Parallel iterative solution of the hermite collocation equations on gpus

•

0 likes•134 views

This document describes research on solving elliptic boundary value problems using a Hermite collocation method. A parallel algorithm was developed using the Schur complement method and BiCGSTAB iterative solver on shared memory architectures with GPUs. Testing on an HP server with Tesla GPUs showed up to 30% acceleration compared to a serial CPU implementation. Future work involves designing an efficient parallel Schur complement algorithm for distributed memory systems with GPUs.

Parallel Iterative solution of the
Hermite Collocation Equations
on GPUs
Emmanuel N. Mathioudakis
Co-Authors : Elena Papadopoulou – Yiannis Saridakis – Nikolaos Vilanakis
TECHNICAL UNIVERSITY OF CRETE
DEPARTMENT OF SCIENCES
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
73100 CHANIA - CRETE - GREECE

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Talk Overview
 Hermite Collocation for elliptic BVPs &
Derivation of the Collocation linear system
 Development of a parallel algorithm for the
Schur Complement method on Shared
Memory Parallel Architectures
 Parallel implementation on multicore computers
with GPUs

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
( , ) ( , ) , ( , )
( , ) ( , ) , ( , )
u x y f x y x y
u x y g x y x y
 
 
L
BBVP
( , ) ( , ) 0 , ( , )
:
( , ) ( , ) 0 , (
,
, )
y y yx x x
i j i j i j
y y yx x x
i j i j i j
a
i j
u f
u g
     
     
  
  
L
B
Hermite Collocation Method

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Hermite Collocation Method…

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Red – Black Collocation Linear system

with 0
( , ) ( , ) ( , ) , ( , )
( , ) ( , ) , ( , )
u x y u x y f x y x y
u x y g x y x y 


  
 
2
Model
Problem
TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY

Helmholtz
Collocation
Matrix
TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY

 The collocation matrix is large, sparse and enjoys no pleasant
properties (e.g. symmetric, definite)
Iterative
+
Parallel
TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Iterative Solution
with

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Iterative Solution

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Eigenvalues of Collocation matrix

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Schur Complement Iterative Solution

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Parallel Iterative Solution of Collocation Linear system
on Shared Memory Architectures
 Uniform Load Balancing between core threads
 Minimal Idle Cycles of core threads
 Minimal Communication

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
case of ns = 2p
( )
1
Rx ( )
2
Rx ( )
3
Rx ( )
4
Rx

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
case of ns = 2p
( )
1
B
p
x 
( )
2
B
p
x 
( )
3
B
p
x 
( )
4
B
p
x 

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
case of ns = 2p
V1
( )
1
Rx ( )
2
Rx ( )
3
Rx ( )
4
Rx( )
1
B
p
x 
( )
2
B
p
x 
( )
3
B
p
x 
( )
4
B
p
x 
V2 V3 V4 V5 V6 V7 V8

Mapping into a fixed size Architecture of N Cores
case of k = 2p/N even 2k virtual threads
l=(j-1)k+1,…,jk
l=2p+(j-1)k+1,…,2p+jk
j=1,…,N
( )B
l
x
( )R
l
x
V2 . . .
Pj

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Parallel Schur Complement Iterative Solution

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Parallel BiCGSTAB

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
The Dirichlet Helmholtz Problem
2100( 0.1)2with
( , ) 10 ( ) ( ) , ( , ) [0,1] [0,1]
( ) ( ) x
u x y x y x y
x x x e
 
  
  
 

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
HP SL390s
6 core Xeon@2.8GHz
24GB memory
Oracle Linux 6.3 x64
PGI 13.5 Fortran
PCI-e gen2 x16
HP SL390s – Tesla M2070 GPUs
+ 2 x

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Realization on HP SL390s Tesla GPU machine
Iterations / Error measurements
ns
BiCGSTAB
Iterations
|| b – Axn||2
256 294 6.06e-11
512 589 2.85e-11
1024 1161 1.39e-11
2048 3726 9.59e-12

TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY
Realization on HP SL390s Tesla GPU machine
Time measurements

Conclusions
• A new parallel algorithm implementing the Schur
complement with BiCGSTAB iterative method for
Hermite Collocation equations has been developed.
• The algorithm is realized on Shared Memory multi-core
machines with GPUs .
• A performance acceleration of up to 30% is observed.
TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY

Future work
• Design an efficient parallel Schur complement
algorithm of the Hermite Collocation equations
for Multiprocessor /Grid machines with GPUs.
TECHNICAL UNIVERSITY OF CRETE
APPLIED MATHEMATICS AND COMPUTERS LABORATORY

The document discusses using an inoculation approach to solve the problem of rescheduling railway trains following a small perturbation like a delay. The inoculation approach involves first running the algorithm with an "empty" incident to solve initial constraints and find a reasonable starting permutation, then using that solution to initialize the population for solving the actual problem. Experimental results show the inoculation approach finds good solutions faster than alternatives for both easy and hard instances of the scheduling problem.

Introduction to cosmology and numerical cosmology (with the Cactus code) (2/2)

SEENET-MTP

This document discusses using the Cactus code to model cosmological simulations numerically. It introduces the Cosmo and RealSF thorns developed to solve Einstein's equations for cosmological models within the Cactus framework. The Cosmo thorn provides initial data and boundary conditions for the Friedmann-Robertson-Walker metric. The RealSF thorn evolves a scalar field by solving the Klein-Gordon equation. Examples are presented of simulations using these thorns to model pure FRW cosmologies and those with a cosmological constant or scalar field.

Artificial Neural Networks for Storm Surge Prediction in North Carolina

Anton Bezuglov

Applications of vector space ksks

MuhammadMujtabaBashi

Linear algebra is important for physics, engineering, and computer graphics. It has applications in quantum mechanics, control theory, cryptography, solving partial differential equations, and analyzing sounds and music. Many software programs use linear operators and matrices to perform tasks like generating drawings, exchanging secret information, and modeling how materials interact with different frequencies of sound. Linear algebra provides powerful mathematical tools with diverse applications across multiple fields.

Java review-2

University of Massachusetts Amherst

This document contains notes from a Java review lecture. It discusses the Java API and various classes and interfaces in the java.util package for common data structures like lists, maps, and sets. It explains that these are interfaces, not classes, and describes some common implementations like LinkedList, ArrayList, HashMap, and HashSet. It also covers generics and how to define parameterized types to avoid casting. Finally, it discusses the for-each loop as a cleaner way to iterate over elements in collections compared to a traditional for loop.

cis98006

perfj

This document summarizes three parallel algorithms for solving geometric problems on coarse-grained multicomputers: 1. The implicit row maxima problem on a totally monotone n x n matrix can be solved in O((n/p)logp) time if n >= p^2. 2. The all-farthest-neighbors problem for a convex n-gon can be solved in O(n/pp) time if n >= 4p^2. 3. The maximum-perimeter triangle inscribed in a convex n-gon can be found in O(nlogn/pp) time if n >= p^2.

用 Python 玩 LHC 公開數據

Yuan CHAO

Big Data and Small Devices by Katharina Morik

BigMine

How can we learn from the data of small ubiquitous systems? Do we need to send the data to a server or cloud and do all learning there? Or can we learn on some small devices directly? Are smartphones small? Are navigation systems small? How complex is learning allowed to be in times of big data? What about graphical models? Can they be applied on small devices or even learned on restricted processors? Big data are produced by various sources. Most often, they are distributedly stored at computing farms or clouds. Analytics on the Hadoop Distributed File System (HDFS) then follows the MapReduce programming model. According to the Lambda architecture of Nathan Marz and James Warren, this is the batch layer. It is complemented by the speed layer, which aggregates and integrates incoming data streams in real time. When considering big data and small devices, obviously, we imagine the small devices being hosts of the speed layer, only. Analytics on the small devices is restricted by memory and computation resources. The interplay of streaming and batch analytics offers a multitude of configurations. In this talk, we discuss opportunities for using sophisticated models for learning spatio-temporal models. In particular, we investigate graphical models, which generate the probabilities for connected (sensor) nodes. First, we present spatio-temporal random fields that take as input data from small devices, are computed at a server, and send results to -possibly different — small devices. Second, we go even further: the Integer Markov Random Field approximates the likelihood estimates such that it can be computed on small devices. We illustrate our learning models by applications from traffic management.

The document introduces a novel reconfigurable hardware architecture for computing polynomial matrix multiplication (PMM) of polynomial matrices and/or vectors. The architecture exploits an extension of the fast convolution technique to multiple inputs and outputs. It is implemented using a highly pipelined and partly systolic FPGA architecture designed with the Xilinx system generator tool. Hardware simulations show the architecture accurately computes PMM with low execution times and limited FPGA resources.

A Gomez T Tat Cesga

Miguel Morales

TimTrack is a software for tracking charged particles. It uses C language for its speed and flexibility. Previous versions used LAPACK or Intel IPP libraries for linear algebra operations. The newest version TimTrack v2.0 uses LAPACK and is 23.6 seconds faster than earlier versions for tracking 1 million particles. Future plans include parallelizing with OpenMP, MPI, and implementing on GPUs using CUDA.

Aritra Sarkar - Search and Optimisation Algorithms for Genomics on Quantum Ac...

Tom Hubregtsen

This document discusses using quantum computing to accelerate genomics algorithms. It proposes using quantum search and optimization algorithms to solve problems like genome assembly and alignment that are challenging for classical computers due to the large amount of genomic data. The document outlines algorithms like Grover's search and the Quantum Approximate Optimization Algorithm that could be applied and benchmarks their performance on test problems and real genomic data. It also discusses the software tools and computing platforms needed to develop and test these quantum genomics applications.

Relaxation methods for the matrix exponential on large networks

David Gleich

A Highly Parallel Semi-Dataflow FPGA Architecture for Large-Scale N-Body Simu...

NECST Lab @ Politecnico di Milano

This document proposes a highly parallel semi-dataflow FPGA architecture for accelerating large-scale N-body simulations. The key aspects of the proposed design are: 1) A hardware/software partitioning that accelerates the computationally intensive force calculation step on the FPGA; 2) An optimized data transfer approach to reduce memory traffic; 3) A semi-dataflow architecture providing high parallelism through 48 computation pipelines; and 4) A tiling approach to further improve performance and resource utilization. Experimental results show the design achieves up to 4400 million particle-pairs per second, outperforming CPU and GPU implementations in terms of performance and performance-per-watt.

User guided discovery of declarative process models

fmaggi

This document discusses user-guided declarative process discovery. Declarative process discovery describes process behavior as rules rather than explicitly specifying all possible behaviors. The core algorithm discovers constraints from event logs. Users can guide discovery by tuning parameters like percentage of events and instances. The paper also discusses techniques for handling truncated logs and detecting vacuously satisfied constraints. The goal is to discover accurate declarative models while allowing user steering toward desired properties.

FMRIPREP - robust and easy to use fMRI preprocessing pipeline

Krzysztof Gorgolewski

FMRIPREP is an fMRI preprocessing tool that provides robust, easy-to-use preprocessing of MRI data including denoising and normalization. It performs T1-weighted preprocessing like N4 bias field correction and tissue segmentation. It also performs EPI preprocessing including motion correction, skull stripping, coregistration to T1, and estimation of noise confounds. FMRIPREP outputs preprocessed data in native and MNI space. It is designed to be transparent and works on any BIDS-formatted dataset through either Docker or Singularity containers.

6. Implementation

bibbidy N-BObbiDY boo at NECST

This document describes a hardware acceleration approach for N-body simulations using an FPGA. It presents a semi-dataflow architecture and tiling technique to efficiently compute pairwise forces with reduced data transfers. Evaluation shows the FPGA implementation achieves 4400 million particle-pairs per second, outperforming a CPU and achieving high performance per watt compared to other platforms. Future work involves connecting to the host via PCIe and further optimizing performance per watt.

Parallel Optimization in Machine Learning

Fabian Pedregosa

Prototyping is an attitude

With Company

The document discusses prototyping and provides examples of different types of prototypes including paper prototypes, digital prototypes, storyboards, role plays, and space prototypes. It explains that prototyping is used to make ideas tangible and test reactions from users in order to gain insights. Prototypes should be iterated on and fail early to push ideas further and save time and money. Both low and high fidelity prototypes are mentioned as ways to test ideas at different stages of the design process.

10 Insightful Quotes On Designing A Better Customer Experience

Yuan Wang

In an ever-changing landscape of one digital disruption after another, companies and organisations are looking for new ways to understand their target markets and engage them better. Increasingly they invest in user experience (UX) and customer experience design (CX) capabilities by working with a specialist UX agency or developing their own UX lab. Some UX practitioners are touting leaner and faster ways of developing customer-centric products and services, via methodologies such as guerilla research, rapid prototyping and Agile UX. Others seek innovation and fulfilment by spending more time in research, being more inclusive, and designing for social goods. Experience is more than just an interface. It is a relationship, as well as a series of touch points between your brand and your customer. Here are our top 10 highlights and takeaways from the recent UX Australia conference to help you transform your customer experience design. For full article, continue reading at https://yump.com.au/10-ways-supercharge-customer-experience-design/

Learn BEM: CSS Naming Convention

In a Rocket

How to Build a Dynamic Social Media Plan

Post Planner

SEO: Getting Personal

Kirsty Hulse

The document discusses how personalization and dynamic content are becoming increasingly important on websites. It notes that 52% of marketers see content personalization as critical and 75% of consumers like it when brands personalize their content. However, personalization can create issues for search engine optimization as dynamic URLs and content are more difficult for search engines to index than static pages. The document provides tips for SEOs to help address these personalization and SEO challenges, such as using static URLs when possible and submitting accurate sitemaps.

Lightning Talk #9: How UX and Data Storytelling Can Shape Policy by Mika Aldaba

ux singapore

How can we take UX and Data Storytelling out of the tech context and use them to change the way government behaves? Showcasing the truth is the highest goal of data storytelling. Because the design of a chart can affect the interpretation of data in a major way, one must wield visual tools with care and deliberation. Using quantitative facts to evoke an emotional response is best achieved with the combination of UX and data storytelling.

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Stanford GSB Corporate Governance Research Initiative

This document summarizes a study of CEO succession events among the largest 100 U.S. corporations between 2005-2015. The study analyzed executives who were passed over for the CEO role ("succession losers") and their subsequent careers. It found that 74% of passed over executives left their companies, with 30% eventually becoming CEOs elsewhere. However, companies led by succession losers saw average stock price declines of 13% over 3 years, compared to gains for companies whose CEO selections remained unchanged. The findings suggest that boards generally identify the most qualified CEO candidates, though differences between internal and external hires complicate comparisons.

Parallelising Dynamic Programming

Raphael Reitzig

This talk was based on my Master's thesis which I had completed earlier that year. It gives an overview on how certain parallel dynamic programming can be computed in parallel efficiently, and what we want that to mean here. The plots in "Performance Examples" show speedup S on the left and efficiency E on the right, both against input size. Read more over here: http://reitzig.github.io/publications/Reitzig2012

Experiences of numerical simulations on a PC cluster

Antti Vanne

Virus, Vaccines, Genes and Quantum - 2020-06-18

Aritra Sarkar

This document discusses using a quantum computer to simulate DNA-based vaccines by indexing and aligning short DNA reads to a reference genome. It describes superimposing the reference genome segmented into short reads and evolving via controlled operations to the Hamming distance against the short read. The maximum probability entry indicates the alignment index. Steps include 1) superposing the indexed reference segments, 2) evolving via controlled operations to the Hamming distance, and 3) finding the maximum probability entry indicating the alignment index.

The CAOS framework: democratize the acceleration of compute intensive applica...

NECST Lab @ Politecnico di Milano

The increasing demand for computing power in fields such as biology, finance, machine learning is pushing the adoption of reconfigurable hardware in order to keep up with the required performance level at a sustainable power consumption. Within this context, FPGA devices represent an interesting solution as they combine the benefits of power efficiency, performance and flexibility. Nevertheless, the steep learning curve and experience needed to develop efficient FPGA-based systems represents one of the main limiting factor for a broad utilization of such devices. In this talk, we present CAOS, a framework which helps the application designer in identifying acceleration opportunities and guides through the implementation of the final FPGA-based system. The CAOS platform targets the full stack of the application optimization process, starting from the identification of the kernel functions to accelerate, to the optimization of such kernels and to the generation of the runtime management and the configuration files needed to program the FPGA.

Evolutionary Algorithms and Self-Organised Systems

Natalio Krasnogor

An Optimized Parallel Algorithm for Longest Common Subsequence Using Openmp –...

IRJET Journal

This document summarizes research on developing parallel algorithms to optimize solving the longest common subsequence (LCS) problem. LCS is commonly used for sequence comparison in bioinformatics. Traditional sequential dynamic programming algorithms have complexity of O(mn) for sequences of lengths m and n. The document reviews parallel algorithms developed using tools like OpenMP and GPUs like CUDA to reduce computation time. It proposes the authors' own optimized parallel algorithm for multi-core CPUs using OpenMP.

What's hot

Novel reconfigurable hardware architecture for polynomial matrix multiplications

I3E Technologies

A Gomez T Tat Cesga

Miguel Morales

Aritra Sarkar - Search and Optimisation Algorithms for Genomics on Quantum Ac...

Tom Hubregtsen

Relaxation methods for the matrix exponential on large networks

David Gleich

A Highly Parallel Semi-Dataflow FPGA Architecture for Large-Scale N-Body Simu...

NECST Lab @ Politecnico di Milano

User guided discovery of declarative process models

fmaggi

FMRIPREP - robust and easy to use fMRI preprocessing pipeline

Krzysztof Gorgolewski

6. Implementation

bibbidy N-BObbiDY boo at NECST

Parallel Optimization in Machine Learning

Fabian Pedregosa

What's hot (9)

Novel reconfigurable hardware architecture for polynomial matrix multiplications

A Gomez T Tat Cesga

Aritra Sarkar - Search and Optimisation Algorithms for Genomics on Quantum Ac...

Relaxation methods for the matrix exponential on large networks

A Highly Parallel Semi-Dataflow FPGA Architecture for Large-Scale N-Body Simu...

User guided discovery of declarative process models

FMRIPREP - robust and easy to use fMRI preprocessing pipeline

6. Implementation

Parallel Optimization in Machine Learning

Viewers also liked

Prototyping is an attitude

With Company

10 Insightful Quotes On Designing A Better Customer Experience

Yuan Wang

Learn BEM: CSS Naming Convention

In a Rocket

How to Build a Dynamic Social Media Plan

Post Planner

SEO: Getting Personal

Kirsty Hulse

Lightning Talk #9: How UX and Data Storytelling Can Shape Policy by Mika Aldaba

ux singapore

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Stanford GSB Corporate Governance Research Initiative

Viewers also liked (7)

Prototyping is an attitude

10 Insightful Quotes On Designing A Better Customer Experience

Learn BEM: CSS Naming Convention

How to Build a Dynamic Social Media Plan

SEO: Getting Personal

Lightning Talk #9: How UX and Data Storytelling Can Shape Policy by Mika Aldaba

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!

SOFTTECHHUB

As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.

20240605 QFM017 Machine Intelligence Reading List May 2024

Matthew Sinclair

National Security Agency - NSA mobile device best practices

Quotidiano Piemontese

Monitoring Java Application Security with JDK Tools and JFR Events

Ana-Maria Mihalceanu

Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf

Paige Cruz

Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack. While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack. I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:

GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...

Neo4j

Dr. Sean Tan, Head of Data Science, Changi Airport Group Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.

Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI

Vladimir Iglovikov, Ph.D.

Presented by Vladimir Iglovikov: - https://www.linkedin.com/in/iglovikov/ - https://x.com/viglovikov - https://www.instagram.com/ternaus/ This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation. Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners. This case study covers various aspects, including: People: The contributors and community that have supported Albumentations. Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions. Challenges: The hurdles in monetizing open-source projects and measuring user engagement. Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration. Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community. Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations. Mental Health: Maintaining balance and not feeling pressured by user demands. Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth. Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects. Explore more about Albumentations and join the community at: GitHub: https://github.com/albumentations-team/albumentations Website: https://albumentations.ai/ LinkedIn: https://www.linkedin.com/company/100504475 Twitter: https://x.com/albumentations

Recently uploaded (20)

Essentials of Automations: The Art of Triggers and Actions in FME

Climate Impact of Software Testing at Nordic Testing Days

Removing Uninteresting Bytes in Software Fuzzing

TrustArc Webinar - 2024 Global Privacy Survey

Data structures and Algorithms in Python.pdf

Building RAG with self-deployed Milvus vector database and Snowpark Container...

Artificial Intelligence for XMLDevelopment

Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf

Microsoft - Power Platform_G.Aspiotis.pdf

How to use Firebase Data Connect For Flutter

Full-RAG: A modern architecture for hyper-personalization

Generative AI Deep Dive: Advancing from Proof of Concept to Production

RESUME BUILDER APPLICATION Project for students

Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!

20240605 QFM017 Machine Intelligence Reading List May 2024

National Security Agency - NSA mobile device best practices

Monitoring Java Application Security with JDK Tools and JFR Events

Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf

GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...

Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI

Parallel iterative solution of the hermite collocation equations on gpus

1. Parallel Iterative solution of the Hermite Collocation Equations on GPUs Emmanuel N. Mathioudakis Co-Authors : Elena Papadopoulou – Yiannis Saridakis – Nikolaos Vilanakis TECHNICAL UNIVERSITY OF CRETE DEPARTMENT OF SCIENCES APPLIED MATHEMATICS AND COMPUTERS LABORATORY 73100 CHANIA - CRETE - GREECE

2. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Talk Overview  Hermite Collocation for elliptic BVPs & Derivation of the Collocation linear system  Development of a parallel algorithm for the Schur Complement method on Shared Memory Parallel Architectures  Parallel implementation on multicore computers with GPUs

3. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY ( , ) ( , ) , ( , ) ( , ) ( , ) , ( , ) u x y f x y x y u x y g x y x y     L BBVP ( , ) ( , ) 0 , ( , ) : ( , ) ( , ) 0 , ( , , ) y y yx x x i j i j i j y y yx x x i j i j i j a i j u f u g                   L B Hermite Collocation Method

4. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Hermite Collocation Method…

5. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Hermite Collocation Method…

6. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

7. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

8. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

9. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

10. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

11. with 0 ( , ) ( , ) ( , ) , ( , ) ( , ) ( , ) , ( , ) u x y u x y f x y x y u x y g x y x y         2 Model Problem TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY

12. Helmholtz Collocation Matrix TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY

13. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Red – Black Collocation Linear system

14.  The collocation matrix is large, sparse and enjoys no pleasant properties (e.g. symmetric, definite) Iterative + Parallel TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY

15. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Iterative Solution with

16. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Iterative Solution

17. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Eigenvalues of Collocation matrix

18. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Schur Complement Iterative Solution

19. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Parallel Iterative Solution of Collocation Linear system on Shared Memory Architectures  Uniform Load Balancing between core threads  Minimal Idle Cycles of core threads  Minimal Communication

20. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY case of ns = 2p ( ) 1 Rx ( ) 2 Rx ( ) 3 Rx ( ) 4 Rx

21. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY case of ns = 2p ( ) 1 B p x  ( ) 2 B p x  ( ) 3 B p x  ( ) 4 B p x 

22. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY case of ns = 2p V1 ( ) 1 Rx ( ) 2 Rx ( ) 3 Rx ( ) 4 Rx( ) 1 B p x  ( ) 2 B p x  ( ) 3 B p x  ( ) 4 B p x  V2 V3 V4 V5 V6 V7 V8

23. V1 V2 V3 V4 V5

24. Mapping into a fixed size Architecture of N Cores case of k = 2p/N even 2k virtual threads l=(j-1)k+1,…,jk l=2p+(j-1)k+1,…,2p+jk j=1,…,N ( )B l x ( )R l x V2 . . . Pj

25. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Parallel Schur Complement Iterative Solution

26. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Parallel BiCGSTAB

27. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Parallel BiCGSTAB

28. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY The Dirichlet Helmholtz Problem 2100( 0.1)2with ( , ) 10 ( ) ( ) , ( , ) [0,1] [0,1] ( ) ( ) x u x y x y x y x x x e          

29. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY HP SL390s 6 core Xeon@2.8GHz 24GB memory Oracle Linux 6.3 x64 PGI 13.5 Fortran PCI-e gen2 x16 HP SL390s – Tesla M2070 GPUs + 2 x

30. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Iterations / Error measurements ns BiCGSTAB Iterations || b – Axn||2 256 294 6.06e-11 512 589 2.85e-11 1024 1161 1.39e-11 2048 3726 9.59e-12

31. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

32. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

33. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

34. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

35. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

36. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY Realization on HP SL390s Tesla GPU machine Time measurements

37. Conclusions • A new parallel algorithm implementing the Schur complement with BiCGSTAB iterative method for Hermite Collocation equations has been developed. • The algorithm is realized on Shared Memory multi-core machines with GPUs . • A performance acceleration of up to 30% is observed. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY

38. Future work • Design an efficient parallel Schur complement algorithm of the Hermite Collocation equations for Multiprocessor /Grid machines with GPUs. TECHNICAL UNIVERSITY OF CRETE APPLIED MATHEMATICS AND COMPUTERS LABORATORY

Parallel iterative solution of the hermite collocation equations on gpus

Recommended

Recommended

More Related Content

What's hot

What's hot (9)

Viewers also liked

Viewers also liked (7)

Similar to Parallel iterative solution of the hermite collocation equations on gpus

Similar to Parallel iterative solution of the hermite collocation equations on gpus (20)

Recently uploaded

Recently uploaded (20)

Parallel iterative solution of the hermite collocation equations on gpus