The document discusses accelerated computing through Quasi-Monte Carlo (QMC) constructions for numerical integration. It covers the theoretical background of Monte Carlo (MC), Quasi-Monte Carlo (QMC), and Randomized QMC (RQMC) methods. It also provides examples of using QMC and polynomial lattice rules for applications like option pricing in finance. The CEO presents on developing efficient software for QMC using a C++ implementation.
This document describes a software tool called Polynomial Lattice Builder that is used to construct rank-1 polynomial lattice rules for quasi-Monte Carlo integration. It discusses the motivation for the tool, theoretical background on rank-1 polynomial lattice rules and figures of merit, an overview of the tool's features, examples of its applications, and conclusions. The tool allows users to choose parameters like the number of points, polynomial, dimension and construction method to generate optimal rank-1 polynomial lattice rules for their application.
Presentation by Tommy Lofstedt, Associated Professor at Umeå University (Sweden), at the FogGuru Workshop on linking with other disciplines in October 2019.
Ba4201 quantitative techniques for decision making l t p cPrasanna E
This document outlines the course objectives and units for a quantitative techniques course. The course aims to apply quantitative techniques to solve business problems. It covers topics like linear programming and extensions, decision and game theories, inventory and replacement models, and queuing theory and simulation. The course objectives are to understand how to use these techniques for product mix decisions, logistics and job allocation, decision making, inventory management, and optimizing real-time scenarios.
Conditional random fields (CRFs) are probabilistic models for segmenting and labeling sequence data. CRFs address limitations of previous models like hidden Markov models (HMMs) and maximum entropy Markov models (MEMMs). CRFs allow incorporation of arbitrary, overlapping features of the observation sequence and label dependencies. Parameters are estimated to maximize the conditional log-likelihood using iterative scaling or tracking partial feature expectations. Experiments show CRFs outperform HMMs and MEMMs on synthetic and real-world tasks by addressing label bias problems and modeling dependencies beyond the previous label.
The document provides an overview of support vector machines (SVMs), including:
1) SVMs are a supervised learning method for classification and regression based on Vapnik-Chervonenkis theory that aims to find a separating hyperplane with maximum margin between classes.
2) The SVM formulation can be expressed as an optimization problem to maximize the margin, which results in a convex quadratic problem.
3) The dual formulation allows SVMs to be applied to non-linearly separable data using kernel methods that implicitly map inputs to higher dimensional feature spaces.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
(Prefer mailing. Call in emergency )
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating profit/net-sales, profit/manufacturing cost, etc. In this paper, we proposed the transformation characteristics to solve the multi objective linear fractional programming (MOLFP) problems. If a MOLFP problem with both the numerators and the denominators of the objectives are linear functions and some technical linear restrictions are satisfied, then it is defined as a multi objective linear fractional programming problem MOLFPP in this research. The transformation characteristics are illustrated and the solution procedure and numerical example are presented.
This document describes a software tool called Polynomial Lattice Builder that is used to construct rank-1 polynomial lattice rules for quasi-Monte Carlo integration. It discusses the motivation for the tool, theoretical background on rank-1 polynomial lattice rules and figures of merit, an overview of the tool's features, examples of its applications, and conclusions. The tool allows users to choose parameters like the number of points, polynomial, dimension and construction method to generate optimal rank-1 polynomial lattice rules for their application.
Presentation by Tommy Lofstedt, Associated Professor at Umeå University (Sweden), at the FogGuru Workshop on linking with other disciplines in October 2019.
Ba4201 quantitative techniques for decision making l t p cPrasanna E
This document outlines the course objectives and units for a quantitative techniques course. The course aims to apply quantitative techniques to solve business problems. It covers topics like linear programming and extensions, decision and game theories, inventory and replacement models, and queuing theory and simulation. The course objectives are to understand how to use these techniques for product mix decisions, logistics and job allocation, decision making, inventory management, and optimizing real-time scenarios.
Conditional random fields (CRFs) are probabilistic models for segmenting and labeling sequence data. CRFs address limitations of previous models like hidden Markov models (HMMs) and maximum entropy Markov models (MEMMs). CRFs allow incorporation of arbitrary, overlapping features of the observation sequence and label dependencies. Parameters are estimated to maximize the conditional log-likelihood using iterative scaling or tracking partial feature expectations. Experiments show CRFs outperform HMMs and MEMMs on synthetic and real-world tasks by addressing label bias problems and modeling dependencies beyond the previous label.
The document provides an overview of support vector machines (SVMs), including:
1) SVMs are a supervised learning method for classification and regression based on Vapnik-Chervonenkis theory that aims to find a separating hyperplane with maximum margin between classes.
2) The SVM formulation can be expressed as an optimization problem to maximize the margin, which results in a convex quadratic problem.
3) The dual formulation allows SVMs to be applied to non-linearly separable data using kernel methods that implicitly map inputs to higher dimensional feature spaces.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
(Prefer mailing. Call in emergency )
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating profit/net-sales, profit/manufacturing cost, etc. In this paper, we proposed the transformation characteristics to solve the multi objective linear fractional programming (MOLFP) problems. If a MOLFP problem with both the numerators and the denominators of the objectives are linear functions and some technical linear restrictions are satisfied, then it is defined as a multi objective linear fractional programming problem MOLFPP in this research. The transformation characteristics are illustrated and the solution procedure and numerical example are presented.
APPLYING TRANSFORMATION CHARACTERISTICS TO SOLVE THE MULTI OBJECTIVE LINEAR F...ijcsit
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating
profit/net-sales, profit/manufacturing cost, etc. In this paper, we proposed the transformation characteristics to solve the multi objective linear fractional programming (MOLFP) problems. If a MOLFP problem with both the numerators and the denominators of the objectives are linear functions and some
technical linear restrictions are satisfied, then it is defined as a multi objective linear fractional programming problem MOLFPP in this research. The transformation characteristics are illustrated and the solution procedure and numerical example are presented.
The document discusses two examples of applying optimization in accounting and auditing. In the first exercise, the document finds the minimum average cost of producing x units of a product. The minimum is found to be $0.59 at x=3 units. In the second exercise, the document uses a demand function to maximize revenue from bus fares. It determines that revenue is maximized at a fare of 40 cents, with maximum revenue of $200,000 and 5,000 passengers per hour. Overall, the document shows how optimization techniques can help find maximum and minimum values to help with decision making.
This document provides information about getting solved assignments for the MBA Semester 2 Operations Research subject. It includes 6 questions related to operations research concepts like linear programming, transportation problems, assignment problems, and simulation. Students can get assignments solved at Rs. 125 each by emailing or calling the provided contact information. The questions cover topics like the framework of operations research, graphical and algebraic methods for linear programming problems, important terms in transportation problems, the Hungarian method for assignment problems, Monte Carlo simulation, assumptions of game theory, characteristics of Markov chains, and job prioritization rules.
The aim of this talk is to introduce two pieces of software, ``Lattice Builder'' and ``Stochastic Simulation in Java'' (SSJ). They allow to easily conduct experiments that rely on Monte Carlo (MC), quasi-Monte Carlo (QMC), or randomized QMC (RQMC). Lattice Builder is a C++ library designed to efficiently search, produce, and examine rank-1-lattices as well as polynomial lattices. It allows the user to choose from a broad palette of search criteria, types of weights, and construction methods, which can be accessed through a graphical interface as well as through a command line tool. Its structure also facilitates the implementation of own extensions and encourages to combine Lattice Builder with other programming languages.
SSJ is a Java library covering an extensive set of tools for stochastic simulations. It is particularly useful for experiments relying on MC and (R)QMC, ranging from integration problems over RQMC for Markov Chains (Array-RQMC) to density estimation.
This talk gives an introductory tour through the interfaces of Lattice Builder, shows how to integrate it into SSJ, and provides first steps in SSJ based on an example for mean estimation with RQMC.
Sampling-Based Planning Algorithms for Multi-Objective MissionsMd Mahbubur Rahman
multiobjective path planning has Increasing demand in military missions, rescue operations, construction job-sites.
There is Lack of robotic path planning algorithm that compromises multiple
objectives. Commonly no solution that optimizes all the objective functions. Here we modify RRT, RRT* sampling based algorithm.
In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis. The basic SVM takes a set of input data and predicts, for each given input, which of two possible classes forms the output, making it a non-probabilistic binary linear classifier.
This document provides information about getting fully solved assignments from an assignment help service. It lists an email address and phone number to contact for assistance with assignments. It also provides details about the available programs, subjects, semesters, credits, and other assignment details like word count requirements. Students are advised to mail their request with details of their semester and specialization to get solved assignments. Calling is listed as an emergency option.
dynamic programming complete by Mumtaz Ali (03154103173)Mumtaz Ali
The document discusses dynamic programming, including its meaning, definition, uses, techniques, and examples. Dynamic programming refers to breaking large problems down into smaller subproblems, solving each subproblem only once, and storing the results for future use. This avoids recomputing the same subproblems repeatedly. Examples covered include matrix chain multiplication, the Fibonacci sequence, and optimal substructure. The document provides details on formulating and solving dynamic programming problems through recursive definitions and storing results in tables.
This document discusses quantum computing business in the Japanese market. It introduces MDR Inc., a Japanese quantum computing startup founded in 2008. MDR develops full-stack quantum computing, from software to hardware, and works with over 20 clients in industries like banking, automotive, materials and more. Some applications discussed include quantum simulation, optimization, and machine learning. The document also provides an overview of the quantum computing developer community and ecosystem in Japan.
SURVEY ON POLYGONAL APPROXIMATION TECHNIQUES FOR DIGITAL PLANAR CURVESZac Darcy
This document summarizes and compares three techniques for polygonal approximation of digital planar curves:
1) Masood's technique which iteratively deletes redundant points and uses a stabilization process to optimize point locations.
2) Carmona's technique which suppresses redundant points using a breakpoint suppression algorithm and threshold.
3) Tanvir's adaptive optimization algorithm which focuses on high curvature points and applies an optimization procedure.
The techniques are evaluated on standard shapes using measures like number of points, compression ratio, error, and weighted error. Masood's technique generally had lower error while Tanvir's often achieved the highest compression.
This document discusses dynamic programming techniques. It covers matrix chain multiplication and all pairs shortest paths problems. Dynamic programming involves breaking down problems into overlapping subproblems and storing the results of already solved subproblems to avoid recomputing them. It has four main steps - defining a mathematical notation for subproblems, proving optimal substructure, deriving a recurrence relation, and developing an algorithm using the relation.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed
method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem
transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will
optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency
of the suggested method, a lot of test problems have been solved using this method. Comparing the results
of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable
for solving the multi-objective problems.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency of the suggested method, a lot of test problems have been solved using this method. Comparing the results of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable for solving the multi-objective problems.
This document discusses variations of the interval linear assignment problem. It begins with an introduction to assignment problems and defines them as problems that assign resources to activities to minimize cost or maximize profit on a one-to-one basis. It then provides the mathematical model for standard assignment problems and discusses variations such as non-square matrices, maximization/minimization objectives, constrained assignments, and alternate optimal solutions. The document also gives examples of managerial applications and provides two numerical examples solving interval linear assignment problems using an interval Hungarian method.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
This is an introduction to computational economics by dynamic programming. MATLAB program is provided to understand how it is solved by using statistical computing.
We introduce in this paper a new flow-driven and constraint-based approach for error localization. The input is a faulty program for which a counter-example and a postcondition are provided. To identify helpful information for error location, we generate a constraint system for the paths of the control flow graph for which at most k conditional statements may be erroneous. Then, we calculate Minimal Correction Sets (MCS) of bounded size for each of these paths. The removal of one of these sets of constraints yields a maximal satisfiable subset, in other words, a maximal subset of constraints satisfying the post condition. To compute the MCS, we extend the algorithm proposed by Liffiton and Sakallah in order to handle programs with numerical statements more efficiently. The main advantage of this flow-driven approach is that the computed sets of suspicious instructions are small, each of them being associated to an identified path. Moreover, the constraint-programming based framework allows mixing Boolean and numerical constraints in an efficient and straightforward way. Preliminary experiments are quite encouraging.
Faster Interleaved Modular Multiplier Based on Sign DetectionVLSICS Design
Data Security is the most important issue nowadays. A lot of cryptosystems are introduced to provide security. Public key cryptosystems are the most common cryptosystems used for securing data communication. The common drawback of applying such cryptosystems is the heavy computations which degrade performance of a system. Modular multiplication is the basic operation of common public key cryptosystems such as RSA, Diffie-Hellman key agreement (DH), ElGamal and ECC. Much research is now
directed to reduce overall time consumed by modular multiplication operation. Abd-el-fatah et al. introduced an enhanced architecture for computing modular ultiplication of two large numbers X and Y modulo given M. In this paper, a modification on that architecture is introduced. The proposed design computes modular multiplication by scanning two bits per iteration instead of one bit. The proposed design for 1024-bit precision reduced overall time by 38% compared to the design of Abd-el-fatah et al.
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...SAJJAD KHUDHUR ABBAS
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies
3.2.3.3. Quasi-Newton (QN) Methods
These methods represent a very important class of techniques because of their extensive use in practical alqorithms. They attempt to use an approximation to the Jacobian and then update this at each step thus reducing the overall computational work.
The QN method uses an approximation Hk to the true Jacobian i and computes the step via a Newton-like iteration. That is,
SAJJAD KHUDHUR ABBAS
Ceo , Founder & Head of SHacademy
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
Microservice Teams - How the cloud changes the way we workSven Peters
A lot of technical challenges and complexity come with building a cloud-native and distributed architecture. The way we develop backend software has fundamentally changed in the last ten years. Managing a microservices architecture demands a lot of us to ensure observability and operational resiliency. But did you also change the way you run your development teams?
Sven will talk about Atlassian’s journey from a monolith to a multi-tenanted architecture and how it affected the way the engineering teams work. You will learn how we shifted to service ownership, moved to more autonomous teams (and its challenges), and established platform and enablement teams.
APPLYING TRANSFORMATION CHARACTERISTICS TO SOLVE THE MULTI OBJECTIVE LINEAR F...ijcsit
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating
profit/net-sales, profit/manufacturing cost, etc. In this paper, we proposed the transformation characteristics to solve the multi objective linear fractional programming (MOLFP) problems. If a MOLFP problem with both the numerators and the denominators of the objectives are linear functions and some
technical linear restrictions are satisfied, then it is defined as a multi objective linear fractional programming problem MOLFPP in this research. The transformation characteristics are illustrated and the solution procedure and numerical example are presented.
The document discusses two examples of applying optimization in accounting and auditing. In the first exercise, the document finds the minimum average cost of producing x units of a product. The minimum is found to be $0.59 at x=3 units. In the second exercise, the document uses a demand function to maximize revenue from bus fares. It determines that revenue is maximized at a fare of 40 cents, with maximum revenue of $200,000 and 5,000 passengers per hour. Overall, the document shows how optimization techniques can help find maximum and minimum values to help with decision making.
This document provides information about getting solved assignments for the MBA Semester 2 Operations Research subject. It includes 6 questions related to operations research concepts like linear programming, transportation problems, assignment problems, and simulation. Students can get assignments solved at Rs. 125 each by emailing or calling the provided contact information. The questions cover topics like the framework of operations research, graphical and algebraic methods for linear programming problems, important terms in transportation problems, the Hungarian method for assignment problems, Monte Carlo simulation, assumptions of game theory, characteristics of Markov chains, and job prioritization rules.
The aim of this talk is to introduce two pieces of software, ``Lattice Builder'' and ``Stochastic Simulation in Java'' (SSJ). They allow to easily conduct experiments that rely on Monte Carlo (MC), quasi-Monte Carlo (QMC), or randomized QMC (RQMC). Lattice Builder is a C++ library designed to efficiently search, produce, and examine rank-1-lattices as well as polynomial lattices. It allows the user to choose from a broad palette of search criteria, types of weights, and construction methods, which can be accessed through a graphical interface as well as through a command line tool. Its structure also facilitates the implementation of own extensions and encourages to combine Lattice Builder with other programming languages.
SSJ is a Java library covering an extensive set of tools for stochastic simulations. It is particularly useful for experiments relying on MC and (R)QMC, ranging from integration problems over RQMC for Markov Chains (Array-RQMC) to density estimation.
This talk gives an introductory tour through the interfaces of Lattice Builder, shows how to integrate it into SSJ, and provides first steps in SSJ based on an example for mean estimation with RQMC.
Sampling-Based Planning Algorithms for Multi-Objective MissionsMd Mahbubur Rahman
multiobjective path planning has Increasing demand in military missions, rescue operations, construction job-sites.
There is Lack of robotic path planning algorithm that compromises multiple
objectives. Commonly no solution that optimizes all the objective functions. Here we modify RRT, RRT* sampling based algorithm.
In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis. The basic SVM takes a set of input data and predicts, for each given input, which of two possible classes forms the output, making it a non-probabilistic binary linear classifier.
This document provides information about getting fully solved assignments from an assignment help service. It lists an email address and phone number to contact for assistance with assignments. It also provides details about the available programs, subjects, semesters, credits, and other assignment details like word count requirements. Students are advised to mail their request with details of their semester and specialization to get solved assignments. Calling is listed as an emergency option.
dynamic programming complete by Mumtaz Ali (03154103173)Mumtaz Ali
The document discusses dynamic programming, including its meaning, definition, uses, techniques, and examples. Dynamic programming refers to breaking large problems down into smaller subproblems, solving each subproblem only once, and storing the results for future use. This avoids recomputing the same subproblems repeatedly. Examples covered include matrix chain multiplication, the Fibonacci sequence, and optimal substructure. The document provides details on formulating and solving dynamic programming problems through recursive definitions and storing results in tables.
This document discusses quantum computing business in the Japanese market. It introduces MDR Inc., a Japanese quantum computing startup founded in 2008. MDR develops full-stack quantum computing, from software to hardware, and works with over 20 clients in industries like banking, automotive, materials and more. Some applications discussed include quantum simulation, optimization, and machine learning. The document also provides an overview of the quantum computing developer community and ecosystem in Japan.
SURVEY ON POLYGONAL APPROXIMATION TECHNIQUES FOR DIGITAL PLANAR CURVESZac Darcy
This document summarizes and compares three techniques for polygonal approximation of digital planar curves:
1) Masood's technique which iteratively deletes redundant points and uses a stabilization process to optimize point locations.
2) Carmona's technique which suppresses redundant points using a breakpoint suppression algorithm and threshold.
3) Tanvir's adaptive optimization algorithm which focuses on high curvature points and applies an optimization procedure.
The techniques are evaluated on standard shapes using measures like number of points, compression ratio, error, and weighted error. Masood's technique generally had lower error while Tanvir's often achieved the highest compression.
This document discusses dynamic programming techniques. It covers matrix chain multiplication and all pairs shortest paths problems. Dynamic programming involves breaking down problems into overlapping subproblems and storing the results of already solved subproblems to avoid recomputing them. It has four main steps - defining a mathematical notation for subproblems, proving optimal substructure, deriving a recurrence relation, and developing an algorithm using the relation.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed
method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem
transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will
optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency
of the suggested method, a lot of test problems have been solved using this method. Comparing the results
of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable
for solving the multi-objective problems.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency of the suggested method, a lot of test problems have been solved using this method. Comparing the results of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable for solving the multi-objective problems.
This document discusses variations of the interval linear assignment problem. It begins with an introduction to assignment problems and defines them as problems that assign resources to activities to minimize cost or maximize profit on a one-to-one basis. It then provides the mathematical model for standard assignment problems and discusses variations such as non-square matrices, maximization/minimization objectives, constrained assignments, and alternate optimal solutions. The document also gives examples of managerial applications and provides two numerical examples solving interval linear assignment problems using an interval Hungarian method.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
This is an introduction to computational economics by dynamic programming. MATLAB program is provided to understand how it is solved by using statistical computing.
We introduce in this paper a new flow-driven and constraint-based approach for error localization. The input is a faulty program for which a counter-example and a postcondition are provided. To identify helpful information for error location, we generate a constraint system for the paths of the control flow graph for which at most k conditional statements may be erroneous. Then, we calculate Minimal Correction Sets (MCS) of bounded size for each of these paths. The removal of one of these sets of constraints yields a maximal satisfiable subset, in other words, a maximal subset of constraints satisfying the post condition. To compute the MCS, we extend the algorithm proposed by Liffiton and Sakallah in order to handle programs with numerical statements more efficiently. The main advantage of this flow-driven approach is that the computed sets of suspicious instructions are small, each of them being associated to an identified path. Moreover, the constraint-programming based framework allows mixing Boolean and numerical constraints in an efficient and straightforward way. Preliminary experiments are quite encouraging.
Faster Interleaved Modular Multiplier Based on Sign DetectionVLSICS Design
Data Security is the most important issue nowadays. A lot of cryptosystems are introduced to provide security. Public key cryptosystems are the most common cryptosystems used for securing data communication. The common drawback of applying such cryptosystems is the heavy computations which degrade performance of a system. Modular multiplication is the basic operation of common public key cryptosystems such as RSA, Diffie-Hellman key agreement (DH), ElGamal and ECC. Much research is now
directed to reduce overall time consumed by modular multiplication operation. Abd-el-fatah et al. introduced an enhanced architecture for computing modular ultiplication of two large numbers X and Y modulo given M. In this paper, a modification on that architecture is introduced. The proposed design computes modular multiplication by scanning two bits per iteration instead of one bit. The proposed design for 1024-bit precision reduced overall time by 38% compared to the design of Abd-el-fatah et al.
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...SAJJAD KHUDHUR ABBAS
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies
3.2.3.3. Quasi-Newton (QN) Methods
These methods represent a very important class of techniques because of their extensive use in practical alqorithms. They attempt to use an approximation to the Jacobian and then update this at each step thus reducing the overall computational work.
The QN method uses an approximation Hk to the true Jacobian i and computes the step via a Newton-like iteration. That is,
SAJJAD KHUDHUR ABBAS
Ceo , Founder & Head of SHacademy
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
Microservice Teams - How the cloud changes the way we workSven Peters
A lot of technical challenges and complexity come with building a cloud-native and distributed architecture. The way we develop backend software has fundamentally changed in the last ten years. Managing a microservices architecture demands a lot of us to ensure observability and operational resiliency. But did you also change the way you run your development teams?
Sven will talk about Atlassian’s journey from a monolith to a multi-tenanted architecture and how it affected the way the engineering teams work. You will learn how we shifted to service ownership, moved to more autonomous teams (and its challenges), and established platform and enablement teams.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Mobile App Development Company In Noida | Drona InfotechDrona Infotech
Looking for a reliable mobile app development company in Noida? Look no further than Drona Infotech. We specialize in creating customized apps for your business needs.
Visit Us For : https://www.dronainfotech.com/mobile-application-development/
Zoom is a comprehensive platform designed to connect individuals and teams efficiently. With its user-friendly interface and powerful features, Zoom has become a go-to solution for virtual communication and collaboration. It offers a range of tools, including virtual meetings, team chat, VoIP phone systems, online whiteboards, and AI companions, to streamline workflows and enhance productivity.
Software Engineering, Software Consulting, Tech Lead, Spring Boot, Spring Cloud, Spring Core, Spring JDBC, Spring Transaction, Spring MVC, OpenShift Cloud Platform, Kafka, REST, SOAP, LLD & HLD.
Artificia Intellicence and XPath Extension FunctionsOctavian Nadolu
The purpose of this presentation is to provide an overview of how you can use AI from XSLT, XQuery, Schematron, or XML Refactoring operations, the potential benefits of using AI, and some of the challenges we face.
E-commerce Development Services- Hornet DynamicsHornet Dynamics
For any business hoping to succeed in the digital age, having a strong online presence is crucial. We offer Ecommerce Development Services that are customized according to your business requirements and client preferences, enabling you to create a dynamic, safe, and user-friendly online store.
Takashi Kobayashi and Hironori Washizaki, "SWEBOK Guide and Future of SE Education," First International Symposium on the Future of Software Engineering (FUSE), June 3-6, 2024, Okinawa, Japan
Transform Your Communication with Cloud-Based IVR SolutionsTheSMSPoint
Discover the power of Cloud-Based IVR Solutions to streamline communication processes. Embrace scalability and cost-efficiency while enhancing customer experiences with features like automated call routing and voice recognition. Accessible from anywhere, these solutions integrate seamlessly with existing systems, providing real-time analytics for continuous improvement. Revolutionize your communication strategy today with Cloud-Based IVR Solutions. Learn more at: https://thesmspoint.com/channel/cloud-telephony
UI5con 2024 - Boost Your Development Experience with UI5 Tooling ExtensionsPeter Muessig
The UI5 tooling is the development and build tooling of UI5. It is built in a modular and extensible way so that it can be easily extended by your needs. This session will showcase various tooling extensions which can boost your development experience by far so that you can really work offline, transpile your code in your project to use even newer versions of EcmaScript (than 2022 which is supported right now by the UI5 tooling), consume any npm package of your choice in your project, using different kind of proxies, and even stitching UI5 projects during development together to mimic your target environment.
Atelier - Innover avec l’IA Générative et les graphes de connaissancesNeo4j
Atelier - Innover avec l’IA Générative et les graphes de connaissances
Allez au-delà du battage médiatique autour de l’IA et découvrez des techniques pratiques pour utiliser l’IA de manière responsable à travers les données de votre organisation. Explorez comment utiliser les graphes de connaissances pour augmenter la précision, la transparence et la capacité d’explication dans les systèmes d’IA générative. Vous partirez avec une expérience pratique combinant les relations entre les données et les LLM pour apporter du contexte spécifique à votre domaine et améliorer votre raisonnement.
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A dynamic process unfolds in the intricate realm of software development, dedicated to crafting and sustaining products that effortlessly address user needs. Amidst vital stages like market analysis and requirement assessments, the heart of software development lies in the meticulous creation and upkeep of source code. Code alterations are inherent, challenging code quality, particularly under stringent deadlines.
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May Marketo Masterclass, London MUG May 22 2024.pdfAdele Miller
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May Marketo Masterclass, London MUG May 22 2024.pdf
ACCELERATED COMPUTING
1. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Accelerated computing through
Quasi-Monte Carlo (QMC) constructions
Polynomial lattice builder, C++ software
Mohamed Hanini,
CEO & Chief Scientist at Koïos Intelligence
Research & development
Montreal, January 28th 2020,
mohamed.hanini@koiosintelligence.ca
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 1 / 42
2. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Summary
1 Motivation
2 Theoretical approach
3 C++ practicing examples
4 Asian Option Pricing
5 Asian Option Valuation powered by Polylatbuilder
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 2 / 42
3. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Why you should care?
For many applications, using QMC structured points is a very
efficient way to cut infrastructure and development costs.
Applications
1 Accelerate your training algorithms (Initializing
parameters),
2 Bayesian Deep Learning (reducing the variance of the
gradient estimator),
3 Parallel computing (optimize your GPU),
4 Evaluate complex financial applications,
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 3 / 42
4. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Why you should care?
For many applications, using QMC structured points is a very
efficient way to cut infrastructure and development costs.
Applications
1 Accelerate your training algorithms (Initializing
parameters),
2 Bayesian Deep Learning (reducing the variance of the
gradient estimator),
3 Parallel computing (optimize your GPU),
4 Evaluate complex financial applications,
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 3 / 42
5. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Why you should care?
For many applications, using QMC structured points is a very
efficient way to cut infrastructure and development costs.
Applications
1 Accelerate your training algorithms (Initializing
parameters),
2 Bayesian Deep Learning (reducing the variance of the
gradient estimator),
3 Parallel computing (optimize your GPU),
4 Evaluate complex financial applications,
We need an efficient software!
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 3 / 42
6. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Multidimensional Integration Methods
Methods
1 Monte Carlo (MC),
2 Quasi-Monte Carlo (QMC) and Randomized QMC
(RQMC).
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 4 / 42
7. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Multidimensional Integration Methods
Methods
1 Monte Carlo (MC),
2 Quasi-Monte Carlo (QMC) and Randomized QMC
(RQMC).
Practical approach QMC-RQMC
1 Start a search for a given construction method to find a
generator vector. How to find this vector?
2 Build a Low Discrepancy sequence (grid points)
Different representations:
Worst case error, square error (QMC),
Variance (RQMC),
Others?
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 4 / 42
8. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Multidimensional Integration Methods
Methods
1 Monte Carlo (MC),
2 Quasi-Monte Carlo (QMC) and Randomized QMC
(RQMC).
Practical approach QMC-RQMC
1 Start a search for a given construction method to find a
generator vector. How to find this vector?
2 Build a Low Discrepancy sequence (grid points)
Different representations:
Worst case error, square error (QMC),
Variance (RQMC),
Others?
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 4 / 42
9. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Monte Carlo (MC)
Multidimensional Integration Methods
Monte Carlo (MC), Quasi-Monte Carlo (QMC) et Randomized
QMC (RQMC)
Let X be a system (output) with real values
X = f(u1, u2, . . . , ud).
µ = E[X] = E[f(u1, u2, . . . , ud)] =
∫
[0,1)d
f(u) du
Inversion technique Example: X ∼ exp(λ)
F(X) = 1 − exp(−λX) = U ∼ U[0, 1)
⇒ X = − log(U)/λ.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 5 / 42
10. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Monte Carlo (MC)
Multidimensional Integration Methods
Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
Randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
standard MC: generate n random values i.i.d of U(0, 1)d, let
U0, . . . , Un−1;
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 6 / 42
11. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Monte Carlo (MC)
Multidimensional Integration Methods
Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
Randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
standard MC: generate n random values i.i.d of U(0, 1)d, let
U0, . . . , Un−1; estimate µ with ˆµn,MC = 1
n
∑n−1
i=0 f(Ui),
En = ˆµn,MC − µ converges with Op(n−1/2).
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 6 / 42
12. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Monte Carlo (MC)
Multidimensional Integration Methods
Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
Randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
standard MC: generate n random values i.i.d of U(0, 1)d, let
U0, . . . , Un−1; estimate µ with ˆµn,MC = 1
n
∑n−1
i=0 f(Ui),
En = ˆµn,MC − µ converges with Op(n−1/2).
Var[ˆµn,MC] = E[(ˆµn,MC − µ)2)] = σ2/n = O(n−1) converges
slowly as a function of n, where σ2 =
∫
[0,1)d f(u) du − µ2.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 6 / 42
13. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Monte Carlo (MC)
Multidimensional Integration Methods
Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
Randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
standard MC: generate n random values i.i.d of U(0, 1)d, let
U0, . . . , Un−1; estimate µ with ˆµn,MC = 1
n
∑n−1
i=0 f(Ui),
En = ˆµn,MC − µ converges with Op(n−1/2).
Var[ˆµn,MC] = E[(ˆµn,MC − µ)2)] = σ2/n = O(n−1) converges
slowly as a function of n, where σ2 =
∫
[0,1)d f(u) du − µ2.
central limit theorem
√
n(ˆµn,MC − µ)/σ ⇒ N(0, 1) when n → ∞.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 6 / 42
14. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Quasi-Monte Carlo (QMC)
Multidimensional Integration Methods
µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
We replace independent points MC by u0, . . . , un−1 a set of n
structured points () which covers the hypercube (0, 1)d more
uniformly than MC points.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 7 / 42
15. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Quasi-Monte Carlo (QMC)
Multidimensional Integration Methods
µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
We replace independent points MC by u0, . . . , un−1 a set of n
structured points () which covers the hypercube (0, 1)d more
uniformly than MC points.
Integration methods: Polynomial lattice rules and digital nets.
Construction methods: example Korobov, CBC etc...
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 7 / 42
16. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Quasi-Monte Carlo (QMC)
Multidimensional Integration Methods
µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
We replace independent points MC by u0, . . . , un−1 a set of n
structured points () which covers the hypercube (0, 1)d more
uniformly than MC points.
Integration methods: Polynomial lattice rules and digital nets.
Construction methods: example Korobov, CBC etc...
Worst case error dans O(n−α+ϵ) where α ≥ 1
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 7 / 42
17. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Randomized Quasi-Monte Carlo (RQMC)
Multidimensional Integration Methods
RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
To create variance, we need to randomize the n structurted
points u0, . . . , un−1 : by shifting them individually. Each point
Ui ∼ U[0, 1)d.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 8 / 42
18. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Randomized Quasi-Monte Carlo (RQMC)
Multidimensional Integration Methods
RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
To create variance, we need to randomize the n structurted
points u0, . . . , un−1 : by shifting them individually. Each point
Ui ∼ U[0, 1)d. With m i.i.d estimators, we can construct an
unbiased estimator ˆµn,RQMC = 1
m
∑m
l=1 ˆµn,RQMC,l.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 8 / 42
19. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Randomized Quasi-Monte Carlo (RQMC)
Multidimensional Integration Methods
RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
To create variance, we need to randomize the n structurted
points u0, . . . , un−1 : by shifting them individually. Each point
Ui ∼ U[0, 1)d. With m i.i.d estimators, we can construct an
unbiased estimator ˆµn,RQMC = 1
m
∑m
l=1 ˆµn,RQMC,l.
Var[ˆµn,RQMC] = Var[f(ui)]
n + 2
n2
∑
i<j Cov[f(ui), f(uj)].
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 8 / 42
20. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Primal lattice
Ld =
v =
d∑
j=1
hjvj such that each hj ∈ Z
,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 9 / 42
21. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Primal lattice
Ld =
v =
d∑
j=1
hjvj such that each hj ∈ Z
,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
A rank 1 lattice: ui = iv1 mod 1 for i = 0, 1, . . . , n − 1.
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 9 / 42
22. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Primal lattice
Ld =
v =
d∑
j=1
hjvj such that each hj ∈ Z
,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
A rank 1 lattice: ui = iv1 mod 1 for i = 0, 1, . . . , n − 1.
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Example: Korobov construction a = (1, a, a2 mod n, . . . , ad−1
mod n) = O(n), because 0 ≤ a ≤ n − 1.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 9 / 42
23. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Primal lattice
Ld =
v =
d∑
j=1
hjvj such that each hj ∈ Z
,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
A rank 1 lattice: ui = iv1 mod 1 for i = 0, 1, . . . , n − 1.
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Example: Korobov construction a = (1, a, a2 mod n, . . . , ad−1
mod n) = O(n), because 0 ≤ a ≤ n − 1.
Dual lattice
L⊥
d =
{
h ∈ Rd
: hT
v ∈ Z for each v ∈ Ld
}
⊆ Zd
.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 9 / 42
24. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Primal lattice
Ld =
v =
d∑
j=1
hjvj such that each hj ∈ Z
,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
A rank 1 lattice: ui = iv1 mod 1 for i = 0, 1, . . . , n − 1.
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Example: Korobov construction a = (1, a, a2 mod n, . . . , ad−1
mod n) = O(n), because 0 ≤ a ≤ n − 1.
Dual lattice
L⊥
d =
{
h ∈ Rd
: hT
v ∈ Z for each v ∈ Ld
}
⊆ Zd
.
A dual representation
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 9 / 42
25. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
How to choose the generator vector?
Naive approach
1 Consider all the possibilities of finding a generator vector,
2 minimize the worst case error or the variance
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 10 / 42
26. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
How to choose the generator vector?
Naive approach
1 Consider all the possibilities of finding a generator vector,
2 minimize the worst case error or the variance
Practical approach
1 Refine the search space g(z)
2 minimize an error function (cost function)
With a construction method
Component by component(CBC),
Fast component by component (Fast-CBC),
Korobov
Others?
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 10 / 42
27. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
How to choose the generator vector?
Naive approach
1 Consider all the possibilities of finding a generator vector,
2 minimize the worst case error or the variance
Practical approach
1 Refine the search space g(z)
2 minimize an error function (cost function)
With a construction method
Component by component(CBC),
Fast component by component (Fast-CBC),
Korobov
Others?
We clearly need a software (Polylatbuilder)!
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 10 / 42
28. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Example of a rank 1 lattice rule
Rank1 lattice
Pn =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
,
Rank1 lattice rule = 1
n
∑n−1
i=0 f(ui),
Number of points: n,
Generator vector: a ∈ Zd.
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
u1
u2
shifted
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 11 / 42
29. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Example of a rank 1 lattice rule
Rank1 lattice
Pn =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
,
Rank1 lattice rule = 1
n
∑n−1
i=0 f(ui),
Number of points: n,
Generator vector: a ∈ Zd.
Randomized lattice rule
Pn,∆ = {Ui = (ui + U) mod 1 : 0 ≤ i ≤ n − 1} ,
Random point : U ∼ U[0, 1)d.
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
U
u1
u2
shifted
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 11 / 42
30. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Example of a rank 1 lattice rule
Rank1 lattice
Pn =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
,
Rank1 lattice rule = 1
n
∑n−1
i=0 f(ui),
Number of points: n,
Generator vector: a ∈ Zd.
Randomized lattice rule
Pn,∆ = {Ui = (ui + U) mod 1 : 0 ≤ i ≤ n − 1} ,
Random point : U ∼ U[0, 1)d.
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
U
u1
u2
shifted
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 11 / 42
31. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration lattice
Projection of the lattice points
⇒ We ideally want n distinct values for each one-dimensional
projection Pn({j}) of Pn so the components of a are found in :
Un = {a ∈ Z : 1 ≤ a ≤ n − 1 et pgcd(a, n) = 1}.
For n prime:
There are n − 1 choices for each component of a,
there are (n − 1)d possibilities to generate the vector a.
For each subset ν ⊆ {1, 2, . . . , d}, the projection of Ld(ν) under
Ld is also a lattice, having a set points Pn(ν).
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 12 / 42
32. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration polynomial lattice
Primal lattice
Ld =
v(z) =
d∑
j=1
hjvj as each hj ∈ Zb[z]
,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
b (a set of Laurent series) where
vj(z) = gj(z)/P(z), which P(z) ∈ Zb[z] a polynomial with degree
m. Pn = φ(Ld) ∩ [0, 1)d.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 13 / 42
33. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration polynomial lattice
Primal lattice
Ld =
v(z) =
d∑
j=1
hjvj as each hj ∈ Zb[z]
,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
b (a set of Laurent series) where
vj(z) = gj(z)/P(z), which P(z) ∈ Zb[z] a polynomial with degree
m. Pn = φ(Ld) ∩ [0, 1)d.
Dual lattice
L∗⊥
d =
h ∈ Nd
0 :
d∑
j=1
hj(z)vj(z) ∈ Zb[z] for each v(z) ∈ Ld
,
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 13 / 42
34. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Integration polynomial lattice
Primal lattice
Ld =
v(z) =
d∑
j=1
hjvj as each hj ∈ Zb[z]
,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
b (a set of Laurent series) where
vj(z) = gj(z)/P(z), which P(z) ∈ Zb[z] a polynomial with degree
m. Pn = φ(Ld) ∩ [0, 1)d.
Dual lattice
L∗⊥
d =
h ∈ Nd
0 :
d∑
j=1
hj(z)vj(z) ∈ Zb[z] for each v(z) ∈ Ld
,
Dual representation
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 13 / 42
35. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Construction of points
Structure of the points for a rank 1 polynomial lattice rule
ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
such asφ : Lb → R.
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 14 / 42
36. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Construction of points
Structure of the points for a rank 1 polynomial lattice rule
ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
such asφ : Lb → R.
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,
Practical approach
1 Truncate Laurent’s series to L ui,j =
∑L
l=w xlb−l,
2 Find the xl ∈ {0, 1} for a base b = 2.
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 14 / 42
37. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Construction of points
Structure of the points for a rank 1 polynomial lattice rule
ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
such asφ : Lb → R.
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,
Practical approach
1 Truncate Laurent’s series to L ui,j =
∑L
l=w xlb−l,
2 Find the xl ∈ {0, 1} for a base b = 2.
ui = (u1,i, u2,i, . . . , uj,i, . . . , ui,d) the i-th point
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Example: rank 1 polynomial lattice rules
Rank 1 polynomial lattice rules
Pn =
{
ui =
qi(z)g(z) mod P(z)
P(z)
: q(z) ∈ Zb[z]
}
rank 1 polynomial lattice rules = 1
n
∑n−1
i=0 f(ui)
prime polynomial : P(z) with degree m.
generator vector : g(z) ∈ Zb[z]d
Example (particular case): Korobov rules
g(z) = (1, g(z), g(z)2 mod P(z), . . . g(z)d−1 mod P(z))
Ṟandomized Rank 1 polynomial lattice rules
Pn,∆ = {ui = ui + ∆ : 0 ≤ i ≤ n − 1}
random point: ∆ ∼ U[0, 1)d, ˆui,j = (ui,j + ∆i,j) mod b
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Example: rank 1 polynomial lattice rules
Rank 1 polynomial lattice rules
Pn =
{
ui =
qi(z)g(z) mod P(z)
P(z)
: q(z) ∈ Zb[z]
}
rank 1 polynomial lattice rules = 1
n
∑n−1
i=0 f(ui)
prime polynomial : P(z) with degree m.
generator vector : g(z) ∈ Zb[z]d
Example (particular case): Korobov rules
g(z) = (1, g(z), g(z)2 mod P(z), . . . g(z)d−1 mod P(z))
Ṟandomized Rank 1 polynomial lattice rules
Pn,∆ = {ui = ui + ∆ : 0 ≤ i ≤ n − 1}
random point: ∆ ∼ U[0, 1)d, ˆui,j = (ui,j + ∆i,j) mod b
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40. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Example: rank 1 polynomial lattice rules
Rank 1 polynomial lattice rules
Pn =
{
ui =
qi(z)g(z) mod P(z)
P(z)
: q(z) ∈ Zb[z]
}
rank 1 polynomial lattice rules = 1
n
∑n−1
i=0 f(ui)
prime polynomial : P(z) with degree m.
generator vector : g(z) ∈ Zb[z]d
Example (particular case): Korobov rules
g(z) = (1, g(z), g(z)2 mod P(z), . . . g(z)d−1 mod P(z))
Ṟandomized Rank 1 polynomial lattice rules
Pn,∆ = {ui = ui + ∆ : 0 ≤ i ≤ n − 1}
random point: ∆ ∼ U[0, 1)d, ˆui,j = (ui,j + ∆i,j) mod b
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Example: Rank 1 polynomial lattice rules
Quadratic error QMC
Could converge more fast the MC depending on:
the regularity of the function to be integrated
the uniformity of points in the polynomial lattice (choice of
parameter g(z))
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Example: Rank 1 polynomial lattice rules
Quadratic error QMC
Could converge more fast the MC depending on:
the regularity of the function to be integrated
the uniformity of points in the polynomial lattice (choice of
parameter g(z))
Polynomial lattice of
n = 64 points
P(z) = z6 + z + 1 in
2D
0 1
0
1
Good lattice
g(z) = (1, z5
+
z3
+ z2
+ z + 1)
0 1
0
1
Very bad lattice
g(z) = (1, 1)
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43. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Réseaux digitaux
Définition
Let a prime number b ≥ 2, called base, a digital net in base b
formed by n = bm points in d dimensions is defined by selecting
d generating matrices C1, C2, . . . , Cd, where each Cj is an
∞ × m. The matrix whose elements belong to the finite field
Fb = 0, 1, 2, . . . , b − 1. ⇒ The matrix Cj determines the
coordinate j of all the points.
We define ui pour i = 0, 1, 2, . . . bm − 1,
we write the binary representation of i in base b,
the vector of the binary coefficient is multiplied by Cj
mod b to obtain the binary coefficients of ui,j.
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44. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Réseaux digitaux
Définition
Let a prime number b ≥ 2, called base, a digital net in base b
formed by n = bm points in d dimensions is defined by selecting
d generating matrices C1, C2, . . . , Cd, where each Cj is an
∞ × m. The matrix whose elements belong to the finite field
Fb = 0, 1, 2, . . . , b − 1. ⇒ The matrix Cj determines the
coordinate j of all the points.
⇒ Similar to the component gj(z) of the vector g(z)
We define ui pour i = 0, 1, 2, . . . bm − 1,
we write the binary representation of i in base b,
the vector of the binary coefficient is multiplied by Cj
mod b to obtain the binary coefficients of ui,j.
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45. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Digital Nets
Construction of points
Let
i =
k−1∑
l=0
ai,lbl
,
and let
ui,j,1
ui,j,2
...
= Cj
ai,0
ai,1
...
ai,k−1
mod b.
ui,j =
∞∑
l=1
ui,j,lb−l
et ui =
(
ui,1, ui,2, . . . , ui,d
)
.
⇒ The resulting set of points is a basic digital net in base b. In
practice we truncate the expansion ui,j to L. For b = 2, the Cj
are the L × m binary matrcies.
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46. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Walsh transform
f(u) =
∑
h∈Nd
0
ˆf(h) walb,h(u),
where
ˆf(h) =
∫
[0,1)d f(u)walb,h(u)du et walb,h(u) = exp (2πih · u)/b.
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47. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Walsh transform
f(u) =
∑
h∈Nd
0
ˆf(h) walb,h(u),
where
ˆf(h) =
∫
[0,1)d f(u)walb,h(u)du et walb,h(u) = exp (2πih · u)/b.
Integration error (Sloan’s property)
En =
1
n
n−1∑
i=0
f(ui) −
∫
[0,1)d
f(u) du =
∑
0̸=h∈L∗⊥
d
ˆf(h).
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48. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Expression of variance
For a randomized lattice, the variance can be expressed:
Var[ˆµn,RQMC] =
∑
0̸=h∈L∗⊥
d
|ˆf(h)|2
.
From a variance reduction perspective, an optimal network
for f minimize Df(Pn)2 = Var[ˆµn,RQMC].
Not widely used as criterion solution ?
⇒ Worst case error
e(HK; Pn) = sup
f∈HK,||f||K≤1
|Qn(f) − I(f)|.
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Practical criteria
Consider the following class of functions:
Eν,d(c) = {f : [0, 1]d
→ R : |ˆf(h)| ≤ c b−νµ1(h)
pour tout h ∈ Nd
0},
où ν > 1 et c > 0. We thus obtain:
|Qn(f) − I(f)| ≤
∑
0̸=h∈L∗⊥
d
c b−νµ1(h)
, ν > 1,
The quantity
Tν =
∑
0̸=h∈L∗⊥
d
c b−νµ1(h)
, ν > 1,
Depends only on the function class Eν,d(1).
To implement!
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51. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
What do we need?
To find good digital and polynomial lattice
Figures of merit and weights adapted to a problem,
for a number of points n, for a polynomial and a dimension
d,
several construction methods,
Options: like normalization etc.
Our research work
Establish merit figures for digital networks and
polynomial,
compare (application) the performance of our software
integration methods compared to other integration
methods,
define new criteria for non-functional classes,
considered (explored) ⇒ unbounded functions!?
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52. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
What already exists
Dirk Nuyens’ Code Matlab
a construction method (Fast CBC)
Limited choice of parameters (ex: only ten primitive
polynomials)
only one type of weight (product weight)
No QMC / RQMC points generated and tested! no
application!!
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53. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
What already exists
Dirk Nuyens’ Code Matlab
a construction method (Fast CBC)
Limited choice of parameters (ex: only ten primitive
polynomials)
only one type of weight (product weight)
No QMC / RQMC points generated and tested! no
application!!
Generator vector tables
J. Dick
D. Nuyens
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54. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Structure of Polylatbuilder
Points of the
polynomial and
digital nets
Figures of merit Weight type
Construction
methods
Error in a
Walsh space
t-value
Spectral test
other heuris-
tic??
Product weight
Order depend-
ing weights
General weight
Product and
order depen-
dent weights
(POD)
CBC
Fast CBC
Random Ko-
robov
Korobov
Exhaustive
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55. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Quality measures (Discrepancies)
Implemented figure of merit
E(f, g(z))1/q
) = e(g(z); n; ||.||p,α,γ)
The worst function:
χ(g(z); n; ||.||p,α,γ) =
∑
0̸=v⊂1:t
γv
q/2
∏
j∈v
ω(uj)
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56. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Quality measures (Discrepancies)
Implemented figure of merit
E(f, g(z))1/q
) = e(g(z); n; ||.||p,α,γ)
The worst function:
χ(g(z); n; ||.||p,α,γ) =
∑
0̸=v⊂1:t
γv
q/2
∏
j∈v
ω(uj)
The kernel
ω(uj) =
∞∑
h=1
wal2,h(uj)
2qα⌊log2hj⌋
= 12(
1
6
− 2⌊log)2uj⌋−1
)
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57. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Construction methods
CBC Constructions
Knowing n, we construct a generator vector
g(z) = (g1(z), g2(z), . . . , gd(z)).
We set g1(z) = 1,
g1(z) remains fixed, we choose g2(z) from the subset of the
components of generator vectors to minimize the desired
error criterion in 2 dimensions,
With g1(z), g2(z) remain fixed, we choose g3(z) from the
subset of the components of generator vectors to
minimize the desired error criterion in 3 dimensions,
ed(gd(z); Pn; ||.||p,α,γ)q
= ed−1(g(z)d−1; n; ||.||p,α,γ)q
+ Θd(g(z)d)
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Setup intger to bit
#include "PolyType.h"
namespace PolyLatBuilder {
// choose the type of lattice
std::ostream& operator<<(std::ostream&
os, PolyLatType polylatType)
{ return os << (polylatType ==
PolyLatType::EMBEDDED ? "embedded" :
"ordinary"); }
}
}
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59. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
C++code
boost::dynamic_bitset<>
PolyLatBuilder::FastCBC::getbin(int
integer){
unsigned long long dec;
dec = integer;
std::string str;
for(unsigned long long d = dec; d>0;
d/=2)
str.insert(str.begin(),
boost::lexical_cast<char>(d&1));
boost::dynamic_bitset<>
binaryNumber(str);
std::cout << "The number " << dec << "
in binary is: " << binaryNumber<<'n';
return binaryNumber;
}
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60. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Setup Bit vector to polynomial
std::string PolyLatBuilder::FastCBC::
pretty(const GF2X& p) {
std::ostringstream os;
if ( deg(p)<0 ) os << "0" ;
for (int i = deg(p); i >= 0; i--) {
if (rep(coeff(p, i))) {
if (i < deg(p)) os << " + ";
if (i >= 2)
os << "x^" << i;
else if (i == 1)
os << "x";
else
os << "1";
}
}
return os.str();
}
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61. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Perform the mapping function Vm(g(z)q(z)modP(z)
GF2X ww(INIT_SIZE,1);
ww.SetLength(1);
std::vector <double> w(rr.length());
std::vector <double> w1(rr.length());
for ( long j= 0; j <rr.length(); j++) {
w[j]=0;
double b = 1.0;
for( int k=0; k< m; k++){
ww[k] = rr[j][m-k-1];
for ( int l =0; l <k; l++){
ww[k] = ww[k] - ww[l] *
P2[m+(l-k)];
}
b *= 0.5;
if (IsOne(ww[k]))
w[j] += b;
}
std::cout << w[j] << std::endl;
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62. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Perform the vector of real values by applying the
Kernel
fftw<double>::real_vector usob(n-1);
for (long i = 0 ; i < w.size(); i++){
usob[i] = 1.0/6 - pow(2,
floor(log2(w[i]))-1);
}
float usob_0=1.0/6;
fftw<double>::complex_vector fft_psi =
fftw<double>::fft(usob);
}
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64. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
QMC square error
Built in C++
Encouraging results
Good uniformity of the polynomial lattice rules (good
parameter g(z))
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65. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
QMC square error
Built in C++
Encouraging results
Good uniformity of the polynomial lattice rules (good
parameter g(z))
Polynomial lattice
rules points
n = 1023, P(z) =
z10 + z3 + 1 en 2D
0 1
0
1
good lattice
g(z)
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66. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Financial application: asian option
Stochastic differential equation(SDE)
dSt = µSt dt + σSt dWt
The SDE solution St = S0 exp(r−σ2/2)t+σBt
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67. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Financial application: asian option
Stochastic differential equation(SDE)
dSt = µSt dt + σSt dWt
The SDE solution St = S0 exp(r−σ2/2)t+σBt
Option pricing
The payoff of the option X = e−rt max(0, 1
d
∑s
j=1 Stj − K),
and the price
c = E[X |Ft]
v(s0, T) = E[e−rt
f(S1 . . . St) |Ft]
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68. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Asian purchase option - Numerical results
v(s0, T) = e−rt
∫
(0,1)d
max(0,
1
d
d∑
i=1
S0 exp [(r − σ2
)ti
+σ
√
ti − ti−1
i∑
j=1
Φ−1
(uj)] − K)du1 . . . dud
Variance reduction techniques for MC
control variate CV1 = e−rt max(0, Y − K) where
Y =
∏t
j=1(Stj )1/d
control variate CV2 = e−rt max(0, Y − K) where
Y =
∑t
j=1(Stj )1/d
Antithetic variable (AV)
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69. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Call Asian option– Numerical results (n=1023)
Risk free rate r = 0.08, volatility σ = 0.2, and the strike price K
MC points with variance reduction techniques
Approach PD LGD EAD
No CV 20.83 49.87
CV1 20.90 1.00 10−2
Data CV2 20.90 9.25 10−4
CV1 +CV2 20.90 8.45 10−4
CV1+CV2+AV 20.91 3.23 10−4
No CV 11.23 4.68
1
CV1 11.28 8.79 10−3
Key Features CV2 11.29 4.61 10−1
CV1 +CV2 11.28 8.52 10−3
CV1+CV2+AV 11.30 3.36 10−3
No CV 3.54 22. 58
1
CV1 3.55 5.66 10−3
100 CV2 3.57 4.27
CV1 +CV2 3.55 5.5710−3
CV1+CV2+AV 3.56 1.4010−3
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70. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Call Asian option– Numerical results
Comparison with LatticeBuilder software
Table: Value of the Asian option with a low exercise based on the
following configuration:(j)/10, for j = 1, 2, . . . , d = 10, number of
n = 210
points, and an exercise price K = 100
MC RQMC RQMC
Polylatbuilder LatticeBuilder
Option value 3.37 3.33 3.354
Standard deviation 4.52 0.015 0.025
Variance Ratio 95.17 33.06
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71. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Java Asian Option Valuation (using QMC points)
import umontreal.iro.lecuyer.rng.*;
import umontreal.iro.lecuyer.hups.*;
import umontreal.iro.lecuyer.stat.Tally;
import umontreal.iro.lecuyer.util.Chrono;
public class AsianQMCSSJ extends AsianSSJ {
public AsianQMCSSJ (double r, double
sigma, double strike, double s0, int
s, double[] zeta) {
super (r, sigma, strike, s0, s, zeta);
}
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72. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Java code
public void simulateQMC (int m, PointSet p,
RandomStream noise, Tally statQMC) {
Tally statValue = new Tally ("stat on
value of Asian option");
PointSetIterator stream = p.iterator ();
for (int j=0; j<m; j++) {
simulateRuns (p.getNumPoints(), stream,
statValue);
statQMC.add (statValue.average());
p.addRandomShift (0, 1, noise);
stream.resetStartStream();
}
}
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73. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Java code
public static void main (String[] args) {
try {
int s = 9;
double[] zeta = new double[s+1];
for (int j=0; j<=s; j++)
zeta[j] = 12.0*j/365;
AsianQMCSSJ process = new AsianQMCSSJ
(0.08, 0.2, 100.0, 100.0, s, zeta);
Tally statValue = new Tally ("value of
Asian option");
Tally statQMC = new Tally ("QMC averages
for Asian option");
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74. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Java code
timer.init();
// QMC Polynomial points
PointSet ps2 = new
PointSetFromFile("/Users/QuasiMonteAsian/src/options/pointsl
int m;
m =10;
// Number of QMC randomizations
process.simulateQMC ( m, ps2, new
MRG32k3a(), statQMC);
System.out.println ("QMC point set with
" + n +
" points and affine matrix
scramble:n");
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75. Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Java code
System.out.printf ("Vlaue of Asian with
QMC: %9.4g%n", valueQMC);
double varQMC = ps2.getNumPoints() *
statQMC.variance();
double cpuQMC = timer.getSeconds() / (m *
n);
System.out.printf ("Variance ratio:
%9.4g%n", varMC/varQMC);
System.out.printf ("Efficiency ratio:
%9.4g%n",
(varMC * cpuMC) / (varQMC * cpuQMC));
System.out.printf ("Efficiency ratio 22:
%9.4g%n",
(cpuMC) / (cpuQMC));
System.out.println ("Test CPU time: "+
cpuMC);
System.out.println ("Test CPUQMC time: "+
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