Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
Want More Out of your SharePoint Environment? Extend your SharePoint Environm...EPM Live
SharePoint is the most common collaboration tool on the market today. It was common in the early adoption years to see SharePoint implementations that were intended for content management purposes only. Now, IT organizations are thinking outside of the box. If you must do more with less, what better way to protect your current investment and minimize costs then to leverage the same platform to bring Project Management, Product Development and Service Management to your IT organization?
Join us in a one hour webinar to learn how you can extend an underutilized SharePoint deployment into a full Enterprise project, product and service management solution. Topics includes:
- Common Uses of SharePoint
- SharePoint Deployment Best Practices
- SharePoint Recommendations
- Benefits of Centralization
- SharePoint for IT Planning and Control – Projects, Products and Services
"The Metropolis adjusted Langevin Algorithm
for log-concave probability measures in high
dimensions", talk by Andreas Elberle at the BigMC seminar, 9th June 2011, Paris
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Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Unit 8 - Information and Communication Technology (Paper I).pdf
Gaussseidelsor
1. 5.2 Gauss-Seidel and SOR (Successive Over Relaxation)
As motivation return to the example of the last section:
−ui −1 + 3ui −ui +1 = f i .
Jacobi: uim +1 = ( f i + uim 1 + uim 1 ) / 3
− + i = 1,2,...,n.
Since ui-1 m+1 has already been computed and may be closer to the solution, we should
consider using it in place of ui-1 m. This is the Gauss-Seidel method
Gauss-Seidel: uim +1 = ( f i + uim 1 1 + uim 1 ) / 3
−
+
+ i = 1,2,...,n.
For the Gauss-Seidel scheme and the given example, the iterates one and two are
0 1/3 13/27
0 , u1 = 4 / 9 and u 2 = 53/81 .
u =
0
0
13/27
124/243
Gauss-Seidel exhibits faster convergence.
SOR Algorithm (0 < ω < 2)
Note, ω = 1.0 is Gauss-Seidel, ω > 1.0 is over-relaxation, and ω < 1.0 is
under-relaxation. Consider Ax = d or in component form
∑a
j< i
i, j x j + ai ,i xi + ∑ ai , j x j = di .
j> i
for m = 0, maxm
for i = 1, n
xim +1 / 2 = ( di − ∑ ai , j x m +1 − ∑ ai , j x m ) / ai , i
j j
j< i j >i
xim +1 = (1 − ω ) xim + ω xim +1 / 2
test for convergence.
2. 2
Most applications are sparse (many aij = 0). Gauss-Seidel would not be used on
dense matrices because of the number of calculations required for the lower and upper
sums.
Notes: (i) If A SPD and 0 < ω < 2, then we will show that SOR converges.
(ii) ω > 1.0 is used to significantly accelerate convergence
(The choice of ω could be a problem for some application).
(iii) ω =1.0 (for G-S) is much faster than Jacobi.
SOR may be viewed as a matrix FOFD iteration
* 0 0 0 0 0 0 0 0 * * *
0 * 0 0 * 0 0 0 0 0 * *
A = D − L −U = − −
0 0 * 0 * * 0 0 0 0 0 *
0 0 0 * * * * 0 0 0 0 0
Then xim +1 = (1 − ω ) xim + ω (di − ∑ ai , j x m +1 −∑ ai , j x m ) / ai ,i
j j
j <i j >i
a x m +1
i, i i = (1 − ω )a x + ω (di − ∑ ai , j x m +1 −∑ ai , j x m )
m
i, i i j j
j <i j >i
The matrix form is
Dx m +1 = (1− ω ) Dx m + ω (d + Lxm +1 + Ux m )
( D − ω L) x m +1 = (1 − ω ) Dx m + ωUx m + ω d
( D − ω L ) x m +1 / ω = (1 − ω) Dx m / ω + Ux m + d
x m +1 = [( D − ω L) / ω] −1[(1 − ω ) D / ω + U ] x m + [( D − ω L) / ω ]−1 d
x m +1 = Hω x m + [( D − ω L ) / ω ]−1 d
3. 3
We need || Hω || < 1 to qualify as a convergent FOFD. Moreover, we want ϖ such that
|| Hϖ || ≤ || Hω || so that the convergence will be fast.
Example. Two dimensional, point SOR for -∆u = f.
Unknowns are uij ≅ u(ih,jh) h = ∆x = ∆y.
Use the five point finite difference method:
−ui −1, j −ui , j −1 + 4ui , j −ui +1, j − ui , j +1 = h 2 f i , j .
(lower sum + ai,i xi + upper sum)
In order to cast this problem in the form of Ax = d, make the following
identifications:
i → i,j pair ,
xi → ui,j ,
ai,i = 4 ,
aij = -1 for j identified with the grid pairs (i+1,j), (i-1,j),(i,j+1) and (i,j-1),
n → (N-1)2 and
d → h2 fi,j with h = 1/N.
The classical order of nodes starts with the bottom grid row (j = 1) and moves
from left (i = 1) to the right (i = N-1)
# # #
lower↓ * i, j # ↑upper
* * *
4. 4
Point SOR with u(i,j) Overwritten.
for m = 0, maxm
numi = 0
for j = 1, N-1
for i = 1, N-1
utemp = [h*h*fij +u(i-1,j)+u(i,j-1)+u(i+1,j)+u(i,j+1)]*0.25
utemp = (1-ω)*u(i,j) + ω*utemp
error = u(i,j) - utemp
if abs(error) < ε numi = numi + 1
u(i,j) = utemp
if numi = (N-1)*(N-1) break
See MA402 chapters 3,4 for MATLAB and Fortran 90 implementations.
Example. Two dimensional block SOR.
B −I U1 F1
A = −I U = h2 F
B −I 2 2
−I B U 3
F3
4 −1 u11
−1 4 −1 ,
B = U1 = u21 ,
I3x3
−1 4
u31
−U i −1 + BU i − U i +1 = h 2 Fi .
Here i represents the i grid row (an abuse of notation)
Block component form iss A = [Aij] where A is (N-1)x(N-1) block matrix.
5. 5
AX = F
∑A X
j< i
ij j + Aii X i + ∑ Aij X j = Fi where Aii are tridiagonal matrices.
j> i
Block G-S algorithm.
for m = 1, maxm
for i = 1,N-1
Solve Aii X im +1 = Fi − ∑ Aij X m +1 − ∑ Aij X m
j j
j< i j> i
test for convergence.
Example. Three dimensional point SOR for -∆u = f.
This is similar to the 2D point SOR with an additional loop in z direction.
The 7 point finite difference equation is
−ui −1, j , l −ui , j− 1,l −ui , j ,l −1 + 6ui , j ,l −ui +1, j ,l − ui , j+ 1,l − ui , j ,l + 1 = h 2 f i ,j ,l .
The point SOR scheme will now have three inner nested loops. The coefficient
matrix will be block tridiagonal with the diagonal blocks having a similar
structure as the 2D coefficient matrix.