e astencils overview
play

E aStencils Overview Task: Solve a PDE (efficiently). Multigrid - PowerPoint PPT Presentation

Mar 06, 2023 •223 likes •855 views

The Mathematics of ExaStencils Hannah Rittich joint work with: Matthias Bolten and Karsten Kahl Bergische Universit at Wuppertal April 15, 2015 E aStencils Overview Task: Solve a PDE (efficiently). Multigrid methods is a framework

  1. The Mathematics of ExaStencils Hannah Rittich joint work with: Matthias Bolten and Karsten Kahl Bergische Universit¨ at Wuppertal April 15, 2015 E aStencils

  2. Overview ◮ Task: Solve a PDE (efficiently). ◮ Multigrid methods is a framework for constucting algorithms. ◮ These methods are build from a set of different ingredients. ⇒ Which ingredients to choose? ◮ How do these ingredients influence (the computational and) numerical performance of method? ◮ Estimate the (numerical) performance is needed. ⇒ Local Fourier Analysis. (What is this?)

  3. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1)

  4. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1) Photo by Graham Richardson / CC BY

  5. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1) 0 -0.5 -1 1 1 0.5 0.5 0 0

  6. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1) 0 -0.5 -1 1 1 0.5 0.5 0 0

  7. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1) u : Ω → R y Ω x

  8. Poisson’s Equation ∆ u = ∂ 2 u ∂x 2 + ∂ 2 u ∆ u = f on Ω ∂y 2 where u = g on ∂ Ω Ω = (0 , 1) × (0 , 1) u : Ω → R u h : Ω h → R y y Ω h Ω x x

  9. Multigrid in One Slide ◮ For many simple iterative methods and certain problems we have that the error is 0.2 0.15 0.1 0.05 smooth after a few iterations. 0 -0.05 -0.1 -0.15 -0.2 ◮ Smooth error can be represented on a 1 0.8 1 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0 coarser grid. ⇒ Linear system of reduced size. 0.015 0.01 ◮ Use recursion on the coarse problem. 0.005 0 -0.005 -0.01 choose u 1 0.8 0.2 0.4 0.6 0.8 1 0.6 0.4 0.2 0 repeat 0 smooth u 0.015 e ← coarse approx. of e ˜ 0.01 0.005 0 u ← u + ˜ e -0.005 -0.01 1 smooth u 0.8 0.2 0.4 0.6 0.8 1 0.6 0.4 0.2 0 0 until satisfied

  10. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  11. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  12. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  13. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  14. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  15. Gauß-Seidel Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Bad Parallelization...

  16. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  17. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  18. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  19. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  20. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  21. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  22. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  23. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  24. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  25. Red-Black GS Smoother u x,y = 1 4 ( f x,y − u x,y − 1 − u x − 1 ,y − u x +1 ,y − u x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 2 Msgs.

  26. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  27. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  28. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  29. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  30. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  31. Jacobi Smoother u n ew x,y = 1 4 ( f x,y − u o ld x,y − 1 − u o ld x − 1 ,y − u o ld x +1 ,y − u o ld x,y +1 ) ◮ Parallelizable ◮ Per Iteration: √ √ ◮ ≈ 4 N/ P Words p. Proc. ◮ 1 Msgs.

  32. Smoother Summary GS RB-GS Jacobi √ √ √ √ Words — 4 N/ P 4 N/ P Msgs — 2 1 Sm. 0.499817 0.25 1 2 -lv 0.192336 0.0737621 1 3 -lv 0.198407 0.104202 1

  33. Local Fourier Analysis 1. We seek for � E � for “some” operator E and (since lim n →∞ � E n � 1 /n = ρ ( E ) ) 2. for ρ ( E ) . That is E = ( I − ωD − 1 L ) (Jacobi’s Method), E = I − P ℓ L − 1 ℓ +1 P T ℓ L ℓ (Coarse Grid Correction) or ℓ ( I − P ℓ ( I − E γ E ℓ = M ν 2 ℓ +1 )( L ℓ +1 ) − 1 R ℓ L ℓ ) M ν 1 ℓ E ℓ max = 0 (Multigrid Method).

  34. Local Fourier Analysis 1. We seek for � E � for “some” operator E and (since lim n →∞ � E n � 1 /n = ρ ( E ) ) 2. for ρ ( E ) . That is E = ( I − ωD − 1 L ) (Jacobi’s Method), E = I − P ℓ L − 1 ℓ +1 P T ℓ L ℓ (Coarse Grid Correction) or ℓ ( I − P ℓ ( I − E γ E ℓ = M ν 2 ℓ +1 )( L ℓ +1 ) − 1 R ℓ L ℓ ) M ν 1 ℓ E ℓ max = 0 (Multigrid Method).

  35. Stencil Operators ◮ Let Ω h = h 1 Z × · · · × h d Z . ◮ If L is a linear operator on the grid Ω h it can be written in the form � ( Lu )( x ) = s x ( κ ) · u ( x + κ ) κ ∈ h Z d for x ∈ Ω h . ◮ For many operators: s x ( κ ) x ≈ x ′ ⇒ s x ≈ s x ′ . u ( x ) u ( x + κ ) ◮ To analyze the local behavior it is sufficient to deal with constant stencils.

  36. Local Fourier Analysis 1. We seek for � E � for “some” operator E and (since lim n →∞ � E n � 1 /n = ρ ( E ) ) 2. for ρ ( E ) . ◮ Consider the operator given by the constant stencil � ( Lu )( x ) = s ( κ ) · u ( x + κ ) κ ∈ h Z d ◮ Grid size independent analysis. ◮ “Move boundary conditions to infinity.” ⇒ Consider operators on an infinite grid. ◮ Exploits simple structure in the frequency Domain. ◮ Quantitative estimates.

  37. Discrete Time Fourier Transform (DTFT) F h “Time” Domain Frequency Domain F − 1 h (truncated) time domain frequency domain 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -10 -5 0 5 10 -3 -2 -1 0 1 2 3 � � vol 1 / 2 vol 1 / 2 u ( ϑ ) e i � ϑ,x � d ϑ f ( x ) e i � ϑ,x � h h ˆ (2 π ) d/ 2 (2 π ) d/ 2 Θ h x ∈ Ω h

  38. Discrete Time Fourier Transform (DTFT) F h “Time” Domain Frequency Domain F − 1 h (truncated) time domain frequency domain 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -10 -5 0 5 10 -3 -2 -1 0 1 2 3 � � vol 1 / 2 vol 1 / 2 u ( ϑ ) e i � ϑ,x � d ϑ f ( x ) e i � ϑ,x � h h ˆ (2 π ) d/ 2 (2 π ) d/ 2 Θ h x ∈ Ω h

  39. Discrete Time Fourier Transform (DTFT) F h “Time” Domain Frequency Domain F − 1 h (truncated) time domain frequency domain 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -10 -5 0 5 10 -3 -2 -1 0 1 2 3 � � vol 1 / 2 vol 1 / 2 u ( ϑ ) e i � ϑ,x � d ϑ f ( x ) e i � ϑ,x � h h ˆ (2 π ) d/ 2 (2 π ) d/ 2 Θ h x ∈ Ω h

Recommend

E aStencils ExaSlang and the ExaStencils code generator Christian Schmitt 1 , Stefan Kronawitter 2

E aStencils ExaSlang and the ExaStencils code generator Christian Schmitt 1 , Stefan Kronawitter 2

E aStencils ExaSlang and the ExaStencils code generator Christian Schmitt 1 , Stefan Kronawitter 2 , Sebastian Kuckuk 3 , Frank Hannig 1 , Jrgen Teich 1 , Christian Lengauer 2 , Harald Kstler 3 , Ulrich Rde 3 1 Hardware/Software Co-Design,

1.01k views • 87 slides
OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 SF park overview OVERVIEW PRESENTATION / 2

OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 SF park overview OVERVIEW PRESENTATION / 2

OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 SF park overview OVERVIEW PRESENTATION / 2 Coin and card meters OVERVIEW PRESENTATION / 3 Making garages work better OVERVIEW PRESENTATION / 4 User interface and customer experience OVERVIEW

1.3k views • 24 slides
OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 Acknowledgements OVERVIEW PRESENTATION / 2 SF

OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 Acknowledgements OVERVIEW PRESENTATION / 2 SF

OVERVIEW PRESENTATION / 1 OVERVIEW PRESENTATION / 1 Acknowledgements OVERVIEW PRESENTATION / 2 SF park overview OVERVIEW PRESENTATION / 3 Coin and card meters OVERVIEW PRESENTATION / 4 Improving the customer experience at garages OVERVIEW

568 views • 25 slides
GSM System Overview GSM System Overview GSM System Overview GSM System Overview Phone Lin

GSM System Overview GSM System Overview GSM System Overview GSM System Overview Phone Lin

National Taiwan University Department of Computer Science and Information Engineering GSM System Overview GSM System Overview GSM System Overview GSM System Overview Phone Lin Phone Lin Ph.D. Ph.D. Email: plin@csie.ntu.edu.tw Email:

686 views • 41 slides
i c r o g u a r d s OVERVIEW The m i c r o g u a r d BIOFUELS s OVERVIEW

i c r o g u a r d s OVERVIEW The m i c r o g u a r d BIOFUELS s OVERVIEW

LGE UNICAMP The m i c r o g u a r d s OVERVIEW The m i c r o g u a r d BIOFUELS s OVERVIEW The m i c r o g u a r d ETHANOL s OVERVIEW The m i c Ideal conditions r o g Costs-benefits u a r d

1.39k views • 59 slides
Football First Aid: Football First Aid: An Overview An Overview Steven Richmond 95#

Football First Aid: Football First Aid: An Overview An Overview Steven Richmond 95#

Football First Aid: Football First Aid: An Overview An Overview Steven Richmond 95# Commissioner --BRYC Firefighter II, EMT-B, HTR & HZMT Tech City of Alexandria Fire and EMS Overview Overview Hyperthermia (Heat Related Injuries)

2.61k views • 31 slides
1 Corporate Overview Transaction 1 Corporate Overview Overview of Libre Group Structure

1 Corporate Overview Transaction 1 Corporate Overview Overview of Libre Group Structure

1 Corporate Overview Transaction 1 Corporate Overview Overview of Libre Group Structure Industry Business Model Hotels Overview Disclaimer This presentation does not constitute, or form any part of any offer for sale or subscription

808 views • 32 slides
HabaPalooza T EA M HYPERION COM M 362 A PRIL 29 T H 201 2 Overview Team Overview

HabaPalooza T EA M HYPERION COM M 362 A PRIL 29 T H 201 2 Overview Team Overview

HabaPalooza T EA M HYPERION COM M 362 A PRIL 29 T H 201 2 Overview Team Overview Project Overview Analysis Active Experimentation Question & Answer Team Overview Our Logo Our Tagline Our Mission Statement

1.09k views • 15 slides
Outbound Logistics Overview 1 Outbound Logistics Overview Outbound Overview Four

Outbound Logistics Overview 1 Outbound Logistics Overview Outbound Overview Four

Outbound Logistics Overview 1 Outbound Logistics Overview Outbound Overview Four Dedicated Outbound Carriers MPF Meijer Private Fleet 111 stores serviced in Michigan, northern Ohio, and Indiana. NTB Nationwide Truck

865 views • 15 slides
HabaPalooza T E A M H Y P E R I O N C O M M 3 6 2 A P R I L 2 9 TH 2 0 1 2 Overview Team

HabaPalooza T E A M H Y P E R I O N C O M M 3 6 2 A P R I L 2 9 TH 2 0 1 2 Overview Team

HabaPalooza T E A M H Y P E R I O N C O M M 3 6 2 A P R I L 2 9 TH 2 0 1 2 Overview Team Overview Project Overview Analysis Active Experimentation Question & Answer Team Overview Our Logo Our Tagline Our

934 views • 15 slides
An overview to Maltese An overview to Maltese An overview to Maltese An overview to Maltese

An overview to Maltese An overview to Maltese An overview to Maltese An overview to Maltese

An overview to Maltese An overview to Maltese An overview to Maltese An overview to Maltese Agriculture Agriculture Agriculture characteristics S mall holdings L imited resources and L and Fragmentation. 2 Maltese agriculture in figures

1.04k views • 15 slides
Library Conversations: Talking with Users University of Rochester Overview Overview Used

Library Conversations: Talking with Users University of Rochester Overview Overview Used

Rebecca Bichel December 7, 2007 Library Conversations: Talking with Users University of Rochester Overview Overview Used anthropological & ethnographic Overview Overview methods Guiding research question: What do students

941 views • 44 slides
.NET Overview Objectives Introduce .NET overview languages libraries

.NET Overview Objectives Introduce .NET overview languages libraries

.NET Overview Objectives Introduce .NET overview languages libraries development and execution model Examine simple C# program 2 .NET Overview .NET is a sweeping marketing term for a family of products

388 views • 23 slides
1 Overview Overview Regional demographic overview Regional demographic overview Workforce

1 Overview Overview Regional demographic overview Regional demographic overview Workforce

1 Overview Overview Regional demographic overview Regional demographic overview Workforce supply analysis Regional demand for workers i l d d f k Balancing supply and demand Reviewing key cluster trends Key challenges

533 views • 39 slides
EZ Overview EZ Overview The Rapid Development Platform p p Muse EZ is a

EZ Overview EZ Overview The Rapid Development Platform p p Muse EZ is a

EZ Overview EZ Overview The Rapid Development Platform p p Muse EZ is a registered trademark of Future Designs, Inc . 1 Overview What is EZ? EZ RTOS Engine EZ Four Tier Hierarchy Reusable HAL and Device

600 views • 32 slides
Overview of the Reconstruction Overview of the Reconstruction Overview of Some Legal Issues

Overview of the Reconstruction Overview of the Reconstruction Overview of Some Legal Issues

Overview of the Reconstruction Overview of the Reconstruction Overview of Some Legal Issues Related to Puerto Ricos Debt Restructuring by Sergio M. Marxuach Center for a New Economy February 25, 2020 CNE Puerto Ricos Think Tank

347 views • 13 slides
Overview & Development Table of Contents 1 Simon Owen, CEO Business Overview

Overview & Development Table of Contents 1 Simon Owen, CEO Business Overview

2018 Investor Tour INGENIA COMMUNITIES GROUP Overview & Development Table of Contents 1 Simon Owen, CEO Business Overview Guidance Update Development Overview Latitude One Overview 2 Craig Shepherd, Project

247 views • 22 slides
CSE 291E / EE260C Spring 2002 Overview Overview of Tensilica Overview of XTensa

CSE 291E / EE260C Spring 2002 Overview Overview of Tensilica Overview of XTensa

Configurable Cores: XTensa CSE 291E / EE260C Spring 2002 Overview Overview of Tensilica Overview of XTensa Design Flow Base ISA Optional Components TIE Challenges for Customized Processors What are the

612 views • 25 slides
CSE143 Au04 00-1 Content Overview (1) Content Overview (2) Programming language and libraries

CSE143 Au04 00-1 Content Overview (1) Content Overview (2) Programming language and libraries

Outline for Today Course Overview Goals Administrative details CSE 143 Workload and grading Computer Programming II Resources Background Welcome! Course Overview and Administrivia This information is largely included in

293 views • 5 slides
do: New to BOS? 1. Why BOS? Overview and practical tips for 2. Overview of BOS set-up

do: New to BOS? 1. Why BOS? Overview and practical tips for 2. Overview of BOS set-up

28/02/2017 Overview do: New to BOS? 1. Why BOS? Overview and practical tips for 2. Overview of BOS set-up conducting your survey 3. Creating a questionnaire 4. Distributing the questionnaire - options Joana Lopes 5. Launching the survey

252 views • 3 slides
Covid-19 and Business Interruption: Maximizing Insurance Coverage and Federal Grants Counsel

Covid-19 and Business Interruption: Maximizing Insurance Coverage and Federal Grants Counsel

Covid-19 and Business Interruption: Maximizing Insurance Coverage and Federal Grants Counsel wmclaw.com Overview Overview Overview Overview Catastrophic Losses Overview Overview Overview Overview 95% of Americans Subject to

434 views • 30 slides
Introduction and Process Overview Agenda Introduction Process Overview and Logistics

Introduction and Process Overview Agenda Introduction Process Overview and Logistics

Introduction and Process Overview Agenda Introduction Process Overview and Logistics Capital Plan Overview and Highlights Operating Plan General Fund Overview Department Manager Area Review Existing service level

514 views • 27 slides
COMPANY PRESENTATION MAY 2017 AGENDA 1. Company overview 2. Market overview 3. Operations in

COMPANY PRESENTATION MAY 2017 AGENDA 1. Company overview 2. Market overview 3. Operations in

BofAML EMERGING MARKETS CORPORATE CREDIT CONFERENCE COMPANY PRESENTATION MAY 2017 AGENDA 1. Company overview 2. Market overview 3. Operations in Chile & Peru 4. Growth opportunities 2 1. COMPANY OVERVIEW 3 Company overview Leading

701 views • 26 slides
Agenda 1. Introduction and overview of TPA changes 2. Terminology changes 3. Overview of changes

Agenda 1. Introduction and overview of TPA changes 2. Terminology changes 3. Overview of changes

Ticket Program Agreement (TPA) Call July 20, 2017 3:00 5:00 pm ET Participant Dial-in: (888) 394-8218 | Conference ID: 827-9297 Agenda 1. Introduction and overview of TPA changes 2. Terminology changes 3. Overview of changes to: Part I:

336 views • 20 slides

More recommend