Transcript Arbeitsgruppe „Finanzmathematik und stochastische

CM4SOC
Computational Mathematical Modelling for
advanced System-On-Chip Design with
special Emphasis on Channel Decoding
Algorithms and Statistical Design
CM4SOC
Anwendung von fortgeschrittenen mathematischen Modellierungsund Optimierungstechniken auf den Entwurf von mikroelektronischen
Systemen (System-on-Chip)
 Techniken der ganzzahligen, kombinatorischen Optimierung (AG Hamacher)
 Risikomaße, Abhängigkeitsmodellierung und stochastische Modelle aus der
Finanzmathematik (AG Korn)
Effiziente Dekodieralgorithmen für lineare Blockcodes in der
drahtlosen Kommunikation
Modellierung und statistische Berechnung des Delays und des
Energieverbrauchs in Nanometer CMOS Technologien
(Hardwarebeschleuniger für finanzmathematische
Anwendungen)
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Decoding of Blockcodes
Team
Mayur Punekar
(PhD)
Frank Kienle
(Akad. Rat)
Norbert Wehn
Jie Liang
(Studienarbeit)
Michael Helmling
(HiWI)
Akin Tanatmis
(PhD)
Stefan Ruzika
(Jun-Prof.)
Horst W. Hamacher
Sebastian Heupel
(Diplomand)
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Channel Coding
Noisy
Channel
Channel
coding
Noisy
Channel
Channel
Decoding
4
ML Decoding


y
Let x be the transmitted datablock and be the received noisy data block
Optimal decoder (Maximum Likelihood Decoder)


y as the input x that has the maximum a posteriori
 Decodes the output

probability P(x y)
ˆ

x  arg max P(x y)
x C

 If p(x) is uniform - this is the case for the majority of communication
systems



ˆ
x  arg max P(y x)
xC
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Goals
ML decoding as integer linear programming problem (NP complete)
 Exact algorithms & heuristics
Importance for information theory
 New bounds, code quality e.g. minimum distance
 Decoding algorithms
Mathematical approach
 Investigation of polyhedral structures, binary matroids
 Algorithms e.g. cutting planes
Algorithmic tool box
 Code analysis, code design, decoding algorithm evaluation
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Solution
ILP model
State-of-the-art model
LP relaxation
Techniques
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Results
Irregular Low-Density Parity-Check Code (64,32)
8
Activities
MISP SS 07 “Optimization and Digital Communications”
Discussion on possible interdisciplinary research topics
 ILP/ LP based algorithms for decoding
Seminar/Proseminar topics on LP/IP decoding
 SS 08, WS 08/09, SS 09
Diploma Theses (MAT, EIT)
 S. Heupel: “Cycle Polytopes and their Application in Coding Theory”
 B. Thome: “Linear Programming Based Approaches in Coding Theory”
 J. Liang: “Deoding of Linear Blocks by Ant Algorithms”
Regular meetings
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Talks, Cooperations
Plenary Presentation
A Separation Algorithm for Improved LP-Decoding of Linear Block
Codes, 5th Int. Symp. on Turbo Codes and Related Topics,
Lausanne, 2008.
Talk at TU Kaiserslautern
Rüdiger Stephan and Akin Tanatmis: Polyhedral Components
for LP-Decoding / TU Berlin - AG Grötschel
Cooperations
 Yair Beery: School of Electrical Engineering, Tel Aviv University
 Pascal Vontobel: Information Theory Research Group,
Information and Quantum Systems Laborator Hewlett-Packard
Laboratories Palo Alto
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Interdisciplinary Publications
New Algorithm for improved LP decoding
 A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„A separation algorithm for improved LP-decoding of linear block codes“, Proc. 5th
International Symposium on Turbo Codes and Related Topics, Lausanne
Switzerland, Sept. 1-5, 2008.
 A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„A separation algorithm for improved LP-decoding of linear block codes“,
submitted to IEEE Transactions on Information Theory.
New cut generation algorithm and computation of minimum
distance property of codes
 A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„New Valid Inequalities for the LP Decoding of Binary Linear Block Codes“,
submitted to IEEE International Symposium on Information Theory 2009.
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Progress
2007
2008
2009
MISP seminar
New cut generation algorithm and calculation
of Minimum Distance property of codes
New Integer Programming/
Linear Programming
formulation of the ML
decoding problem
Publication:
A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„New Valid Inequalities for the LP-Decoding of Binary Linear Block Codes“,
submitted to IEEE International Symposium on Information Theory 2009.
New Algorithm for improved LP decoding
Publication:
A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„A separation algorithm for improved LP-Decoding of linear block codes“
submitted to IEEE Transactions on Information Theory
Publication:
A. Tanatmis, S. Ruzika, H.W. Hamacher, M. Punekar, F. Kienle, and N. Wehn
„A separation algorithm for improved LP-Decoding of linear block codes“
Proc. 5th International Symposium on Turbo Codes and Related Topics,
Lausanne Switzerland, Sept. 1-5, 2008
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Roadmap
2009
2010
Paper on LP
decoding of Turbo
codes
2011
Dissertation
Akin Tanatmis
Dissertation
Mayur Punekar
Overview paper
for LP decoding
Toolkit for AG Wehn
Minimum Distance and
ILP decoding framework
Research Goals
• Polynomial time decoding algorithms based on LP
• Library of „optimum decoding“ (Reference) curves
for codes used in current standards e.g. UMTS.
• Low complexity LP decoding algorithms
• Simulation framework
DFG Initiative
- Einzelantrag ?
- SFB ?
- Excellence Initiative
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Statistical SoC Design
Advanced Statistical Methods for Probabilistic Chip Design
 Finance mathematics
Ralf Korn
Nicole Tschauder
Norbert Wehn
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Motivation
10 nm
30 nm
20 nm
Extreme device variations (Leff,Tox)
Relative
100
50
Wider
0
100
120
140 160
180
200
Vt(mV)
 Worst Case / Corner Case Design  Statistical Design
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Mathematical Approach
Leakage current of a SoC: sum of log-normal random variables
Ileak 
Ng
 Ileak ,gatei
i 1

Ng
e
ai biLi  ci Ti  diRi
i 1

Ng
e
Xi
i 1
Li, Lj, Ti, Tj are dependent on each other
Total distribution = Marginal distribution + Dependency
Moment based approximation
 Wilkinson Method, inverse Gamma Method
Bounds
 Frechet-Hoeffding Bounds
Focus on critical regions e.g. high leakage currents
 Tail dependencies
 Gumbel-Copulas
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Mathematical Approach
Risk measures
Quantify the consequences of a distribution, i.e, the risk of a random
variable X




Variance
Value-at-risk, Tail-Value-at-risk
Stop-Loss-Rate
Expected Shortfall
Concept of Comonotonicity
 Allows calculation of bounds for risk measures
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Current Status and Next Steps
Investigated new mathematical approaches
Open issue: performance evaluation with concrete technology data
Set up cooperation with TU München (Prof. Dr. U. Schlichtmann)
 Presentation at TU München 4.11.2009
 Scientific exchange and cooperation agreement
 Decision on same technology platform
Request for Infineon C12 technology data in progress
Performance evaluation with IFX C12 technology
Cooperation TU Munich
DFG Initiative: Einzelantrag / SFB ?
R. Korn: Seminar “Monte-Carlo für Elektroingenieure”
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Hardwareaccelerator
AWGN Channel Simulation
Implementation
Architecture
Throughput
Standard C code with
custom random number generator
Optimized random generator using Intel
Intel Core 2 Duo PC
SSE2 SIMD instruction set, GNU scientific
2.0 GHz,
6 Mbps
library
Cell processor optimized using
3 GB RAM
Cell 3.2 GHz
72 Mbps
IBM Monte Carlo Llibrary
256 MB RAM
FPGA Virtex 5
0.5 Mbps
Dedicted HW solution
150 Mbps
 FPGA based coprocessor for hardware supported Monte-Carlo
based price finding
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AG Wehn
Current projects
 INFINEON Project “Channel coding in Software Defined Radio”
 DFG Excellence Cluster UMIC RWTH Aachen “MIMO & Channel Coding”
 BMBF Project “Autonome integrierte Systeme”
DFG SPP Proposal submitted
 Entwurf und Architekturen verlässlicher eingebetteter Systeme: Ein Grand
Challenge im Nano-Zeitalter (TU Kaiserslautern, TU Karlsruhe, TU München, Univ. Tübingen)
Zugewiesene Mittel
 Bisher: 30.000 €
 Zukünftiger Mittelbedarf aus (CM)2: ein WiMi + Softwarelizenzen
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AG Hamacher
Current projects
 DFG-SPP 1126 “Algorithmik großer und komplexer Netze”
 BMBF-Projekt “REPKA” (mit Siemens, Fraunhofer IIS)
DFG Proposal
 Combinatorial Properties of Multiple Criteria Integer Programming
Problems
 Joint Proposal “Discrete Optimization Methods in Digital Communications”
with AG Wehn in discussion
Zugewiesene Mittel:
 Bisher: 30.000 €
 Zukünftiger Mittelbedarf aus (CM)2: ein WiMi + Softwarelizensen
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AG Korn
Current projects
 DFG-Projekt “Anwendung und Entwicklung neuer Monte Carlo Methoden
bei freien Randwertproblemen und Quasi-Variationsungleichungen in der
Finanzmathematik”
Zugewiesene Mittel
 Bisher: 0 € - Finanzierung von N. Tschauder aus DFG Graduiertenkolleg
Mathematik und Praxis
 Zukünftiger Mittelbedarf aus (CM)2: ein WiMi + Softwarelizenzen
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