-mlmc

, , René Milk (rene.milk@wwu.de)