This document discusses stochastic models for site characterization. It describes several continuous models for generating random fields including the multivariate normal method, LU decomposition method, and turning bands method. The multivariate normal method models a random vector as having a multivariate normal distribution defined by a mean vector and covariance matrix. The LU decomposition method generates a random field with a given covariance structure by decomposing the covariance matrix into lower and upper triangular matrices. It provides numerical examples of applying the LU decomposition method to generate correlated random variables at two points.
Representation of of Stochastic Processes in Stochastic Processes in Real and Spectral Domains Real and Spectral Domains and and Monte Monte-Carlo sampling
Representation of of Stochastic Processes in Stochastic Processes in Real and Spectral Domains Real and Spectral Domains and and Monte Monte-Carlo sampling
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
21st Mediterranean Conference on Control and Automation
The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.
Solving boundary value problems using the Galerkin's method. This is a weighted residual method, studied as an introduction to the Finite Element Method.
This is a part of a series on Advanced Numerical Methods.
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
21st Mediterranean Conference on Control and Automation
The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.
Solving boundary value problems using the Galerkin's method. This is a weighted residual method, studied as an introduction to the Finite Element Method.
This is a part of a series on Advanced Numerical Methods.
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, risk theory, queuing, and supply chain/social/multi-agents networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and propose two alternative importance sampling (IS) approaches [1,2] to improve the Monte Carlo (MC) estimator efficiency. The key challenge in the IS framework is to choose an appropriate change of probability measure to achieve substantial variance reduction, which often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control formulation. We pursue two alternative approaches to mitigate the curse of dimensionality when solving the resulting dynamic programming problem. In the first approach [1], we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. As an alternative, we present in [2] a dimension reduction method, based on mapping the problem to a significantly lower dimensional space via the Markovian projection (MP) idea. The output of this model reduction technique is a low dimensional SRN (potentially one dimension) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained via a discrete $L^2$ regression. By solving a resulting projected Hamilton-Jacobi-Bellman (HJB) equation for the reduced-dimensional SRN, we get projected IS parameters, which are then mapped back to the original full-dimensional SRN system, and result in an efficient IS-MC estimator of the full-dimensional SRN. Our analysis and numerical experiments verify that both proposed IS (learning based and MP-HJB-IS) approaches substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators. [1] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. Learning-based importance sampling via stochastic optimal control for stochastic reaction net-works. Statistics and Computing 33, no. 3 (2023): 58. [2] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. To appear soon.
Chemical dynamics and rare events in soft matter physicsBoris Fackovec
Talk for the Trinity Math Society Symposium. First summarises the approximations leading from Dirac equation to molecular description and then the synthesis towards non-equilibrium statistical mechanics. The relaxation approach to projection of a molecular system to a Markov jump process is discussed.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Generating a high quality Chaotic sequence is crucial to the success of the Superefficient Monte Carlo Simulation methodology. In this slides, we discuss how to numerically generates Chebychev Chaotic Sequence with arbitrary precision, and proposed a highly efficient parallel implementation.
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
Simulation of Tracer Injection from a Well in a Nearly Radial FlowAmro Elfeki
This is a result of Simulation of Tracer Injection from a well in a nearly radial flow using finite difference and particle tracking Radom walks.
To observe the animation of the plume progress in time, one has to download the file.
Aquifer recharge from flash floods in the arid environment: A mass balance ap...Amro Elfeki
Estimation of the infiltration/natural recharge to groundwater from rainfall is an important issue in hydrology, particularly in arid regions. This paper proposes the application of The Natural Resources Conservation Service (NRCS) mass balance model to develop infiltration (F)–rainfall (P) relationship from flash flood events. Moreover, the NRCS method is compared with the rational and the Ф-index methods to investigate the discrepancies between these methods. The methods have been applied to five gauged basins and their 19 sub-basins (representative basins with detailed measurements) in the southwestern part of Saudi Arabia with 161 storms recorded in 4 years. The F–P relationships developed in this study based on NRCS method are: F = 39% P with R2 = 0.932 for the initial abstraction factor, λ = 0.2. However, F = 77% P with R2 = 0.986 for λ = 0.01. The model at λ = 0.01 is the best to fit the data, therefore, it is recommended to use the formula at λ = 0.01. The results show that the NRCS model is appropriate for the estimation of the F–P relationships in arid regions when compared with the rational and the Ф index methods. The latter overestimates the infiltration because they do not take λ into account. There is no significant difference between F–P relationships at different time scales. This helps the prediction of infiltration rates for aquifer recharge at ungauged basins from monthly and annual rainfall data with a single formula.
Basics of Contaminant Transport in Aquifers (Lecture)Amro Elfeki
This is a basic lecture on contaminant transport in aquifers. It covers various aspects. Types of transport in aquifers. Reactive and non-reactive, governing equations of solute transport. Method of solutions and simulations.
This is a lecture on well hydraulics. The basics of flow towards the well in confined and unconfined aquifers. Well interactions. Method of images. Flow nets in case of multiple wells. Superposition theory for multiple wells.
This is a lecture on the hydraulics of gradually varied flow in open channels. It shows the profiles common in the open channels and some numerical examples using numerical integration.
Two Dimensional Flood Inundation Modelling In Urban Area Using WMS, HEC-RAS a...Amro Elfeki
This research presents a two-dimensional flood inundation modelling in urbanized areas when some features such as roads, buildings, and fences have great effect on flood propagation. Wadi Qows located in Jeddah City, Saudi Arabia was chosen as case study area because of the flood occurrence of 2009 causing lots of losses either economic or loss of life. The WMS and HEC-RAS program were used for a hydraulic simulation based on channel geometry built by incorporating urban features into DEM using GIS effectively. A resampling method of DEM 90 × 90 m become 10 × 10 m grid cell sizes was conducted to produce a higher resolution DEM suitable for urban flood inundation modelling. The results show that a higher resolution leads to increasing the average flood depth and decreasing the flood extent. Although the change of the grid cell sizes does not affect its elevation values, this approach is helpful to perform flood simulations in urban areas when high resolution DEM availability is limited. In addition, the integration of WMS, HEC-RAS and GIS are powerful tools for flood modelling in rural, mountainous and urban areas.
https://www.researchgate.net/publication/330004725_Two_Dimensional_Flood_Inundation_Modelling_in_Urban_Areas_Using_WMS_HEC-RAS_and_GIS_Case_Study_in_Jeddah_City_Saudi_Arabia_IEREK_Interdisciplinary_Series_for_Sustainable_Development
Development of Flash Flood Risk Assessment Matrix in Arid Environment: Case S...Amro Elfeki
Risk indices and risk matrix have been used among governmental agencies for assessing risks and ranking alternatives protection measures. The popularity of risk matrices can be associated with its characteristics to quick assessing risk and providing inexpensive solutions. The risk assessment is associated with flood protection as: economic, environmental, and life-safety. Economic risks are reasonably well dealt with by the well-known traditional cost-benefit analysis, insurance, and financial markets. Environmental risks are difficult to assess by traditional methods in flood project evaluation. Environmental consequences cannot be directly measured, while social risks represent the most challenge to quantify. It may be possible to estimate the number of fatalities, and cost of damaged infrastructures, while the social aspects cannot be measured. This paper is proposing a flood risk matrix technique for assessing risks in urban arid and extreme arid regions demonstrated through case-study application on the catchment of Taibah University (TU) and Islamic University (IU) in Medina, KSA. The study focused on assessing the geographical impacts of the flash flood inundated depths for different frequencies from 5 to 100 years return periods leading to identification of the flood channels, and its floodplains that may be vulnerable to different degrees of flood hazards.
Derivation of unit hydrograph of Al-Lith basin in the south west of saudi ar...Amro Elfeki
Mohammad Albishi, Jarbou Bahrawi and Amro Elfeki (2016). Derivation of the Unit Hydrograph of Allith Basin in the South West of Saudi Arabia. 7 International Conference on Water Resources and the Arid Environments (ICWRAE 7): 621-628 4-6 December 2016, Riyadh, Saudi Arabia.
Abstract:
Most studies on unit hydrograph theory is developed for temperate regions and to the best of the authors’ knowledge, there is no studies on this topic in arid regions because of the lack of runoff measurements. This paper presents the derivation of a unit hydrograph of a Allith basin and its S-curve in the south western part of Saudi Arabia to be used to predict flash flood more accurately in this region. The derivation is based on the method of stream flow data that has been collected from measured rainfall and runoff storms in the region. The study resulted in the unit hydrograph of 1 hr duration and the S-curve that is used to transfer the hydrograph to any other durations. This unit hydrograph can be used to predict flash floods in Allith basin and similar watersheds.
Empirical equations for flood analysis in arid zonesAmro Elfeki
Mohammad Albishi, Jarbou Bahrawi, and Amro Elfeki (2016). Empirical Equations for Flood Analysis in Arid Zones. Published in the book of abstracts at IWC 2016 International Water Conference 2016 on Water Resources in Arid Areas: the Way Forward.
Empirical equations for estimation of transmission lossesAmro Elfeki
Transmission losses is caused by infiltration into the streambed in ephemeral streams. The conventional methods for flood routing in wadis is impossible to achieve due to transmission losses. The Muskingum routing procedure in its basic form has two parameters, the channel time lag, Km, and the weighting parameter, x. However, both parameters do not consider transmission losses of floods in channels. O’Donnell 1985 introduced a third parameter, α, in the continuity equation to allow for the lateral movement of floodwater. Elfeki et. Al (2014) carry some modifications of the assumptions such that the negative sign of the parameter α represents transmission losses. In this research, the third-parameter α has been investigated as a tool for estimation of transmission losses using data from Yiba catchment in the Kingdom of Saudi Arabia. A spreadsheet model will be developed to deduce the equations.
Proceedings of the Second International Symposium on Flash Floods in Wadi Systems: Disaster Risk Reduction and Water Harvesting in the Arab Region (2016): 166-170
Representative elementary volume (rev) in porousAmro Elfeki
This presentation shows the concept of Representative elementary volume known in porous media. The presentation is the results of an excel spreadsheet prepared to generate hypothetical porous media in 2D. It calculates the porosity of the medium at different scales until it reaches the rev value.
Merging Heterogeneous Structures at Various Scales by Way of Tree-indexed Mar...Amro Elfeki
Elfeki, A.M., Dekking, F.M., Kraaikamp, C., Bruining, J. and Verlaan, M.L.(1998). Merging Heterogeneous Structures at Various Scales by Way of Tree-indexed Markov Chains. Poster Presentation at the Gordon Research Conferences, Modeling of Flow In Permeable Media, Andover, New Hampshire, USA. (Augest 2-7, 1998).
Modeling Adsorption Kinetics of Chemically Interactive Porous Sediments by a ...Amro Elfeki
Elfeki, A.M., Bruining, J., Dekking, F.M., Kraaikamp, C. and Uffink, G. (2002). Modeling Adsorption Kinetics of Chemically Interactive Porous Sediments by a Two-State Random Walk Particle Method. Poster Presentation on the third meeting on DIOC “Water” mini-symposium on 7th November, 2002..Faculty of Civil Engineering and Geosciences, TU Delft, The Netherlands.
New Approach for Groundwater Detection Monitoring at Landfills. Amro Elfeki
Yenigul, N.B., Elfeki, A. M. and den Akker, C. (2006). New Approach for Groundwater Detection Monitoring at Landfills. Journal of Groundwater Monitoring and Remediation, 26, no. 2/Spring 2006/pp. 79-86.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
4. Multivariate Normal MethodMultivariate Normal Method
The multi-variate normal density function Np(µ,C), p-dimensional normal
random variate, is given by,
( ) ( )⎥
⎦
⎤
⎢
⎣
⎡
µZCµZ
C
Z ---
π||
=f -T
p//
1
221
2
1
exp
)2(
1
)(
where, Np(µ,C) is a multi-variate normal distribution with mean µ, and
covariance matrix C,
p is the number of parameters (nodes of the model),
Z = {Z1, Z2,..., Zp}T, p-dimensional random vector, (p*1),
µ = {µ1, µ2,…, µp}T , p-dimensional mean values vector, (p*1),
T is superscript transpose operation of a matrix,
-1 superscript is inverse operation of a matrix,
C is a pxp covariance matrix given by,
6. Correlation MatrixCorrelation Matrix
The correlation coefficient ρij
σσ
ρ
ZZ
ji
ij
ji
)Z,ZCov(
=
The correlation matrix R
1...
.....
..1..
...1
..1
1
21
112
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
ρ
ρ
ρρ
=
p
p
R
7. General Technique for Generation ofGeneral Technique for Generation of
Multivariate Distribution (CD Approach)Multivariate Distribution (CD Approach)
Let Z = {Z1,Z2,Z3,...Zp}T be the p-dimensional random vector of interest.
)()()...,(),()( 11212112121 zpz|zpz,...z|zpz,...z,z|zpz,...,z,zp p-pn-p-pp −=
The conditional distribution approach involves the following steps:
(1) generate Z1 = z1 from the marginal distribution of Z1 (uni-variate
distribution of Z1);
(2) generate Z2 = z2 from the conditional distribution of Z2 given Z1 = z1;
(3) generate Z3 = z3 from the conditional distribution of Z3 given Z1 = z1
and Z2 = z2...
and so forth through the p steps.
8. Comparison between MultiComparison between Multi--variatevariate StatisticalStatistical
Theory and Random Field TheoryTheory and Random Field Theory
•• Statistical Theory:Statistical Theory:
EachEach variatevariate is considered as a component of ais considered as a component of a
random vector and the covariance matrixrandom vector and the covariance matrix
gives the dependences between componentsgives the dependences between components
of the random vector.of the random vector.
•• Random Field Theory:Random Field Theory:
Each node value in the field is considered as aEach node value in the field is considered as a
component of a random vector. Thecomponent of a random vector. The
dependences between node values aredependences between node values are
described by autodescribed by auto--covariance function.covariance function.
9. Example of CD Approach(1)Example of CD Approach(1)
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
The joint probability of the lattice process can be expressed mathematically as,
)Pr( 11212111 S=Z...,S=Z,S=Z,S=Z,...,S=Z,S=Z pN,Nqi,j-l,ji-ki,j,, yx
where,
Sk is a state of cell (i,j), which is one of the n states describing the geological
system,
Nx is the maximum number of cells in the horizontal direction,
Ny is the maximum number of cells in the vertical direction.
10. Example of CD Approach(2)Example of CD Approach(2)
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
)Pr()Pr(
)Pr(
)Pr(
)Pr(
)Pr(
111111212
11121232122
11111
11111
11212111
S=Z.S=Z|S=Z
.S=Z,S=Z,S=Z|S=Z
...S=Z,...,S=Z,S=Z|S=Z
...S=Z,...,S=Z,S=Z|S=Z
=S=Z...,S=Z,S=Z,S=Z,...,S=Z,S=Z
,,,
,,,f,
,ri,j-l,ji-ki,j
,t-N,NqN,-NpN,N
pN,Nli,j-q,ji-ki,j,,
yxyxyx
yx
where, Pr(z1,1 =S1) is the marginal probability of state S1
11. Example of CD Approach(3)Example of CD Approach(3)
i-1,j i,j
i,j-1
1,1
Nx,Ny
Nx,1
1,Ny
Nx,j
Introducing some nearest neighbour property according to Markov chain theory,
)Pr()Pr(
)Pr(
)Pr()Pr()Pr(
)Pr()Pr(
)Pr(
111111221
111
111212111221312422
1111
11212111
S=Z.S=Z|S=Z
...S=Z|S=Z
.S=Z|S=Z...S=Z|S=Z.S=Z,S=Z|S=Z
...S=Z,S=Z|S=Z...S=Z,S=Z|S=Z
=S=Z...,S=Z,S=Z,S=Z,...,S=Z,S=Z
,,,
x-N,rN,
,,g,-Nd,N,,,
fi,j-t,ji-ki,ja-N,NlN,-NpN,N
pN,Nqi,j-l,ji-ki,j,,
yy
xx
yxyxyx
yx
12. LU Decomposition Method (1)LU Decomposition Method (1)
,
,
i
j
X
Y
0
Z
Z
sij
1
p
,
,
,
2 3
,
The algorithm for generating random fields with a given covariance structure
based on the covariance matrix of the system is as follows:
1) Build the covariance matrix C of the system. The elements of C are denoted
by,
)( Z,Z= Covc jiij
and if i=j the covariances becomes the variances.
...),(
.....
....
...),(
),(..),(
2
1
2
2
12
121
2
2
1
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
p
i
Zp
Z
Z
pZ
ZZCov
ZZCov
ZZCovZZCov
σ
σ
σ
σ
C
13. LU Decomposition Method(2)LU Decomposition Method(2)
,
,
i
j
X
Y
0
Z
Z
sij
1
p
,
,
,
2 3
,
In case of stationary random field, the elements of the covariance matrix are
given as:
)s(=c ij
2
Zij ρσ
σ2
Z is the variance of the process Z,
ρ(sij) is the auto-correlation function, and
sij is the distance vector between point i and point j.
)y-y(+)x-x(=s
2
ji
2
jiij
For pairs of values Zi and Zj with i = 1,...p and j = 1,...p
determine (xi - xj) and (yi - yj)
where, (xi,yi) are the coordinates of point Zi and point (xj,yj) are the
corresponding coordinates for point Zj. The distance sij between two points is,
15. Properties of AutoProperties of Auto--correlation Matrixcorrelation Matrix
1) All the diagonal elements are equal
to one, i.e. correlation between the
point and itself is perfect (complete
correlation).
2) If ρij = 0, this means no correlation
between i and j.
3) All the off-diagonal elements are
called autocorrelation coefficients
and they are less than one.
4) The autocorrelation matrix is
symmetric, i.e., ρij = ρ ji.
5) According to the stationarity
assumption:
ρ12 = ρ23 =...= ρp-1p,
ρ13 = ρ 24 =...= ρp-2p,
and so on.
16. LU Decomposition Method (3)LU Decomposition Method (3)
2) One has to decompose the covariance matrix by the Cholesky factorization
method (Square-Root method),
C = L U
where,
L is a unique lower triangular matrix,
U is a unique upper triangular matrix, and
U is LT , i.e., U is the transpose of L.
1
1
1
1
i1
i1
11
1/2
i
2
ii ii ik
k
i
ij ij ik jk
kjj
ij
c
= ,1 i pl
c
= - ,1 i pl c l
1
= - ,1< j <i pl c l l
l
=0 , i < j pl
−
=
−
=
≤ ≤
⎡ ⎤
≤ ≤⎢ ⎥
⎢ ⎥⎣ ⎦
⎡ ⎤
≤⎢ ⎥
⎣ ⎦
≤
∑
∑
17. LU Decomposition Method (4)LU Decomposition Method (4)
3) Generation of normally distributed p-dimensional sequence of independent
random numbers with zero mean and unit standard deviation N(0,1) which can
be expressed as,
T
1 2{ , ,..., }pε ε ε=ε
where, ε is vector of normally distributed random numbers, and εi is the i-th
random number drawn from N(0,1).
4) Multiplication of the independent random vector ε by the triangular matrix U
to get a vector of auto-correlated random numbers. This vector can be
expressed by matrix multiplication convention as,
X = U ε
where, X is a vector of multi-variate normal random Np(0,I), 0 is zero mean
vector (p*1), and I is the identity matrix (p*p).
Z = µ + X
21. Nearest Neighbour MethodNearest Neighbour Method
Whittle’s Model [1954],
∑≠ij
ijiji ε+ZW=Z
Zi is a random variable satisfying the nearest neighbour relation,
εi is uncorrelated normal random number with E(εi) = 0, and Var(εi) = σi
2 ,
i=1,2,...p, and
Wij are weighting coefficients.
Anisotropic first-order auto-regressive (Smith and Freeze [1979b])
ε+Z+Zα+Z+Zα=Z ijij+ij-yji+ji-xij )()( 1111
α x is an auto-regressive parameter expressing the degree of dependence of
Zij on its two neighbouring values Zi-1j and Zi+1j, (|αx|<1), and
α y is an auto-regressive parameter expressing the degree of dependence of
Zij on its two neighbouring values Zij-1 and Zij+1, (|α y| < 1).
i,j
i,j+1
i+1,j
i,j-1
i-1,j,,
,
,
,
N=2 N=3
N=4
22. Nearest Neighbour Method (1)Nearest Neighbour Method (1)
=Z W Z + ε
W is called the p*p connectivity matrix, or the p*p spatial lag operator of scaled
weights, wkl.
The elements of the connectivity matrix wkl are defined as,
i,j
i,j+1
i+1,j
i,j-1
i-1,j,,
,
,
,
N=2 N=3
N=4
N
w
=w
*
kl
kl
where, k = 1,2,...p, l = 1,2,...p, and k ≠ l,
w*
kl = α x if the blocks k and l are contiguous in the x-direction,
w*
kl = α y if the blocks k and l are contiguous in the y-direction, and
w*
kl = 0 otherwise, i.e., if k = l, or if blocks k and l are not contiguous, and
N is the total number of blocks surrounding block k, i.e.,
N=4 if block k is located inside the domain,
N=3 if block k is located on the boundary of the domain, and
N=2 if block k is located at a corner of the domain.
23. Nearest Neighbour Method (2)Nearest Neighbour Method (2)
+Z = W Z ε
Z ~ 0 and σZ.
ε ~ 0 and σε.
To simulate the predetermined standard deviation σz :
- Start from a random vector ε with σε =1.
- ε is pre-multiplied by an appropriate factor η to yield σz .
η+Z = W Z ε
Solution
( ) η= -1
Z I - W ε
Determination of η
{ }
2
. T
Z
E
σ
=
Z Z
R
24. Nearest Neighbour Method (3)Nearest Neighbour Method (3)
2 2
2
1 1 1
2
2
2
2
1
( ) (( ) ) (( ) ( ) )
1
1
,
1
1. ( )
ε
Z
- - T T -
ε
Z
Z
Z
m
η σ
σ
σ
η
σ
Taking the trace of a matrix
p tr η
σ
σ
η
V
= =
=
=
=
R = V .
V I - W . I - W I - W . I - W
R = V
V
Vm = tr V/p, and the symbol "tr" is the trace of the matrix, tr (V) = Σ vii , i= 1,…,p
25. Nearest Neighbour Method (4)Nearest Neighbour Method (4)
1
1
( )
' ( )
Z
m
Z
z
m
σ
V
σ
V
−
−
= +
Z = I - W ε
Z µ I - W ε
The analysis of the covariance function describing the generated random field
with first-order dependence is approximately an exponential decay function.
The advantage is: at the beginning of any simulation the matrix (I - W) must be
inverted only once. For each realization of the process Z, the inverted matrix (I -
W)-1 is simply multiplied by the generated random vector ηε.
The drawback of this method is computing the inverse matrix.
27. Turning Bands Method(TBM)Turning Bands Method(TBM)
The TBM was first proposed by Matheron [1973] and applied by the school of
mines in Paris.
Its basic concept is to transform a multidimensional simulation into the sum of a
series of equivalent uni-dimensional simulations
28. TBM ProcedureTBM Procedure
TBMTBM is a repetition of a two steps:is a repetition of a two steps:
1. a realization of a random process with a prescribed auto1. a realization of a random process with a prescribed auto--
covariance function and zero mean is generated on one line.covariance function and zero mean is generated on one line.
-- TheThe CholeskyCholesky decomposition method can be used (but withdecomposition method can be used (but with
much smaller correlation matrix dimensions) ormuch smaller correlation matrix dimensions) or
-- AutoAuto--regression methods, like nearest neighbour.regression methods, like nearest neighbour.
2. Orthogonal projection of the generated line process to each p2. Orthogonal projection of the generated line process to each pointoint
in the simulated twoin the simulated two-- or threeor three--dimensional random field.dimensional random field.
The two steps are repeated for a given number of lines and thenThe two steps are repeated for a given number of lines and then aa
final value is assigned to each grid point in the field by takinfinal value is assigned to each grid point in the field by taking ag a
weighted average over the total number of lines.weighted average over the total number of lines.
29. Background of the MethodBackground of the Method
Let Zi(u), i = 1,...L a set of N independent realizations of a one-dimensional,
second order stationarity stochastic process on a line u with an auto-correlation
function ρ1(uo), where uo is the spatial lag on the line.
The values given by the relation,
uZ
L
=x,y,zZ
L
i
is ∑=1
)(
1
)(
is a realization of a two- or three-
dimensional process with zero
mean.
The subscript s represents the
term " simulated " or " synthetic ".
[ ])()(:3 1 uu
du
d
=uρcaseD oo
o
o ρ−
)(:2
0
s
s
s
ρ
πρ
2
=
)u-(
du)u(
caseD
2
o
2
oo1
∫−
30. Spectral Turning Bands Method (STBM)Spectral Turning Bands Method (STBM)
To circumvent the difficulty, an expression for the spectral density function of
the one-dimensional processes as a function of the radial spectral density
function of the two-dimensional process is used.
This expression is given in a Fourier space by,
2
1( ) ( )
2
Z
SS
σω = ω
The spectral density function of
the uni-dimensional process
S1(ω) along the turning bands
lines is given by one half of the
radial spectral density function
S(ω) of the two-dimensional
process multiplied by the
variance of the two-dimensional
process.
31. Implementation of the STBM in 2D (1)Implementation of the STBM in 2D (1)
(1) Generation of One-dimensional Uni-variate Process on The Turning Bands
Line: (Standard Fourier Integration method)
∑∫ ≈
+=
ω
ω
ω
ω
ωω
all
j
ui
all
ui
dWedWe=uX
uiYuZuX
j
)()()(
)()()(
X is the sum of a complex series of sinusoidal functions of varying wavelength,
each magnified by complex random amplitude with zero mean dW(ωj), ωj=j.∆ω.
∑
∫
=
+=
+==
M
j
jji
all
j
udWuZ
udWuZuX
1
)cos()()(
)cos()()()}(Re{
φωω
φωω ω
ω
where, φj represents independent random angles which is uniformly distributed
between 0 and 2π,
M is the number of harmonics used in the calculations, ωj =(j-.5) ∆ω, j =1,2,...M,
and ∆ω is the discretized frequency which is given by ωmax/M, and ωmax is the
maximum frequency used in the calculations.
32. Implementation of the STBM in 2D (2)Implementation of the STBM in 2D (2)
1( ) 4 ( ).j jdW Sω ω ω⎡ ⎤= ∆⎣ ⎦
where, S1(ωj) is the spectral density function of the real process Z(u) on the line.
S1(ω) is assumed to be insignificant outside the region [- ωmax,+ ωmax]
1
1
'
1
1
( ) 2 ( ). cos( )
( ) 2 ( ). cos( )
j
j
M
i j j
j
M
i j j
j
Z u S u
Z u S u
ω ω ω φ
ω ω ω φ
=
=
⎡ ⎤= ∆ +⎣ ⎦
⎡ ⎤= ∆ +⎣ ⎦
∑
∑
where, ω 'j = ωj + δω
The frequency δω is a small random frequency added here in order to avoid
periodicities. δω is uniformly distributed between - ∆ω’ /2 and ∆ω’ /2, where, ∆ω’
is a small frequency, ∆ω’<<∆ω. ∆ω’ is taken equal to ∆ω /20 according to
Shinozuka and Jan [1972]
33. Implementation of the STBM in 2D (3)Implementation of the STBM in 2D (3)
( )
1/ 222
2
2
3/ 22 2 2 2
2 .exp. var
( ) exp
( )
2 1
yx
Z
x y
Z x y
x x y y
D Anistorpic Co iance
ss
Cov
Spectrum
S
σ
λ λ
σ λ λ
π λ ω λ ω
−
⎡ ⎤⎛ ⎞⎧ ⎫⎧ ⎫ ⎪ ⎪⎢ ⎥⎜ ⎟= − +⎨ ⎬ ⎨ ⎬⎢ ⎥⎜ ⎟⎪ ⎪⎩ ⎭ ⎩ ⎭⎝ ⎠⎢ ⎥⎣ ⎦
=
+ +
s
ω
34. Implementation of the STBM in 2D (4)Implementation of the STBM in 2D (4)
(2) Distribution of The Turning Bands Lines and The Number of Lines:
Random orientation versus evenly spaced (it converges faster)
8-16 lines are satisfactory choice in case of isotropic correlation.
(3) Spectral Discretization:
∆ω must be kept small enough.
M must be kept large enough
(M∆ω >50).
Mantoglou and Wilson [1982] WRR
(4) Physical Discretization:
∆u < min {∆x, ∆y, ∆z}
(5) Length of The Turning Bands
Lines:
Minimum length is determined by the
orientation of the line and the domain
size.
35. Spectral Turning Bands Method:Spectral Turning Bands Method:
Projection from many linesProjection from many lines
36. Comparison Between Various MethodsComparison Between Various Methods
Item
Method
〈K 〉
m/day
σK
m/day
〈 Y 〉 σY
-0.8 1.3
1.3
1.3
-0.8
-0.8
λx
M
λy
m
NNG 1 2 αx =.98
1.2
1.2
1.2
αy =.50
0.73
MVG 1 2 0.73
TBG 1 2 0.73
Monte-Carlo=100
Domain dimensions =15 ×15 ms
Domain discretization = 1 ×1 ms
38. MosaicMosaic FaciesFacies ((Discrete) ModelsDiscrete) Models
In this approach one is aiming to construct
- formation geological units, its geometric characteristics.
- lithologies.
- units dimensions (length, thickness, and width),
- orientations and frequency of occurrence, etc..
Types of discrete models:Types of discrete models:
•• ObjectObject--based Models.based Models.
•• Sequential based Models:Sequential based Models:
--Markov Chains in 1Markov Chains in 1--D, 2D, 2--D etc.D etc.
--Markov Random Fields.Markov Random Fields.
--TruncatedTruncated GaussianGaussian Fields.Fields.
--Sequential Indicator Simulation Models.Sequential Indicator Simulation Models.
--Random Lines Models.Random Lines Models.
--Random Sets.Random Sets.
39. ObjectObject--based Modelsbased Models
This model consider only two states (like sand and shale formation).
Two parameters are considered:
1. The density of the random objects per unit of volume.
2. Statistical distribution of the sizes of the objects.
41. Theory of OneTheory of One--dimensional Markov Chaindimensional Markov Chain
SS S
i0 1 i+1i-1 N2
l k q
,:)Pr(
)Pr(
pSZ|SZ
SZ,...,SZ,SZ,SZ|SZ
lkl1-iki
p0r3-in2-il1-iki
===
======
...
.....
....
....
..
1
21
11211
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
nnn
lk
n
pp
p
p
ppp
p
1,...,0
1
pp
n
k
lklk =≥ ∑=
wp k
N
lk
N
=
∞→
)(
lim
1,0
...,
1
1
=≥
==
∑
∑
=
=
n
k
kk
klkl
n
l
ww
n,1k,wpw
Transition prob.
Marginal prob.
61. Markov Random Field ModelMarkov Random Field Model
- Originally developed for image processing.
- Similarity between image description and reservoir description.
- The method does not use variogram or auto-correlation to describe
the relation between the neighbouring locations but it is based on the
theory of conditional probabilities.
Procedure (Simulated annealing or Metropolis algorithm):
1. The states in the system are generated by arbitrary distribution over
the lattice.
2. Two grid points are selected at random from the lattice.
3. Simple exchange of the grid points states based on conditional
probabilities.
4. After each trial, a new permutation of the states of the grid points is
created.
5. The procedure continues iteratively until the marginal and the
transition probabilities of the states in the system is stabilized.
63. Random Lines ModelRandom Lines Model
Poisson Random Lines Model
Switzer [1965].
The method generates random lines
from Poisson distribution on a circle.
The lines are used to represent
boundaries between different soils.