Monte Carlo Sampling Method
Course Specifications
Duration: 22 hours
Number of Sessions: 11 sessions (two hours per session)
Class Schedule: Saturdays & Sundays
How to Participate: Classes will be held online
Course Tuition: 500 USD. No charge for pre-registration; tuition is due upon final registration.
Programming Language: MATLAB
Instructor: Dr. Mahdi Shadabfar
Programming Language: MATLAB
Instructor: Dr. Mahdi Shadabfar
Course Description
Our understanding of the majority of the phenomena that occur in our environment is incomplete. Hence, we are continually confronted with a degree of uncertainty. This encompasses practically all disciplines of research, from physics and engineering to economics and medicine. For example, despite all the advancements in earth science, a geologist cannot discern with certainty the behavior of soil or rock strata in an underground environment. Or, an economist cannot anticipate the behavior of a stock with certainty. For this purpose, it is essential to be familiar with the approaches that enable us to represent uncertainty and provide the findings probabilistically.
The current course is one of the most thorough in the subject of probabilistic modeling. This course is excellent for anyone seeking a foundational understanding of uncertainty modeling. It is designed so that non-experts in the fields of statistics and probability can utilize it without difficulty. This means that you do not need an in-depth understanding of the principles and theorems of statistics and probability to begin this course; a fundamental understanding is sufficient. The majority of this course's subjects are introduced early on, and their specifics are discussed in sequence.
Course Introduction by the Instructor
?Who Should Take This Course
This course is suitable for a diverse audience, including:
- Graduate and Ph.D. Students: This course provides a solid foundation in uncertainty modeling for your research. It’s an excellent opportunity to enhance your understanding of probabilistic methods.
- Researchers: If you’re involved in scientific research, data analysis, or stochastic modeling, the Monte Carlo sampling method can significantly improve your work. Gain insights into uncertainty quantification and apply it to your research.
- Professors and Educators: As an educator, you can benefit from this course by incorporating probabilistic modeling concepts into your teaching. Enhance your ability to explain uncertainty to students and guide them in practical applications.
- Curious Minds: Even if you’re not a specialist in statistics or probability, this course offers accessible content. If you’re intrigued by uncertainty and want to explore its practical implications, you’re welcome to join!
Remember, no deep statistical knowledge is required—just a basic understanding is sufficient.
- Graduate and Ph.D. Students: This course provides a solid foundation in uncertainty modeling for your research. It’s an excellent opportunity to enhance your understanding of probabilistic methods.
- Researchers: If you’re involved in scientific research, data analysis, or stochastic modeling, the Monte Carlo sampling method can significantly improve your work. Gain insights into uncertainty quantification and apply it to your research.
- Professors and Educators: As an educator, you can benefit from this course by incorporating probabilistic modeling concepts into your teaching. Enhance your ability to explain uncertainty to students and guide them in practical applications.
- Curious Minds: Even if you’re not a specialist in statistics or probability, this course offers accessible content. If you’re intrigued by uncertainty and want to explore its practical implications, you’re welcome to join!
Remember, no deep statistical knowledge is required—just a basic understanding is sufficient.
Course Content
Session 1: Introduction to Monte Carlo Sampling Method
1.1. Generating random samples with a uniform distribution
1.1.1. Generating random samples
1.1.1. Generating random samples
1.1.2. Solving integrals using generated random samples
1.1.3. Implementation of integral calculation in MATLAB environment
1.2. Example: elevator failure
1.1.3. Implementation of integral calculation in MATLAB environment
1.2. Example: elevator failure
1.2.1. Problem introduction and description of the random variables involved
1.2.2. Calculation of failure probability using Monte Carlo sampling method
1.3. Various methods available to generate random samples
1.3.1. Inverse method of cumulative distribution function in MATLAB environment
1.3.2. Using library functions in MATLAB environment
1.3.2. Using library functions in MATLAB environment
Session 2: Generating Correlated Random Samples
2.1. Understanding correlation between random variables
2.2. Example: Investment decision in football
2.2.1. Problem introduction and description of the random variables involved
2.2.2. Solving the problem without considering variable correlations
2.2.3. Explaining the problems caused by not taking correlation into account
2.3. Introducing correlation
2.3.1. Correlation coefficient between two variables
2.3.2. Correlation matrix
2.3.3. Explanations on how to generate correlated random samples
2.4. Solving the footbal investment problem considering correlation
2.5. Discussing the effect of correlation on the output of the Monte Carlo sampling method
Session 3: Convergence of the Monte Carlo Sampling Method
3.1. An introduction to the convergence of Monte Carlo sampling method
3.1.1. Accuracy of the Monte Carlo sampling method based on the number of random samples
3.1.2. Demonstrating convergence of the Monte Carlo sampling method
3.2. Example: Deformation of a beam under bending
3.2.1. Problem introduction and description of the random variables involved
3.2.2. Solving the example and implementing it in the MATLAB environment
3.2.3. Extracting convergence graphs
3.3. Introducing the coefficient of variation (CoV) criterion to assess the accuracy of the Monte Carlo sampling method
3.3.1. Calculation of the CoV
3.3.2. CoV for the beam deflection example
3.3.3. Visualizing the results in the MATLAB environment
3.2.2. Solving the example and implementing it in the MATLAB environment
3.2.3. Extracting convergence graphs
3.3. Introducing the coefficient of variation (CoV) criterion to assess the accuracy of the Monte Carlo sampling method
3.3.1. Calculation of the CoV
3.3.2. CoV for the beam deflection example
3.3.3. Visualizing the results in the MATLAB environment
Session 4: Exceedance Probability Diagram
4.1. Structural reliability problem
4.1.1. An introduction to the reliability problem
4.1.2. Limit state function
4.2. Example: Cracking caused by rock explosion
4.2.1. Problem introduction and description of the random variables involved
4.2.2. Solving the established reliability problem in MATLAB using Monte Carlo sampling method
4.3. Exceedance probability diagram
4.3.1. Adding a decision variable to the limit state function
4.3.2. Using the decision variable to generate the exceedance probability diagram
4.3.3. Implementing the solution in Matlab environment
4.4. Histogram sampling method
4.4.1. Discussing the problem of the conventional methods in extracting the exceedance probability diagram
4.4.2. Introducing the histogram sampling method
4.4.3. Implementation of histogram sampling method in MATLAB environment
4.5. Additional notes regarding histogram sampling method
4.1.1. An introduction to the reliability problem
4.1.2. Limit state function
4.2. Example: Cracking caused by rock explosion
4.2.1. Problem introduction and description of the random variables involved
4.2.2. Solving the established reliability problem in MATLAB using Monte Carlo sampling method
4.3. Exceedance probability diagram
4.3.1. Adding a decision variable to the limit state function
4.3.2. Using the decision variable to generate the exceedance probability diagram
4.3.3. Implementing the solution in Matlab environment
4.4. Histogram sampling method
4.4.1. Discussing the problem of the conventional methods in extracting the exceedance probability diagram
4.4.2. Introducing the histogram sampling method
4.4.3. Implementation of histogram sampling method in MATLAB environment
4.5. Additional notes regarding histogram sampling method
Session 5: Reliability Sensitivity Analysis
5.1. Deterministic sensitivity analysis
5.1.1. Application of sensitivity analysis and its significance
5.1.2. Providing an example and implementing deterministic sensitivity analysi
5.2. Reliability sensitivity analysis
5.2.1. Advantages of reliability sensitivity analysis compared to deterministic reliability analysis
5.2.2. Steps to implement reliability sensitivity analysis
5.2.3. Implementing reliability sensitivity analysis for the rock explosion problem in the MATLAB environment
5.3. Application of reliability sensitivity analysis in variable ranking
5.4. Application of reliability sensitivity analysis in examining the most important range of random variables
5.2.2. Steps to implement reliability sensitivity analysis
5.2.3. Implementing reliability sensitivity analysis for the rock explosion problem in the MATLAB environment
5.3. Application of reliability sensitivity analysis in variable ranking
5.4. Application of reliability sensitivity analysis in examining the most important range of random variables
Session 6: Back Analysis and Surrogate Modeling using Artificial Neural Networks
6.1. Surrogate modeling
6.1.1. Introduction of surrogate modeling and its application
6.1.2. low- and high-fidelity models
6.1.3. steps to develop a surrogate model based on low- and high-fidelity models
6.2. Example: Ground deformation resulting from tunnel excavation
6.2.1. Problem introduction and description of the random variables involved
6.2.2. Implementing the deterministic form of the given example in the Matlab environment
6.2.3. Conducting reliability analysis on the given example using the Monte Carlo sampling method
6.3. Reliability analysis using surrogate model
6.3.1. Introducing a comprehensive data set of the given example
6.3.2. An introduction to artificial neural networks
6.3.3. Development of a surrogate model with a feed-forward neural network
6.3.4. Combining the developed surrogate model with the Monte Carlo sampling method
6.3.5. Comparison of the obtained results with conventional reliability analysis
6.4. Back analysis
6.4.1. The basics of back analysis and its application
6.4.2. Steps to implement the back analysis
6.4.3. Implementing back analysis for the given example in the MATLAB environment
6.4.4. Evaluating the results
6.1.1. Introduction of surrogate modeling and its application
6.1.2. low- and high-fidelity models
6.1.3. steps to develop a surrogate model based on low- and high-fidelity models
6.2. Example: Ground deformation resulting from tunnel excavation
6.2.1. Problem introduction and description of the random variables involved
6.2.2. Implementing the deterministic form of the given example in the Matlab environment
6.2.3. Conducting reliability analysis on the given example using the Monte Carlo sampling method
6.3. Reliability analysis using surrogate model
6.3.1. Introducing a comprehensive data set of the given example
6.3.2. An introduction to artificial neural networks
6.3.3. Development of a surrogate model with a feed-forward neural network
6.3.4. Combining the developed surrogate model with the Monte Carlo sampling method
6.3.5. Comparison of the obtained results with conventional reliability analysis
6.4. Back analysis
6.4.1. The basics of back analysis and its application
6.4.2. Steps to implement the back analysis
6.4.3. Implementing back analysis for the given example in the MATLAB environment
6.4.4. Evaluating the results
Session 7: Bootstrap Sampling Method
7.1. An introduction to data with limited dispersion
7.2. The necessity to use the bootstrap sampling
7.3. Steps to implement the bootstrap sampling method
7.4. Example: the problem of tunnel excavation with data
7.4.1. Importing data into the MATLAB environment
7.4.2. Using the bootstrap sampling method to solve the problem
7.4.3. Computing the results and comparing them with the conventional Monte Carlo method
7.4. Additional tips on using the bootstrap sampling method
7.4.2. Using the bootstrap sampling method to solve the problem
7.4.3. Computing the results and comparing them with the conventional Monte Carlo method
7.4. Additional tips on using the bootstrap sampling method
Session 8: Importance Sampling and an Introduction to the rt Software
8.1. Exploring the common issues with conventional sampling methods
8.2. Importance sampling
8.2.1. Basics and formulation of importance sampling method
8.2.2. Design point and its application in importance sampling method
8.2.3. Steps to implement importance sampling method
8.2.4. Illustrating an example and solving it with the importance sampling method in the MATLAB environment
8.3. rt software
8.3.1. Modeling the beam deformation example in the rt environment
8.3.2. Solving the example with the FORM method
8.3.3. Solving the example with the importance sampling method
8.4. Exploring other types of importance sampling methods
8.2.4. Illustrating an example and solving it with the importance sampling method in the MATLAB environment
8.3. rt software
8.3.1. Modeling the beam deformation example in the rt environment
8.3.2. Solving the example with the FORM method
8.3.3. Solving the example with the importance sampling method
8.4. Exploring other types of importance sampling methods
(RBO) Session 9: Reliability-based Optimization
9.1. An introduction to conventional optimization problems
9.1.1. Unconstrained optimization problems
9.1.2. Constrained optimization problems
9.1.3. Optimization problems with probabilistic constraints
9.2. Reliability-based optimization
9.2.1. A review of different methods in solving reliability-based optimization
9.2.2. Application of the Monte Carlo sampling method in reliability-based optimization problems
9.3. Example: A truss roof deformation
9.3.1. Problem introduction and description of the random variables involved
9.3.2. Importing the problem into the Microsoft Excel environment
9.3.3. Excel settings to enable optimization
9.3.4. Problem solving in Excel environment
9.4. Further explanations for solving different reliability-based optimization problems
9.2.1. A review of different methods in solving reliability-based optimization
9.2.2. Application of the Monte Carlo sampling method in reliability-based optimization problems
9.3. Example: A truss roof deformation
9.3.1. Problem introduction and description of the random variables involved
9.3.2. Importing the problem into the Microsoft Excel environment
9.3.3. Excel settings to enable optimization
9.3.4. Problem solving in Excel environment
9.4. Further explanations for solving different reliability-based optimization problems
Session 10: Time-variant System Reliability Analysis (Part I)
10.1. Comparison of component and system reliability problems
10.2. Example: Corrosion in sewage pipelines
10.2.1. Description of the problem
10.2.2. Introducing the corrosion model in the pipe
10.2.3. Implementing the deterministic form of the problem in the MATLAB environment
10.3. Reliability model of the problem
10.3.1. Probabilistic characteristics of variables
10.3.2. Utilizing the Monte Carlo sampling method to solve the established reliability problems in the MATLAB environment
10.3.3. Extracting the exceedance probability diagram
10.6. Additional notes regarding the results
10.3.2. Utilizing the Monte Carlo sampling method to solve the established reliability problems in the MATLAB environment
10.3.3. Extracting the exceedance probability diagram
10.6. Additional notes regarding the results
Session 11: Time-variant System Reliability Analysis (Part II)
10.1. Different types of failure modes in sewage pipelines
10.1.1. Bending failure mode
10.1.2. Shear failure mode
10.1.3. Excessive cracking
10.1.4. Cover loss
10.2. System reliability problem
10.2.1. Implementing failure modes in MATLAB environment
10.2.2. Integrating failure modes and formulating the system reliability problem
10.2.3. Solving the system reliability problem using conditional sampling
10.3. Time-variant system reliability analysis
10.3.1. Incorporating time into the formulation of the system reliability problem
10.3.2. Handling time in the system reliability problem
10.3.3. Solving the problem in a general form and generating a three-dimensional diagram of “corrosion-failure probability-time”
10.4. Additional considerations in solving time-variant reliability problems
10.2.2. Integrating failure modes and formulating the system reliability problem
10.2.3. Solving the system reliability problem using conditional sampling
10.3. Time-variant system reliability analysis
10.3.1. Incorporating time into the formulation of the system reliability problem
10.3.2. Handling time in the system reliability problem
10.3.3. Solving the problem in a general form and generating a three-dimensional diagram of “corrosion-failure probability-time”
10.4. Additional considerations in solving time-variant reliability problems
Pre-registration deadline
Pre-registration is currently available, and you can access the pre-registration link at the bottom of this page.
Pre-registration
Enrolling in this comprehensive course on Monte Carlo sampling will not only prepare you extensively for various types of research in this field, but also enable you to become an expert in reliability analysis and probabilistic modeling. Pre-registration for this course is now open. To enroll, please register your details using the following link. Once the class has met the required number of participants, you will receive an email notification to pay the tuition fee and complete your registration. The date, time, and link for attending the lessons, along with other important details, will be communicated to you via email.
Pre-registration
(FAQ) Frequently Asked Questions
What is the cost of the course and how can it be paid?
The tuition fee for this course is 500 USD. Upon pre-registration, the course administrator will email you and provide a deposit slip for payment. You have the option to pay the course fee through bank e-transfer or PayPal.
As a student, the tuition fee for this course is quite high for me. Is it possible to pay in multiple installments?
Certainly! We welcome students to join our courses and embark on the path to becoming professional researchers even during their student years. If you are unable to pay the full tuition upfront, you may request to pay in two or three installments.
If the class schedule doesn’t align with my availability, how can I ensure I don’t miss out on the class?
All classes are recorded, and the video recordings of each week’s classes will be shared with all course participants early next week. This way, even if you are unable to attend the live session online, you can catch up by watching the class video. However, we do recommend attending the class online.
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