petabricks
play

PetaBricks A Language and Compiler for Algorithmic Choice Jason - PowerPoint PPT Presentation

Jan 31, 2024 •332 likes •1.87k views

PetaBricks A Language and Compiler for Algorithmic Choice Jason Ansel Cy Chan Yee Lok Wong Marek Olszewski Qin Zhao Alan Edelman Saman Amarasinghe MIT - CSAIL June 16, 2009 Jason Ansel (MIT) PetaBricks June 16, 2009 1 / 47

  1. Introduction Language & Compiler Overview Sort algorithm timings 1 0.0025 InsertionSort QuickSort 0.002 0.0015 Time (s) 0.001 0.0005 0 0 250 500 750 1000 1250 1500 1750 Input Size 1 On an 8-way Xeon E7340 system Jason Ansel (MIT) PetaBricks June 16, 2009 8 / 47

  2. Introduction Language & Compiler Overview Sort algorithm timings 1 0.0025 InsertionSort QuickSort MergeSort 0.002 0.0015 Time (s) 0.001 0.0005 0 0 250 500 750 1000 1250 1500 1750 Input Size 1 On an 8-way Xeon E7340 system Jason Ansel (MIT) PetaBricks June 16, 2009 8 / 47

  3. Introduction Language & Compiler Overview Sort algorithm timings 1 0.0025 InsertionSort QuickSort MergeSort RadixSort 0.002 0.0015 Time (s) 0.001 0.0005 0 0 250 500 750 1000 1250 1500 1750 Input Size 1 On an 8-way Xeon E7340 system Jason Ansel (MIT) PetaBricks June 16, 2009 8 / 47

  4. Introduction Language & Compiler Overview Sort algorithm timings 1 0.0025 InsertionSort QuickSort MergeSort RadixSort 0.002 Autotuned 0.0015 Time (s) 0.001 0.0005 0 0 250 500 750 1000 1250 1500 1750 Input Size 1 On an 8-way Xeon E7340 system Jason Ansel (MIT) PetaBricks June 16, 2009 8 / 47

  5. Introduction Language & Compiler Overview Timings on different architectures Trained on Mobile Xeon 1-way Xeon 8-way Niagara Mobile - 1.09x 1.67x 1.47x Run on Xeon 1-way 1.61x - 2.08x 2.50x Xeon 8-way 1.59x 2.14x - 2.35x Niagara 1.12x 1.51x 1.08x - Jason Ansel (MIT) PetaBricks June 16, 2009 9 / 47

  6. Introduction Language & Compiler Overview Timings on different architectures Trained on Mobile Xeon 1-way Xeon 8-way Niagara Mobile - 1.09x 1.67x 1.47x Run on Xeon 1-way 1.61x - 2.08x 2.50x Xeon 8-way 1.59x 2.14x - 2.35x Niagara 1.12x 1.51x 1.08x - Jason Ansel (MIT) PetaBricks June 16, 2009 9 / 47

  7. Introduction Language & Compiler Overview Timings on different architectures Trained on Mobile Xeon 1-way Xeon 8-way Niagara Mobile - 1.09x 1.67x 1.47x Run on Xeon 1-way 1.61x - 2.08x 2.50x Xeon 8-way 1.59x - 2.35x 2.14x Niagara 1.12x 1.51x 1.08x - Jason Ansel (MIT) PetaBricks June 16, 2009 9 / 47

  8. Introduction Why choices Outline Introduction 1 Motivating Example Language & Compiler Overview Why choices PetaBricks Language 2 Key Ideas Compilation Example Other Language Features Results 3 Benchmarks Scalability Variable Accuracy Conclusion 4 Final thoughts Jason Ansel (MIT) PetaBricks June 16, 2009 10 / 47

  9. Introduction Why choices Early compilers Constrained Input Language (No choices) Parsing Early computers (and compilers) were weak Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 11 / 47

  10. Introduction Why choices Early compilers Constrained Input Language (No choices) Parsing Early computers (and compilers) were weak Code Gen Parsing and code generation dominated compilation Jason Ansel (MIT) PetaBricks June 16, 2009 11 / 47

  11. Introduction Why choices Early compilers Constrained Input Language (No choices) Parsing Early computers (and compilers) were weak Code Gen Parsing and code generation dominated compilation Needed a constrained input language to simplify compilation Jason Ansel (MIT) PetaBricks June 16, 2009 11 / 47

  12. Introduction Why choices Current compilers Constrained Input Language Current computers are much more powerful (No choices) Compilers can do a lot more Parsing Exposing Choices Decisions Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 12 / 47

  13. Introduction Why choices Current compilers Constrained Input Language Current computers are much more powerful (No choices) Compilers can do a lot more Parsing Input language is still constraining Exposing Choices Decisions Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 12 / 47

  14. Introduction Why choices Current compilers Constrained Input Language Current computers are much more powerful (No choices) Compilers can do a lot more Parsing Input language is still constraining Compilation dominated by exposing choices Exposing Choices Decisions Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 12 / 47

  15. Introduction Why choices Current compilers Constrained Input Language Current computers are much more powerful (No choices) Compilers can do a lot more Parsing Input language is still constraining Compilation dominated by exposing choices Input language specifies only one Exposing Choices Algorithmic choice Iteration order choice Parallelism strategy choice Data layout choice Decisions Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 12 / 47

  16. Introduction Why choices Current compilers Constrained Input Language Current computers are much more powerful (No choices) Compilers can do a lot more Parsing Input language is still constraining Compilation dominated by exposing choices Input language specifies only one Exposing Choices Algorithmic choice Iteration order choice Parallelism strategy choice Data layout choice Decisions Compiler must perform heroic analysis to reconstruct Code other choices Gen Jason Ansel (MIT) PetaBricks June 16, 2009 12 / 47

  17. Introduction Why choices PetaBricks compiler Rich Input Language (w/ choices) Parsing We propose explicit choices in the language Exploring Choices & Making Decisions Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 13 / 47

  18. Introduction Why choices PetaBricks compiler Rich Input Language (w/ choices) Parsing We propose explicit choices in the language The programmer defines the space of legal Algorithmic choices Exploring Choices Iteration orders (include parallel) & Making Decisions Data layouts Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 13 / 47

  19. Introduction Why choices PetaBricks compiler Rich Input Language (w/ choices) Parsing We propose explicit choices in the language The programmer defines the space of legal Algorithmic choices Exploring Choices Iteration orders (include parallel) & Making Decisions Data layouts Allow compilers to focus on exploring choices Compiler no longer needs to reconstruct choices Code Gen Jason Ansel (MIT) PetaBricks June 16, 2009 13 / 47

  20. Introduction Why choices Future-proof programs The result: programs can adapt to their environment Jason Ansel (MIT) PetaBricks June 16, 2009 14 / 47

  21. Introduction Why choices Future-proof programs The result: programs can adapt to their environment Choices make programs less brittle Jason Ansel (MIT) PetaBricks June 16, 2009 14 / 47

  22. Introduction Why choices Future-proof programs The result: programs can adapt to their environment Choices make programs less brittle Programs change with architecture, available cores, inputs, etc Jason Ansel (MIT) PetaBricks June 16, 2009 14 / 47

  23. PetaBricks Language Key Ideas Outline Introduction 1 Motivating Example Language & Compiler Overview Why choices PetaBricks Language 2 Key Ideas Compilation Example Other Language Features Results 3 Benchmarks Scalability Variable Accuracy Conclusion 4 Final thoughts Jason Ansel (MIT) PetaBricks June 16, 2009 15 / 47

  24. PetaBricks Language Key Ideas Algorithmic choice in the language Algorithmic choice is the key aspect of PetaBricks Jason Ansel (MIT) PetaBricks June 16, 2009 16 / 47

  25. PetaBricks Language Key Ideas Algorithmic choice in the language Algorithmic choice is the key aspect of PetaBricks Programmer can define multiple rules to compute the same data Jason Ansel (MIT) PetaBricks June 16, 2009 16 / 47

  26. PetaBricks Language Key Ideas Algorithmic choice in the language Algorithmic choice is the key aspect of PetaBricks Programmer can define multiple rules to compute the same data Compiler re-use rules to create hybrid algorithms Jason Ansel (MIT) PetaBricks June 16, 2009 16 / 47

  27. PetaBricks Language Key Ideas Algorithmic choice in the language Algorithmic choice is the key aspect of PetaBricks Programmer can define multiple rules to compute the same data Compiler re-use rules to create hybrid algorithms Can express choices at many different granularities Jason Ansel (MIT) PetaBricks June 16, 2009 16 / 47

  28. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  29. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  30. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make By rows? Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  31. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make By rows? By columns? Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  32. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make By rows? By columns? Diagonal? Reverse order? Blocked? Parallel? Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  33. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make By rows? By columns? Diagonal? Reverse order? Blocked? Parallel? Instead programmer provides explicit producer-consumer relations Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  34. PetaBricks Language Key Ideas Synthesized outer control flow Outer control flow synthesized by compiler Another choice that the programmer should not make By rows? By columns? Diagonal? Reverse order? Blocked? Parallel? Instead programmer provides explicit producer-consumer relations Allows compiler to explore choice space Jason Ansel (MIT) PetaBricks June 16, 2009 17 / 47

  35. PetaBricks Language Compilation Example Outline Introduction 1 Motivating Example Language & Compiler Overview Why choices PetaBricks Language 2 Key Ideas Compilation Example Other Language Features Results 3 Benchmarks Scalability Variable Accuracy Conclusion 4 Final thoughts Jason Ansel (MIT) PetaBricks June 16, 2009 18 / 47

  36. PetaBricks Language Compilation Example Simple example program 1 transform RollingSum 2 from A[ n ] 3 to B[ n ] 4 { 5 // r u l e 0: use the p r e v i o u s l y computed value 6 B. c e l l ( i ) from (A. c e l l ( i ) a , 7 B. c e l l ( i − 1) leftSum ) { 8 return a+leftSum ; 9 } 10 11 // r u l e 1: sum a l l elements to the l e f t 12 B. c e l l ( i ) from (A. region (0 , i ) in ) { 13 return sum( in ) ; 14 } 15 } Jason Ansel (MIT) PetaBricks June 16, 2009 19 / 47

  37. PetaBricks Language Compilation Example Simple example program 1 transform RollingSum 2 from A[ n ] 3 to B[ n ] 4 { 5 // r u l e 0: use the p r e v i o u s l y computed value 6 B. c e l l ( i ) from (A. c e l l ( i ) a , 7 B. c e l l ( i − 1) leftSum ) { 8 return a+leftSum ; 9 } 10 11 // r u l e 1: sum a l l elements to the l e f t 12 B. c e l l ( i ) from (A. region (0 , i ) in ) { 13 return sum( in ) ; 14 } 15 } Jason Ansel (MIT) PetaBricks June 16, 2009 19 / 47

  38. PetaBricks Language Compilation Example Simple example program ... 5 // r u l e 0: use the p r e v i o u s l y computed value 6 B. c e l l ( i ) from (A. c e l l ( i ) a , 7 B. c e l l ( i − 1) leftSum ) { 8 return a+leftSum ; 9 } ... A: B: Jason Ansel (MIT) PetaBricks June 16, 2009 20 / 47

  39. PetaBricks Language Compilation Example Simple example program A: B: ... 11 // r u l e 1: sum a l l elements to the l e f t 12 B. c e l l ( i ) from (A. region (0 , i ) in ) { 13 return sum( in ) ; 14 } ... Jason Ansel (MIT) PetaBricks June 16, 2009 21 / 47

  40. PetaBricks Language Compilation Example Applicable regions Compilation Process Applicable regions Choice grids Choice dependency graph Jason Ansel (MIT) PetaBricks June 16, 2009 22 / 47

  41. PetaBricks Language Compilation Example Applicable regions Compilation Process Applicable regions Choice grids Choice dependency graph // r u l e 0 : use the p r e v i o u s l y computed v a l u e B. c e l l ( i ) from (A. c e l l ( i ) a , B. c e l l ( i − 1) leftSum ) { a+leftSum ; return } Applicable where 1 ≤ i < n Jason Ansel (MIT) PetaBricks June 16, 2009 22 / 47

  42. PetaBricks Language Compilation Example Applicable regions Compilation Process Applicable regions Choice grids Choice dependency graph // r u l e 0 : use the p r e v i o u s l y computed v a l u e B. c e l l ( i ) from (A. c e l l ( i ) a , B. c e l l ( i − 1) leftSum ) { a+leftSum ; return } Applicable where 1 ≤ i < n // r u l e 1 : sum a l l elements to the l e f t B. c e l l ( i ) from (A. region (0 , i ) i n ) { return sum( i n ) ; } Applicable where 0 ≤ i < n Jason Ansel (MIT) PetaBricks June 16, 2009 22 / 47

  43. R1 R0 or R1 0 1 n PetaBricks Language Compilation Example Choice grids Compilation Process Applicable regions Choice grids Choice dependency graph Jason Ansel (MIT) PetaBricks June 16, 2009 23 / 47

  44. R1 R0 or R1 0 1 n PetaBricks Language Compilation Example Choice grids Compilation Process Applicable regions Choice grids Choice dependency graph Divide data space into symbolic regions with common sets of choices Jason Ansel (MIT) PetaBricks June 16, 2009 23 / 47

  45. PetaBricks Language Compilation Example Choice grids Compilation Process Applicable regions Choice grids Choice dependency graph Divide data space into symbolic regions with common sets of choices In this simple example: A: Input (no choices) B: [0 , 1) = rule 1 B: [1 , n ) = rule 0 or rule 1 R1 R0 or R1 0 1 n Jason Ansel (MIT) PetaBricks June 16, 2009 23 / 47

  46. PetaBricks Language Compilation Example Choice grids Compilation Process Applicable regions Choice grids Choice dependency graph Divide data space into symbolic regions with common sets of choices In this simple example: A: Input (no choices) B: [0 , 1) = rule 1 B: [1 , n ) = rule 0 or rule 1 R1 R0 or R1 0 1 n Applicable regions map rules → symbolic data Choice grids map symbolic data → rules Jason Ansel (MIT) PetaBricks June 16, 2009 23 / 47

  47. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  48. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Adds dependency edges between symbolic regions Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  49. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Adds dependency edges between symbolic regions Edges annotated with directions and rules Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  50. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Adds dependency edges between symbolic regions Edges annotated with directions and rules Many compiler passes on this IR to: Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  51. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Adds dependency edges between symbolic regions Edges annotated with directions and rules Many compiler passes on this IR to: Simplify complex dependency patterns Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  52. PetaBricks Language Compilation Example Choice dependency graph Compilation Process Applicable regions Choice grids Choice dependency graph (r0,=,-1) (r1,<=),(r0,=) B.region(1, n) A.region(0, n) Choices: r0, r1 (r1,<=),(r0,=) (r0,=,-1) B.region(0, 1) Choices: r1 Adds dependency edges between symbolic regions Edges annotated with directions and rules Many compiler passes on this IR to: Simplify complex dependency patterns Add choices Jason Ansel (MIT) PetaBricks June 16, 2009 24 / 47

  53. PetaBricks Language Compilation Example Code generation PetaBricks Source Code 1 PetaBricks source code is compiled PetaBricks Compiler Autotuning Final Binary Binary Choice Configuration File Jason Ansel (MIT) PetaBricks June 16, 2009 25 / 47

  54. PetaBricks Language Compilation Example Code generation PetaBricks Source Code 1 PetaBricks source code is compiled PetaBricks Compiler 2 An autotuning binary is created Autotuning Final Binary Binary Choice Configuration File Jason Ansel (MIT) PetaBricks June 16, 2009 25 / 47

  55. PetaBricks Language Compilation Example Code generation PetaBricks Source Code 1 PetaBricks source code is compiled PetaBricks Compiler 2 An autotuning binary is created 3 Autotuning occurs creating a Autotuning Final Binary Binary choice configuration file Choice Configuration File Jason Ansel (MIT) PetaBricks June 16, 2009 25 / 47

  56. PetaBricks Language Compilation Example Code generation PetaBricks Source Code 1 PetaBricks source code is compiled PetaBricks Compiler 2 An autotuning binary is created 3 Autotuning occurs creating a Autotuning Final Binary Binary choice configuration file 4 Choices are fed back into the compiler to create a final binary Choice Configuration File Jason Ansel (MIT) PetaBricks June 16, 2009 25 / 47

  57. PetaBricks Language Compilation Example Autotuning Based on two building blocks: A genetic tuner An n -ary search algorithm Jason Ansel (MIT) PetaBricks June 16, 2009 26 / 47

  58. PetaBricks Language Compilation Example Autotuning Based on two building blocks: A genetic tuner An n -ary search algorithm Flat parameter space Compiler generates a dependency graph describing this parameter space Jason Ansel (MIT) PetaBricks June 16, 2009 26 / 47

  59. PetaBricks Language Compilation Example Autotuning Based on two building blocks: A genetic tuner An n -ary search algorithm Flat parameter space Compiler generates a dependency graph describing this parameter space Entire program tuned from bottom up Jason Ansel (MIT) PetaBricks June 16, 2009 26 / 47

  60. PetaBricks Language Compilation Example Parallel Runtime Library Task-based parallel runtime Thread-local decks of runnable tasks Jason Ansel (MIT) PetaBricks June 16, 2009 27 / 47

  61. PetaBricks Language Compilation Example Parallel Runtime Library Task-based parallel runtime Thread-local decks of runnable tasks Use a work-stealing algorithm similar to that of Cilk Jason Ansel (MIT) PetaBricks June 16, 2009 27 / 47

  62. PetaBricks Language Other Language Features Outline Introduction 1 Motivating Example Language & Compiler Overview Why choices PetaBricks Language 2 Key Ideas Compilation Example Other Language Features Results 3 Benchmarks Scalability Variable Accuracy Conclusion 4 Final thoughts Jason Ansel (MIT) PetaBricks June 16, 2009 28 / 47

  63. PetaBricks Language Other Language Features More PetaBricks features Automatic consistency checking The tunable keyword Call external code Custom training data generators Matrix versions for iterative algorithms Rule priorities where (clause for limiting applicable regions) Template transforms Jason Ansel (MIT) PetaBricks June 16, 2009 29 / 47

  64. PetaBricks Language Other Language Features More PetaBricks features Automatic consistency checking The tunable keyword Call external code Custom training data generators Matrix versions for iterative algorithms Rule priorities where (clause for limiting applicable regions) Template transforms Jason Ansel (MIT) PetaBricks June 16, 2009 29 / 47

  65. PetaBricks Language Other Language Features More PetaBricks features Automatic consistency checking The tunable keyword Call external code Custom training data generators Matrix versions for iterative algorithms Rule priorities where (clause for limiting applicable regions) Template transforms Jason Ansel (MIT) PetaBricks June 16, 2009 29 / 47

  66. PetaBricks Language Other Language Features More PetaBricks features Automatic consistency checking The tunable keyword Call external code Custom training data generators Matrix versions for iterative algorithms Rule priorities where (clause for limiting applicable regions) Template transforms Jason Ansel (MIT) PetaBricks June 16, 2009 29 / 47

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

PetaBricks: A Language and Compiler for Algorithmic Choice Jason Ansel, Cy Chan, Yee Lok Wong,

PetaBricks: A Language and Compiler for Algorithmic Choice Jason Ansel, Cy Chan, Yee Lok Wong,

PetaBricks: A Language and Compiler for Algorithmic Choice Jason Ansel, Cy Chan, Yee Lok Wong, Marek Olszewski, Qin Zhao, Alan Edelman, Saman Amarasinghe Presentation: Thomas Etter Motivating example Sorting numbers Algorithms K-way

791 views • 47 slides
PetaBricks and Julia Kathleen C. Alexander Massachusetts Institute of Technology December 11th,

PetaBricks and Julia Kathleen C. Alexander Massachusetts Institute of Technology December 11th,

PetaBricks and Julia Kathleen C. Alexander Massachusetts Institute of Technology December 11th, 2013 Motivation Motivation Background Approach Results Recommendations Index The Programmers Dilemma a personal example energy

525 views • 38 slides
Autotuning Programs with Algorithmic Choice Jason Ansel MIT - CSAIL December 18, 2013

Autotuning Programs with Algorithmic Choice Jason Ansel MIT - CSAIL December 18, 2013

Introduction PetaBricks OpenTuner Conclusions Autotuning Programs with Algorithmic Choice Jason Ansel MIT - CSAIL December 18, 2013 Parallelism choices s e c i o h c c i m h t i r o g A l Accuracy choices Introduction

1.25k views • 89 slides
An Efficient Evolutionary Algorithm for Solving Incrementally Structured Problems Jason Ansel

An Efficient Evolutionary Algorithm for Solving Incrementally Structured Problems Jason Ansel

An Efficient Evolutionary Algorithm for Solving Incrementally Structured Problems Jason Ansel Maciej Pacula Saman Amarasinghe Una-May OReilly MIT - CSAIL July 14, 2011 Jason Ansel (MIT) PetaBricks July 14, 2011 1 / 30 Who are we? I

770 views • 61 slides
Geometry of First-order Methods Trajectory and Adaptive Acceleration Clarice Poon University of

Geometry of First-order Methods Trajectory and Adaptive Acceleration Clarice Poon University of

Geometry of First-order Methods Trajectory and Adaptive Acceleration Clarice Poon University of Bath Joint work with: Jingwei Liang, University of Cambridge Outline Introduction Trajectory of first-order methods Adaptive acceleration via

1.29k views • 115 slides
M.S. Petrovi , A.I. Strini , N.B . Aleksi and M.R. Beli 1 OR THE TALE OF THE PAPER THAT

M.S. Petrovi , A.I. Strini , N.B . Aleksi and M.R. Beli 1 OR THE TALE OF THE PAPER THAT

Solitons in strongly nonlocal nematic liquid crystals M.S. Petrovi , A.I. Strini , N.B . Aleksi and M.R. Beli 1 OR THE TALE OF THE PAPER THAT REFUSES TO GET PUBLISHED Institute of Physics, P.O.Box 68, 11001 Belgrade, Serbia, 1 Texas

533 views • 28 slides
Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th

Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th

Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 Lecture 20 Last time: The inverse of a matrix Iterative methods for solving sytems of linear equations Gauss-Siedel

582 views • 7 slides
CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative Methods CS475/CM375 (c) 2011 P.

CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative Methods CS475/CM375 (c) 2011 P.

29/11/2011 CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative Methods CS475/CM375 (c) 2011 P. Poupart &amp; J. Wan 1 Richardson Convergence The iteration matrix is given by:

471 views • 8 slides
Analysis of the Influences on Server Power Consumption and Energy Efficiency for CPU-Intensive

Analysis of the Influences on Server Power Consumption and Energy Efficiency for CPU-Intensive

Analysis of the Influences on Server Power Consumption and Energy Efficiency for CPU-Intensive Workloads Jakim v. Kistowski, Hansfried Block, John Beckett, Klaus-Dieter Lange, Jeremy A. Arnold, Samuel Kounev University of Wrzburg, Fujitsu,

404 views • 26 slides
Java on Scalable Memory Architectures University of Crete , 25th of October 2016 Foivos S. Zakkak

Java on Scalable Memory Architectures University of Crete , 25th of October 2016 Foivos S. Zakkak

Java on Scalable Memory Architectures University of Crete , 25th of October 2016 Foivos S. Zakkak Except where otherwise noted, this presentation is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. Third

1.07k views • 95 slides
Assessing the algorithmic scaling behavior of PDE solvers Matthias Bolten University of

Assessing the algorithmic scaling behavior of PDE solvers Matthias Bolten University of

Assessing the algorithmic scaling behavior of PDE solvers Matthias Bolten University of Wuppertal 2015/09/17 Motivation Stationary problems Time-dependent problems Wrap-up Outline Motivation Stationary problems Model problem

901 views • 64 slides
From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014

From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014

From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014 Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Wigners universality idea (1956). Perhaps I am too

256 views • 21 slides
Search Problems Declarative knowledge Agent Perception creates alternatives: Robotics

Search Problems Declarative knowledge Agent Perception creates alternatives: Robotics

Search Problems Declarative knowledge Agent Perception creates alternatives: Robotics (Where reasoning consists of Which pieces of Reasoning knowledge to use? exploring alternatives) Search How to use them? Learning Knowledge

465 views • 12 slides
8-Puzzle State: x00 x01 x02 x10 x11 x12 x20 x21 x22 xij in {0, , 8}, i, j in {0, 1, 2} xij

8-Puzzle State: x00 x01 x02 x10 x11 x12 x20 x21 x22 xij in {0, , 8}, i, j in {0, 1, 2} xij

8-Puzzle State: x00 x01 x02 x10 x11 x12 x20 x21 x22 xij in {0, , 8}, i, j in {0, 1, 2} xij is the number in row I and column j. xij = 0 iff the square is empty. Start state: 724506831 Goal state: 012345678 Successor function: move the

2.79k views • 4 slides
Title: Solving Problems by Searching AIMA: Chapter 3 (Sections 3.1, 3.2 and 3.3)

Title: Solving Problems by Searching AIMA: Chapter 3 (Sections 3.1, 3.2 and 3.3)

B.Y. Choueiry Title: Solving Problems by Searching AIMA: Chapter 3 (Sections 3.1, 3.2 and 3.3) Introduction to Artificial Intelligence 1 CSCE 476-876, Fall 2020 URL: www.cse.unl.edu/~choueiry/F20-476-876 Berthe Y. Choueiry

462 views • 27 slides
Solving Problems by Searching Berlin Chen 2004 Reference: 1. S. Russell and P. Norvig. Artificial

Solving Problems by Searching Berlin Chen 2004 Reference: 1. S. Russell and P. Norvig. Artificial

Solving Problems by Searching Berlin Chen 2004 Reference: 1. S. Russell and P. Norvig. Artificial Intelligence: A Modern Approach . Chapter 3 1 Introduction Problem-Solving Agents vs. Reflex Agents Problem-solving agents : a kind of

690 views • 50 slides
C ONCURRENT E DITING : I NITIAL S TATE (R EVIEW ) 4 0 4 0 PC PC 1 2 1 2 3 4 1 2 3 4

C ONCURRENT E DITING : I NITIAL S TATE (R EVIEW ) 4 0 4 0 PC PC 1 2 1 2 3 4 1 2 3 4

O PERATION T RANSFORMATION Prasun Dewan Department of Computer Science University of North Carolina at Chapel Hill dewan@cs.unc.edu I MMEDIATELY D ELIVERING C ONCURRENT M ESSAGES A tree of message paths exist Create vector time stamp and

1.88k views • 170 slides
Solving Problems by Searching Chapter 3 Ch. 03 p.1/49 Outline Problem-solving agents

Solving Problems by Searching Chapter 3 Ch. 03 p.1/49 Outline Problem-solving agents

Solving Problems by Searching Chapter 3 Ch. 03 p.1/49 Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms Ch. 03 p.2/49 Problem-solving agents

1.4k views • 49 slides
Advanced Topics in Theoretical Computer Science Part 3: Recursive Functions (1) 21.11.2018

Advanced Topics in Theoretical Computer Science Part 3: Recursive Functions (1) 21.11.2018

Advanced Topics in Theoretical Computer Science Part 3: Recursive Functions (1) 21.11.2018 Viorica Sofronie-Stokkermans Universit at Koblenz-Landau e-mail: sofronie@uni-koblenz.de 1 Contents Recapitulation: Turing machines and Turing

881 views • 40 slides
Question 1

Question 1

Question 1 !&quot;# State, state space, search

545 views • 6 slides
Recent Results in Cohesive Powers R. Dimitrov Department of Mathematics and Philosophy, Western

Recent Results in Cohesive Powers R. Dimitrov Department of Mathematics and Philosophy, Western

Recent Results in Cohesive Powers R. Dimitrov Department of Mathematics and Philosophy, Western Illinois University WDCM-2020 July 24, 2020 RD WDCM-2020 07/27/2020 1 / 32 Content Definitions Motivation: [Dimitrov, 2004, 2008, 2009]

496 views • 32 slides
String Matching: Boyer-Moore Algorithm Greg Plaxton Theory in Programming Practice, Fall 2005

String Matching: Boyer-Moore Algorithm Greg Plaxton Theory in Programming Practice, Fall 2005

String Matching: Boyer-Moore Algorithm Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin Notation We abbreviate min { p r | r R } as min( p R ) In general, if

595 views • 16 slides
Productivity, Reuse, and Competition between Generalizations Timothy J. ODonnell MIT Two

Productivity, Reuse, and Competition between Generalizations Timothy J. ODonnell MIT Two

Productivity, Reuse, and Competition between Generalizations Timothy J. ODonnell MIT Two Problems 1. Problem of Competition 2. Problem of Productivity The Problem of Competition When multiple ways of expressing a meaning exist, how do

1.55k views • 143 slides
Non-autonomous valency- changing devices in Chitimacha Daniel W. Hieber University of

Non-autonomous valency- changing devices in Chitimacha Daniel W. Hieber University of

Non-autonomous valency- changing devices in Chitimacha Daniel W. Hieber University of California, Santa Barbara danielhieber.com Workshop on American Indigenous Languages (WAIL), May 6-7, 2016 University of California, Santa Barbara Chief

407 views • 30 slides

More recommend