Institut Polytechnique de Paris

When you want to design and simulate a real issue (be a question of population dynamism in ecology, tumor growth in biomedical engineering, combustion dynamics in new-generation rocket engines - for example at SpaceX, prediction of solar storms and magnetic reconnection phenomena in solar physics, simulation of turbulence and turbulent combustion in fluid mechanics) the engineer or the researcher in applied mathematics or in "computational science" has to use a range of digital methods and must be able to analyze it mathematically, assess in terms of quality and computational efficacity, and finally, implement.

This course offers an introduction to numerical analysis, starting with the mathematical foundations on which numerical methods are based through to the implementation and the use of these methods on Jupyter notebooks (an active field at the Ecole Polytechnique), including their stability in relation to mathematical conditioning of the problems posed. The link is made with applications to understand the extent to which this type of method can be used from a practical point of view. Implementations of these methods in existing digital libraries are also documented.

Every class sessions include an analysis of the mathematical foundations on which a numerical method class is built, a description and analysis of the digital method (with a historical perspective). Those two aspects are covered during the course. The PC offers a review and an in-depth look at some concepts of the course, use of the method in the context of Jupyter notebooks and a description of how students can implement the method in a notebook to be handed in for the next class session.

Partial differential equations play a fundamental role in modeling complex phenomenon in fields as varied as mechanics, physics, and biology. Since the 1950s and the advent of computers, the development and use of numerical methods for the approximate machine calculation of solutions to partial differential equations have become routine in engineering. In automotive engineering, for example, not only deformations of the passenger compartment in the event of an impact but also air conditioning, ambient noise and electromagnetic compatibility are nowadays calculated by computer.

The course aims to highlight the link between classical mechanical and physical models based on partial differential equations, the underlying mathematical analysis, and the development of the finite element method. The common thread will be the variational point of view, which enables problems to be rewritten as minimization problems, making the link with optimization. A significant part of the course will be devoted to machine implementation of the finite element method and to the explicit approximate solution of certain partial differential equations using FreeFem++ software. In particular, students will be asked to carry out a mini-project using this software.

In addition, some of the PCs will illustrate on computer the concepts seen in class.

Evaluation: A written exam, quiz and mandatory mini-project.

Partial differential equations play a fundamental role in modeling complex phenomenon in fields as varied as mechanics, physics, and biology. Since the 1950s and the advent of computers, the development and use of numerical methods for the approximate machine calculation of solutions to partial differential equations have become routine in engineering. In automotive engineering, for example, not only deformations of the passenger compartment in the event of an impact but also air conditioning, ambient noise and electromagnetic compatibility are nowadays calculated by computer.

The course aims to highlight the link between classical mechanical and physical models based on partial differential equations, the underlying mathematical analysis, and the development of the finite element method. The common thread will be the variational point of view, which enables problems to be rewritten as minimization problems, making the link with optimization. A significant part of the course will be devoted to machine implementation of the finite element method and to the explicit approximate solution of certain partial differential equations using FreeFem++ software. In particular, students will be asked to carry out a mini-project using this software.

In addition, some of the PCs will illustrate on computer the concepts seen in class.

Evaluation: A written exam, quiz and mandatory mini-project.

More and more companies are actively investing human and financial resources in  data management  technologies in order to improve and making effective decision making. This is mainly because a  wide class of data analytics operations (e.g., statistical analysis, machine learning, ad-hoc queries) for supporting decision makers requires to process massive data sets in order to ensure high precision of analysis result. In addition, companies are more and more interested in extracting, transforming, store and analyse data involving massive and heterogeneous data collections coming from external sources (social networks, competitors' movements, market trends, etc.), typically made available by Web applications.

The  main aim of this course is to give students a deep and solid understanding of the state of the art of Big Data systems and programming paradigms, and to enable them to devise and implement efficient algorithms for analysing massive data sets. The focus will be on paradigms based on distribution and shared-nothing parallelism, which are crucial to enable the implementation of algorithms that can be run on clusters of computers, scale as the size of input data increases, and can be safely executed even in the presence of system failures.

Lectures will give articular emphasis to the MapReduce paradigm and the internal aspects of its related runtime support Hadoop, as well as to MapReduce-based systems, with a particular focus on Spark that provides users with powerful programming tools and efficient execution support for performing operations related to complex data flows. The attention will be then given to mechanisms and algorithms for both both iterative and interactive data processing. A particular attention will be given to SQL-like data querying,  graph analysis, and the development of  machine learning algorithms.

A large part of the course consists of lab-sessions where students develop parallel algorithms for data querying and analysis, including algorithms for relational database operators, matrix operations, graph analysis, and clustering. Lab-sessions rely on the use of both desktop computers and Hadoop clusters on Google Cloud.

More and more companies are actively investing human and financial resources in  data management  technologies in order to improve and making effective decision making. This is mainly because a  wide class of data analytics operations (e.g., statistical analysis, machine learning, ad-hoc queries) for supporting decision makers requires to process massive data sets in order to ensure high precision of analysis result. In addition, companies are more and more interested in extracting, transforming, store and analyse data involving massive and heterogeneous data collections coming from external sources (social networks, competitors' movements, market trends, etc.), typically made available by Web applications.

The  main aim of this course is to give students a deep and solid understanding of the state of the art of Big Data systems and programming paradigms, and to enable them to devise and implement efficient algorithms for analysing massive data sets. The focus will be on paradigms based on distribution and shared-nothing parallelism, which are crucial to enable the implementation of algorithms that can be run on clusters of computers, scale as the size of input data increases, and can be safely executed even in the presence of system failures.

Lectures will give articular emphasis to the MapReduce paradigm and the internal aspects of its related runtime support Hadoop, as well as to MapReduce-based systems, with a particular focus on Spark that provides users with powerful programming tools and efficient execution support for performing operations related to complex data flows. The attention will be then given to mechanisms and algorithms for both both iterative and interactive data processing. A particular attention will be given to SQL-like data querying,  graph analysis, and the development of  machine learning algorithms.

A large part of the course consists of lab-sessions where students develop parallel algorithms for data querying and analysis, including algorithms for relational database operators, matrix operations, graph analysis, and clustering. Lab-sessions rely on the use of both desktop computers and Hadoop clusters on Google Cloud.

A key issue of the Applied Mathematics is to modeling and simulate complex systems to understand their qualitative and quantitative behavior.

This cours introduces effective pobabilistical methods of calcul and simulation, mainly focused on continuous time processes, with linear then non-linear dynamics ("interactions between particles"). A constant concern is their validation, their digital efficacity and their illustration in concrete situations, notably from financial engineering, ecology, communication networks, fluid mechanics, physics and chemistry, among others... These methods have become particularly important in the various strategical applicative fields.

Evaluation: continuous control and final exam

Course language: French
Course blog: http://montecarlo-polytechnique.blogspot.fr

The objective of this course is to provide tools for a better understanding of networks, in the broad sense of the word, from communication networks which underlie the Internet to social networks (online or not).

The aim is to propose:

• models and phenomenon of interest,
• algorithmic methods for their control (ideally decentralized),
• mathematics tools to analyse these systems' behavior and performance.

Bibliographic reference:
"Réseaux : contrôle distribué et phénomènes émergents", Laurent Massoulié, 2016

The course will be taught in English

Partie I: Cours sur économie du climat

Résumé: Ce module introduit les fondements de la modélisation macroéconomique appliquée aux enjeux climatiques. Il s’articule en trois séances de quatre heures chacune, combinant environ une heure de cours magistral et une heure de mise en pratique par le code (sous Python ou Dynare). L’objectif est de familiariser les étudiants avec les modèles intégrés climat-économie (IAM) et leur articulation avec la théorie de la croissance et les modèles d’équilibre général.

Organisation:

1. La structure du modèle de croissance optimale de Ramsey-Cass-Koopmans et son extension à la question environnementale. Les étudiants apprennent à coder et résoudre ce modèle, à interpréter les conditions du premier ordre, et à relier les trajectoires de capital, consommation et émissions.

2. Le modèle DICE et sa variante moderne : la dynamique du cycle du carbone, la relation entre température et dommages économiques, et les politiques d’atténuation (taxe carbone, subventions vertes). 

3. Extensions contemporaines: la détermination du cout social du carbone, et les controverses associées.

Acquisition de compétence:

1/ Comprendre les fondements théoriques de l’économie du climat
2/ Maîtriser les outils numériques permettant de résoudre et analyser ces modèles

TITLE: INTRODUCTION TO FINANCIAL MARKETS

Lecturer: Dorinel BASTIDE

Session split: 24 hours split into 6 session of 4 hours (including 4 hours test)

Objectives:

The first objective is to provide Students with an understanding of the underlying context of financial system usage through investments facilitation. This will require introducing main accounting principles, business activities development and economic rationale of funding needs. The financial network structure and various intermediaries will then be presented with an emphasis on underlying needed IT infrastructures including.

These sessions will allow for the second objective to present Students with the various types of risk that can materialize through conducting activities within the financial markets. A specific focus will be brought on credit risk, how it materializes and how it can be managed through dedicated qualitative and quantitative assessments. Armed with this knowledge, specific focus will be made on climate change with economic and financial impacts and transition risk management. The various financial markets will be covered along with main basics for cashflows, standard and OTC derivatives contract valuation and hedging giving rise to numerous types of risks.

The third objective is to present Students with the various regulatory requirements touching on those for new developing risks including XVA costs, Cyber, AI and Quantum Computing risks as well.

Knowledge outcome:

• Understanding the economic rationale of financial markets and their various actors
• Having basic knowledge of the numerous risks related to activities within financial markets
• Be able to approach cashflow and contract valuation
• Understanding the underlying infrastructure for conducting financial markets activities
• Understanding various types of financial data
• Be sensitized to the climate change, its impacts on economic and financial markets
• Understanding the development of sustainable finance and materialization of transition risks
• Having basic knowledge of the various regulatory requirements and implication for financial actors
• Understanding XVA costs and how they can be cascaded through the financial network
• Be sensitized to new risk topic trends w.r.t. AI, quantum computing with possible applications and Cyber risk

Skills outcomes:

• Mathematical formulation of financial cashflow and contract valuation
• Identify risk types given a simple financial portfolio
• Identify data types that can be required for financial analysis
• Have a system view of funding needs and how they can be supported by the financial markets

Teaching format:

Each of the 5 lecture sessions may start by making Students think about the topics that will be covered during the beginning of the session through dedicated questions. The first half of the session can be quite interactive, pushing Students to question the place taken by Finance in general in the World society and what the various means underlying the financial markets such us business development requiring investments, IT infrastructure implementation, service platform setup. Then the topics are formalized, and practical applications will be designed so as to apply the presented topics. End of session will be dedicated to collect all thoughts of the applications and wrap-up with main key take-always. To sum up:

• Questions and reflection on session topics
• Lecture formalizing the session topics
• Applications
• Restitution and wrap-up

Resources:

Michel Crouhy, Dan Galai & Robert Mark, The Essentials of Risk Management, 2^nd Edition (2014)

Philippe Jorion, Financial Risk Manager Handbook (2005)

Jon Gregory, The XVA Challenge: Counterparty Risk, Funding, Collateral, Capital and Initial Margin (2020)

Jon Gregory, Central Counterparties: Mandatory Central Clearing and Initial Margin Requirements for OTC Derivatives (2014)

John Hull, Options, Futures, and Other Derivatives 9^th Edition (2018)

Tiziano Bellini, Stress Testing and Risk Integration in Banks: A Statistical Framework and Practical Software Guide (in Matlab and R) 1^st Edition (2016)

Short Biography

Dorinel Bastide is a 18-year Experienced Senior Quantitative Analyst at BNP Paribas with a demonstrated history of working in the banking industry, in particular risk modelling, with strong research professional skills in topics such as CCP, Counterparty, Market, Operational & Credit Risks and Stress Test modelling including extensive coding experience in Python, C#, C++ and VBA. He is the Coordinator for BNP Paribas of the Research Chair Stress Test, RISK Management & Financial Steering since its creation in 2018. He holds a PhD in Applied Mathematics from University Paris-Saclay and is a GARP FRM certified professional.