Iterative Methods - Science topic

Under the theme “Mathematics for All, Mathematics as a Path to Development,” one of the best examples of math's impact on daily life is its application in medical imaging, such as CT and CBCT scanners.

How do these devices work?

Mathematics is at the heart of it! When X-rays are emitted from a source, they pass through tissues and are captured by detectors. Along their path, these rays undergo absorption or attenuation, which depends on the properties of the tissue they traverse. Each ray's path represents a unique equation, where the unknowns are the attenuation values of the points it passes through.

As the source rotates around the object, it generates multiple paths, creating a system of n equations with n unknowns. Sophisticated software solves this complex system using advanced algorithms, such as the Radon Transform or iterative methods, to reconstruct the internal structure of the object as a detailed 3D image.

How does this contribute to development?

Mathematics is the backbone of such groundbreaking technologies, enabling precise diagnostics and effective treatment planning. This practical application of equations demonstrates how math can transform our world.

Now it’s your turn:

What other applications of mathematics in medicine and engineering do you think are underexplored? How can we further harness the power of math as a tool for development?

Let’s explore the incredible role of mathematics in everyday life together in this discussion!

The importance of finding a powerful numerical solution for the Laplacian matrix A in the Laplace and Poisson partial differential equations is obvious.

The classical conventional numerical procedure for solving the Laplace partial differential equation is based on the discretization of the 1D, 2D, 3D geometric space into n equidistant free nodes and on the use of the finite difference method FDM supplemented by the Dirichlet boundary conditions (vector b) to obtain a system of algebraic equations of order n prime in the matrix form A,

A . U(x,y,z,t) = b

The solution to the above equation is U(x,y,z,t) = A^-1.b which is often quite complicated since the matrix A is singular.

It is worth mentioning that common numerical iteration methods such as Gaussian elimination and Gauss-Seidel methods are complicated and require the use of ready-made algorithms such as those in Matlab or Python..etc .

We assume that there exists another SIMPLE statistical numerical solution expressed by:

A^-1=(I-A)^-1=A^0+A+A+A^2+....+A^N

Where A^0=I and N is the number of iterations or time steps dt.

As simple as that!

I have a large sparse matrix A which is column rank-defficient. Typical size of A is over 100000x100000. In my computation, I need the matrix W whose columns span the null space of A. But I do not know how to fastly compute all the columns of W.

If A is small-scale, I know there are several numerical methods based on matrix factorization, such as LU, QR, SVD. But for large-scale matrices, I can not find an efficient iterative method to do this.

Could you please give me some help?

For large-scale matrix Aand vector v, I want to to compute R^{-1]v, where A=QR is the QR factorization of A. The explicit computation of QR and R^{-1} is unaffordable.

Is there any iterative method to approximate R^{-1]v implicitly?

I need some papers links in which we have a solution for system of first order odes using the Laplace variational iteration method (VIM)?

Which controller is best in trajectory tracking, i am getting same response.

please share your experience

Suppose there is a linear system of equation Ax =b, x is required. A is a large symmetric matrix.

Which method is faster to compute x?

1. If I solve it through the calculation of inverse of A and compute x = A^-1b.

2. If I solve the linear equations through some iterative method (like Gauss-elimination, Gauss-seidel).

I am working on a numerical problem that involves large banded matrices, I am trying to solve the problem using an ADI scheme however the problem is really sensitive to the acceleration parameters I use (i.e. most choices lead to $\rho(T_{w_n})>1$), the problem tends to converge fast for the first iterations but diverges afterwards.

I tried to use some of the values of $w_n$ I found in literature and the one provided in Ref. [1]. seems to be the closest fit for my problem but still leads to a $\rho(T_{w_n})>1$.

For small problems I can use the power iteration and the Simulated Annealing proposed by Koutserimpas et al. [2] (since their suboptimal optimization parameters don't seems to work for me and leads immediately to a diverging result), but it becomes impractical for larger ones.

In Ref. [2] a linear dependence on the discretization for $w_n$ is proposed but I am not sure if it is a general rule or a quirk in their simulations. Since I was working with preconditioner on a multigrid as an alternative to the ADI scheme I was curious to know if I could extrapolate a general relationship between the spectral radius of a matrix for a discretization of $\Omega_h$ and it's upscaled smoothed version for $\Omega_{2h}$.

Is the linear dependence found in Ref. [2] a coincidence for their problems? Or is there some theorem I can use?

[1] Hadley, G. Ronald. "Numerical simulation of reflecting structures by solution of the two-dimensional Helmholtz equation." Optics letters 19.2 (1994): 84-86.

[2] Koutserimpas, Theodoros T., Aristeides D. Papadopoulos, and Elias N. Glytsis. "Applicability and optimization of the alternating-direction-implicit iterative method for the 2-D finite-difference frequency-domain solution of scattering problems." IEEE Transactions on Antennas and Propagation 65.12 (2017): 7166-7173.

What are the most recommended publishers for submitting a book in the field of Iterative Methods for solving fractional Differential Equations?

BICGSTAB-FFT can be used in DDA because the special form of its interaction matrix, which makes the time complexity reduce from O(N^2) to O(NlogN). Iterative method such as bicgstab can also be used to solve the matrix in finite element method, but it seems FFT(fast fourier transform) can not be implemented in this case.

Does this mean, suppose with the same number of elements (or matrix size), finite element method will principally be more time consuming than DDA simply because the more time consuming matrix vector multiplication in bicgstab (suppose bicgstab runs the same number of iterations).

I have implemented a BiCGSTAB iterative solver, for a certain linear equation, the Jacobi preconditioner actually deteriorates the convergence rate. The matrix A is positive definite and nothing really special. Based on my understanding of BiCGSTAB, with a Jacobi preconditioner, it should be always easier to solve such a linear equation than a BiCGSTAB solver without a preconditioner. However, when I remove the preconditioner, it converges much faster.

So can anyone tell me what is going on here? Do you understand the BiCGSTAB correctly?

Hi

Yes as it's, the system is not easy and requires so much work and tears. I will summarize my question to make it easier to understand. So I have system of coupled pdes. The unknown variables are dependent on time and space: du/dt=f(u'', u', u, v'', v', v, s'', s', s), dv/dt=f(u'', u', u, v'', v', v, s'', s', s), ds/dt=f(u'', u', u, v'', v', v, s'', s', s). So I have used finite difference method to discretize the derivative terms. However, since they are nonlinear, I should choose an iterative method to compute the solution. I was thinking to implement Picard method, but skeptical that the iteration will never reach the minimum tolerance between the solution and initial guesses. So which method would you recommend to use?

Thank you

I want to use LES to simulate the flow past a sphere at high Re by non-iterative PISO method, which is much faster than iterative method. However, the result of non-iterative PISO is far from the result in literatures, especially in Strouhal number. Can this method stable and accurate enough to apply to LES in turbulent flow? How can I improve the accuracy?

For example, Broyden’s method, Newton iteration method, ... .

I did not write much program with it. Also, I do not have much time.

I have attached some explanations.

Thank you

Nonlinear algebraic or transcendental scalar equations are usually solved by using iteration methods. Because some of them are so intricate to solve analytically or an approximate numerical solution is required. Nonlinear algebraic equations can occur during several modeling processes or when solving nonlinear BVPs using FDMs......

I wonder that if anyone tried SOR method for unsteady problem discretized by finite difference or finite volume method. Does it converge faster AND solve faster when it is compared to Gauss-Seidel and Jacobi solver?

As is well known, iterative methods for solving linear systems such as Successive Over Relaxation and the like, are very attractive for solving many problems such as sparse matrices. These methods, in general, are formulated in the context of determined system in which the number of equations is equal to the unknowns. Now and for sake of simplicity, let us assume that we have one additional observation and We need to update the previous solution. In other words, now we have an over determined system with the provision of this additional observation. The question is how to include this observation to the to update the previously computed parameters. Indeed, the theory of parameter estimation provides a lot of guidelines to handle this task to get an optimal solution in the sense of least squares. But let us assume that we need to stick to the iterative approach for parameters. Then with assumption how we could handle the additional observation for parameters update and what kind of errors that we need to minimize.

What is the relationship between iterative methods and line search techniques?

I'm new DIANA user and now I am working in numerical analysis of reinforcement concrete infilled frames with DIANA.

So I have a problem when it comes to running the model. After results these errors appears

SEVERITY : FATAL

ERROR CODE: /DIANA/AP/LB40/0053

ERRORMSG.F: Divergence occurred, iteration method failed.

Use smaller load steps, different iteration procedure or different

control procedure. Restart from a saved FILOS file or rerun the

complete job.

SEVERITY : ABORT

ERROR CODE: /DIANA/LB/ER50/0098

ERRORMSG.A: 1 fatal error messages occurred during previous analysis.

Can't continue.

DIANA-JOB ABORTED

I let you check my model whether you can bring your experience in.

Thanks in advance

How can i solve system of differential equation using picard iterative method?

I am looking for a new iterative method or analytic-approximate method to solve ODE\PDE not the already known methods such as:ADM, VIM, HAM, HPM, DJM, DTM,,,,,but new method.

Best regards,

such as homotopy analysis method

He's Variational Iteration method is applied in my paper for obtaining an approximate and analytical solution of non-linear differential equation containing a non-linear term related to enzymatic reaction.

I have read the article "A-Optimal Projection for Image Representation" ().

The authors used an iterative method to solve the optimization problem, i.e. obtain the transformation matrix A. But the number of iterations hasn't been discussed in the paper, and I fail to find the code of the paper thus not know the performance in practice.

How to determine the number of iterations of such a form of iterative method?

The proportional fitting procedure (IPFP,) is an iterative algorithm for estimating expected cell values [M_ijk] of a contingency table such that the marginal conditions are met.

For iterative computation of the M_ij, using MLE for the model, other Algorithm exist , and most common are for 2D while am interested in that of 3D, Assuming model of independence is equivalent to the model that all odds ratios equal one. I need references of examples showing manual calculation, and how to do more advance ones on R.

I am programming a code for an iterative method of solving a PDE (partial differential equations) problem under Matlab using the finite element method. The solution I get changes depending on the mesh I choose. I know that this is illogical. I wonder if anyone of you knows the reason of this problem.

Hello researchers,

I have been working recently on numerical analysis questions and came across a problem to which i am stuck. The image is attached.

Can you please help?

Thank you.

Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately? What happen in the case where we have no pre-smoothing steps and more post -smoothing and viceversa.

The difficulty is that the observation data is not simultaneous, thus a direct empirical estimation is not feasible. Desroziers has proposed an iterative method based on posterior residual diagnostic but its convergence can be difficult to obtain, especially in a highly non linear dynamical system (e.g. Lorenz)

I am wondering if other methods have been tested, like a regression (observation data -> matrix) or an optimization modelling (find the matrix that minimize the output error in a dynamical system)

Dear the experts,

I have seen a lot of studies using deep learning into tomographic imaging, and improving the computational time of it.

I wonder if there is another measure for comparing computational costs.

If I am right, I found that FLOPS is widely used metric in deep learning models.

But in the iterative methods, it would not be appropriate because there can be the computations beyond the simple multiplication and summation.

If there is a good way to, I would like to ask you about it.

Thank you in advance

BFGS algorithm is known to be an iterative method for solving unconstrained nonlinear optimization problems. Will different presentations of objective functions affect the applicability/feasibility/convergence of the BFGS algorithm?

On the other hand, SQP is known to be an iterative method for solving constrained nonlinear optimization problems. Will different presentations of objective functions and/or constraints affect the applicability/feasibility/convergence of the SQP?

What are the rationales behind?

I am writing a code for solving Eliashberg equations(self-consistant) by simply iterating method( x=f(x)), but not getting convergence.

Can someone suggest some mixing or any oyher method for getting fast convergence?

I'm working on iterative phase tomography which combines the phase retrieval and image tomography together, while few papers could be found till now. Can you recommend me some of them?

Thanks!

How can we apply adomian decompostion method when there is no linear term in differential eqation

These two methods (1) Picard Iterative Method(PIM)

(2) Temini and Ansari Method (TAM)

seen almost similar both in principles and applications especially when they are applied to initial-value problems. Can someone tell me the differences ?

Dear friends

I have a serious problem in order to regenerate the results of the attached paper. I do exactly mentioned method (FEM) in the paper but I don't get correct results.

This paper about gas face seal and compressible zeroth and first order perturbed Reynolds equation. I Solved zeroth order with FEM and FDM correctly and I got same results that bring out in the paper (Opening force for various pressure ratio) and I solve first order equation with 1 pressure ratio (figure 5 and 6) (I am thankful to you if look at the paper), but in other figure I can't get the paper's results. I try many method to solve linear system of equation (Ax=b) such as direct method, iterative method (i.e. gauss-sidel, cgs, gmres, bicg, pcg ...) but I failed, also I try many network grid and there is no achievement.

So what should I do?

I really know my question is general, but I really don't know about the errors.

As newton raphson method find one root it a time while simultaneous iterative methods approximate all root at time are the above procedure hold for it?

Suppose we've found the dominant e.value & corresponding e.vector by power method and Rayleigh quotient. Then what about the others?!

I am currently working on a project in which we aim to produce a thermo-labile receptor. in the course of my preliminary research I came across the B FITTER program which calculates the amino acid B-factor as an average of the B-factors of all of its atoms excluding hydrogen, then generates an output file with a ranked classification of all the residues according to the B-factor. (described here: Reetz MT, Carballeira JD. 2007. Iterative saturation mutagenesis (ISM) for rapid directed evolution of functional enzymes. Nat Protoc 2:891–903.)

My problem is that some of the PDB files generate a proper output file, while others turn up a blank file (example: 3K90 produces a proper file, while 3KB0 doesn't). Could this be caused by a difference in the PDB file format? is there a way around this issue?

Many thanks in advance,

Dan.

Just for test; I applied a fixed (known) transformation to a point cloud, afterwards I applied ICP to estimate my initial transform. I noticed that the values obtained from the estimation with ICP were not same with my initial transform. If ICP doesn't provide a unique solution, how can we ensure its accuracy?

**Edit-1:

I compared both initial and estimated transforms in terms of Euler's angles rather than conventional Rotation matrices.

There are two computational aspects in computing the zeros of orthogonal polynomials(Gauss quadrature nodes)

1: Eigenvalue approach

2: Root finding Iterative methods like as Newton's method.

Which one of the above schemes do not suffer from round-off errors?

Which one of these two technique is suitable specially for large values of

approximation degree?

Best.

Hi,

I am trying to implement MLEM algorithm for SPECT image reconstruction in 3D. But I am confused with probability matrix. How can I calculate probability matrix?

Thanks in advance

Manu

I need to do iterative more than 1000 simulation of a job with just small change in each run, so i do not want to check for warnings and errors in each run, as it is taking A LOT OF TIME, and directly just submit and run the analysis each time..

Large scale sparse matrix iterative solution- 1 million Order level

For sure, this simple mapping has been explored:

Y_{n+1} = y (1+Y_n)/(1+ Y_n +y).

For y < -0.25 it oscillates. Can anybody point me to a study? 

Hi everyone!

Is someone have sucessfully implemented a DG code ( in matlab, fortran, python or alike... not in C/C++) for simulate highly convective viscous incompressible flows?

I did not have success in doing it.

Let me explain:

I have used the DG code by Hesthaven & Warburton (J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods - Algorithms, Analysis, and Applications, Spinger, (2008)). This code only works under low Reynolds number flows ( at maximum about 100 Re ). So I begun to try extend this code so it runs high-Reynolds number flows.

I adopted the ideas from Krank eta al article (A high-order semi-explicit DG solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow - arXiv ). These authors made a C++ DG code ( using projection method "a la Karniadakis" ) and it seems they was successfull in simulating high-Reynolds number viscous incompressible flows. Until now, I have implemented the new convective-step part ( from the three parts that compounds the projective method ) and kepting all othes program's subroutines untouched. Unfortunately I dont had sucess. The program blows up, even running low Reynolds number flows!? It seems strange to me, for I would expect this unique program's change do not affected the entire program behaviour, as I have changed the old convective step ( that works for just low Rynolds number flows) by a new (better!) one, that was conceived to run highly convective flows!?

What is happening?

Can someone help-me?

hello everyone;

i'm trying to implement lattice FIR filter and RII lattice filter in c.

my question is how to make the first g(n-1), some one can help me please! i know that z-1 represente a delay time so how can i program it ?, I thought to take each time the previous sampler, but it will not work for the first samples because we will have negative values

you can find the algorithm here

I'm trying to solve the multi-objective optimization problem 

%% optimisation multiobjective en utilisant la méthode de pondération%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [f] = objfun2(x,w1,w2)

% X1 : Entrée en électricité

% X2 : Entrée en gaz naturel

% X3 : Entrée en biomasse

% X4 : biogaz à exportation

% X5 : Facteur split 1

% X6 : Facteur split 2

% X7 : Facteur split 3

% X8 : Puissance échangée avec l'élement du stockage d'hydrogène

% X9 : Energie stockée

% D_ch :Demande en chaleur

% D_ele : demande en électricité

% Pour le scenarion 1:

n_ee = 0.90; % Trasformateur

n_EL_H2 = 0.85 ; % Eléctrolyseur

n_EL_TH =0.15 ; % Eléctrolyseur

n_GB_TH = 0.90; % Gas boiler

n_FC_EL = 0.55; % Fuel cell

n_FC_TH = 0.45; % Fuel cell

n_RF_H2 =0.7 ; % Reformer

n_FR_GN = 0.7; % fermenteur

n_H2_CH =0.8 ; % charge H2

n_H2_DE =0.8; % Décharge H2

C1 =0.8; % Cout spécifique Transformateur

C2 =0.20; % Cout spécifique chauddière à gaz

C3 =0.6; % Cout spécifique reformer

C4 =2; % Cout spécifique fermenteur

C5 =2; % Cout spécifique électrolyseur

C6 =2; % Cout spécifique pile à combustible

C7 =2; % Cout spécifique élément du stockage

f1=0;

for j=1:52

f1 = f1 + x(1)*C1+x(2)*x(5)*C2+(x(2)*(1-x(5))+(x(3)*n_FR_GN)*(1-x(6)))*C3+x(3)*C4+(x(1)*n_ee*(1-x(7)))*C5+((x(1)*n_ee*(1-x(7)))+(x(1)*(1-x(5))+x(3)*n_FR_GN*(1-x(6)))*n_FR_GN)*C6+x(8)*C7;

end

e1 =0.8; % facteur d'émission lié à l'usage d'électricité du réseau

e2 =0.20; % facteur d'émission lié à l'usage du gaz naturel dans la chadière

e3 =0.6; % facteur d'émission lié à l'usage du gaz naturel pour le réformage

e4 =2; % facteur d'émission lié à l'usage du la biomasse dans le fermentreur

f2=0;

for j=1:52

f2 = f2 + e1* x(1+9*(j-1))+ e2* x(2+9*(j-1))*x(5) + e3* x(2+9*(j-1))*(1-x(5+9*(j-1)))+e4*x(3+9*(j-1));

end

f = w1*f1+w2*f2;

end

%% Mes contraintes d'égalité et d'inégalité %%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Mes contraintes d'égualité:

% Définir mes variables :

% X1 : Entrée en électricité

% X2 : Entrée en gaz naturel

% X3 : Entrée en biomasse

% X4 : biogaz à exportation

% X5 : Facteur split 1

% X6 : Facteur split 2

% X7 : Facteur split 3

% X8 : Puissance échangée avec l'élement du stockage d'hydrogène

% X9 : Energie stockée

% D_ch :Demande en chaleur

% D_ele : demande en électricité

function [c,c_eq] = mycon3(x,vout)

% Pour le scenarion 1:

n_ee = 0.90; % Trasformateur

n_EL_H2 = 0.85 ; % Eléctrolyseur

n_EL_TH =0.15 ; % Eléctrolyseur

n_GB_TH = 0.90; % Gas boiler

n_FC_EL = 0.55; % Fuel cell

n_FC_TH = 0.45; % Fuel cell

n_RF_H2 =0.7 ; % Reformer

n_FR_GN = 0.7; % fermenteur

n_H2_CH =0.8 ; % charge H2

n_H2_DE =0.8; % Décharge H2

% c_eq1 = x(1)*(n_ee*x(7)+n_ee*(1-x(7))*n_EL_H2) + x(2)*(1-x(5))*n_FR_GN*n_FC_EL + x(3)*(n_FR_GN*(1-x(6))*n_FC_EL*n_RF_H2);

% c_eq2 = x(1)*(n_ee*(1-x(7))*n_EL_TH+n_ee*(1-x(7))*n_EL_H2) + x(2)*(x(5)*n_GB_TH+n_RF_H2*(1-x(5)))+ x(3)*((1-x(6)*n_RF_H2*n_FR_GN) +x(8);

% c_eq3 = x(3)* n_FR_GN * x(6);

for j = 1:52

c_eq(j) = vout(j)-x(1+9*(j-1))*(n_ee*x(7+9*(j-1))+n_ee*(1-x(7+9*(j-1)))*n_EL_H2) - x(2+9*(j-1))*(1-x(5+9*(j-1)))*n_FR_GN*n_FC_EL - x(3+9*(j-1))*(n_FR_GN*(1-x(6+9*(j-1)))*n_FC_EL*n_RF_H2);

end

for i = 1:52

if x(8+9*(i-1)) >= 0

eh = n_H2_CH;

else

eh = 1/n_H2_DE;

c_eq(52+i) = vout(52+i)-x(1+9*(i-1))*(n_ee*(1-x(7+9*(i-1)))*n_EL_TH+n_ee*(1-x(7+9*(i-1)))*n_EL_H2) - x(2+9*(i-1))*(x(5+9*(i-1))*n_GB_TH+n_RF_H2*(1-x(5+9*(i-1))))- x(3+9*(i-1))*((1-x(6+9*(i-1))*n_RF_H2*n_FR_GN) +x(8+9*(i-1)))*eh;

end

for j = 1:52

c_eq(104+j) = x(4+9*(j-1))-x(3+9*(j-1))* n_FR_GN * x(6+9*(j-1));

end

% contraintes sur l'évolution de la charge des stockages

if x(8) >= 0

eh = n_H2_CH;

else

eh = 1/n_H2_DE;

end

c_eq(157) = x(9) - x(468) + 0.05 - eh*x(8);

for j=2:52

if x(8+9*(j-1)) >= 0

eh = n_H2_CH;

else

eh = 1/n_H2_DE;

end

c_eq(156+j) = x(9+9*(j-1)) - x(9+9*(j-2)) + 0.05 - eh*x(8+8*(j-2));

end

c = [];

end

%% Optimimisation %%%% Optimimisation %%

clear allclc

%% scénario demande en électricité et en chaleur par les batiments A-D

        %demande en Electricité batiments A B C D        scenario= [1   208816.92   164334.53   63948.44   14547.17;    2   208816.92   164334.53   63948.44   14547.17;  3   208816.92   164334.53   63948.44   14547.17;    4   208816.92   164334.53   63948.44   14547.17;    5   208816.92   164334.53   63948.44   14547.17;    6   208816.92   164334.53   63948.44   14547.17;    7   208816.92   164334.53   63948.44   14547.17;    8   208816.92   164334.53   63948.44   14547.17;    9   208816.92   164334.53   63948.44   14547.17;   10   208816.92   164334.53   63948.44   14547.17;    11   208816.92   164334.53   63948.44   14547.17;    12   208816.92   164334.53   63948.44   14547.17;    13   208816.92   164334.53   63948.44   14547.17;    14   208816.92   164334.53   63948.44   14547.17;    15   208816.92   164334.53   63948.44   14547.17;    16   208816.92   164334.53   63948.44   14547.17;    17   208816.92   164334.53   63948.44   14547.17;    18   208816.92   164334.53   63948.44   14547.17;    19   208816.92   164334.53   63948.44   14547.17;    20   208816.92   164334.53   63948.44   14547.17;    21   208816.92   164334.53   63948.44   14547.17;    22   208816.92   164334.53   63948.44   14547.17;    23   208816.92   164334.53   63948.44   14547.17;    24   208816.92   164334.53   63948.44   14547.17;    25   208816.92   164334.53   63948.44   14547.17;    26   208816.92   164334.53   63948.44   14547.17;    27   208816.92   164334.53   63948.44   14547.17;    28   208816.92   164334.53   63948.44   14547.17;    29   208816.92   164334.53   63948.44   14547.17;    30   208816.92   164334.53   63948.44   14547.17;    31   208816.92   164334.53   63948.44   14547.17;    32   208816.92   164334.53   63948.44   14547.17;    33   208816.92   164334.53   63948.44   14547.17;    34   208816.92   164334.53   63948.44   14547.17;    35   208816.92   164334.53   63948.44   14547.17;    36   208816.92   164334.53   63948.44   14547.17;    37   208816.92   164334.53   63948.44   14547.17;    38   208816.92   164334.53   63948.44   14547.17;    39   208816.92   164334.53   63948.44   14547.17;    40   208816.92   164334.53   63948.44   14547.17;    41   208816.92   164334.53   63948.44   14547.17;    42   208816.92   164334.53   63948.44   14547.17;   43   208816.92   164334.53   63948.44   14547.17;    44   208816.92   164334.53   63948.44   14547.17;    45   208816.92   164334.53   63948.44   14547.17;    46   208816.92   164334.53   63948.44   14547.17;    47   208816.92   164334.53   63948.44   14547.17;    48   208816.92   164334.53   63948.44   14547.17;    49   208816.92   164334.53   63948.44   14547.17;    50   208816.92   164334.53   63948.44   14547.17;    51   208816.92   164334.53   63948.44   14547.17;    52   208816.92   164334.53   63948.44   14547.17];  

        %demande en chaleur batiments A B C D          scenario2 = [1 1427594.60   1284109.59   523483.67   190031.15   2 982931.18   872385.05   368411.96   134151.09   3 1135105.91   966735.18   413340.25   149325.03   4 1495313.57   1299812.17   503369.62   188862.93   5 1325288.52   1059053.42   417684.96   151609.55   6 995935.34   729842.68   326075.69   110981.31   7 895859.46   748401.59   324866.08   114330.22   8 1064518.63   897955.07   376706.41   134196.02   9 884647.89   701571.18   300466.56   108722.74   10 800896.54   531547.79   265629.86   93068.54   11 690779.38   574932.49   284292.38   95440.39   12 630955.39   491976.27   218766.20   71651.09   13 451613.74   302690.85   182270.61   59536.17   14 282668.92   188690.11   127527.22   39082.41   15 365144.19   231710.71   164852.26   49844.24   16 375957.22   212325.89   141524.11   44132.92   17 300533.75   182593.74   146811.82   44003.12   18 279095.43   189308.38   150077.76   42990.65   19 139571.14   69729.07   83462.93   23753.89   20 199881.76   83818.97   115863.14   31542.06   21 85175.05   52589.89   70917.57   18691.59   22 93955.75   44086.60   45412.13   12980.27   23 131220.95   0             15552.10   3115.26   24 108755.75   0             17107.31   2076.84   25 116913.61   59.38       15552.10   2076.84   26 53402.44   0           12441.68   3115.26   27 47767.21   0           12441.68   2336.45   28 24543.93   0           12441.68   0     29 110250.54   0           12441.68   1038.42   30 151976.55   0           12441.68   3115.26   31 69422.12   0             12441.68   3115.26   32 86918.05   0             12441.68   3115.26   33 34192.03   0           12441.68   4672.90   34 20738.80   0             12441.68   6230.53   35 39519.77   0             12441.68   4153.69   36 68539.80   0             12441.68   3115.26   37 55234.83   0             12441.68   3115.26   38 292989.19   0             30482.12   3115.26   39 187184.72   126185.65   18662.52   23260.64   40 183608.26   143900.02   154789.13   48286.60   41 287919.61   240537.02   122083.98   56464.17   42 677292.12   605297.97   280715.40   112409.14   43 884163.86   742486.71   368584.76   123831.78   44 661414.79   501422.13   273999.72   89304.26   45 775772.10   632838.22   276827.37   110072.69   46 714229.28   673670.97   298859.51   108870.31   47 1109646.28   1082949.28   386823.13   169781.93   48 1387665.61   1405851.38   538102.64   214330.22   49 1178721.87   1101958.19   481832.32   169781.93   50 1500936.04   1296977.48   495889.80   213914.85   51 1437692.34   1143034.01   554765.61   182717.05   52 1504741.00   1394642.85   497874.55   192951.71];

for i = 1:52    vout(i) = scenario(i,2)+scenario(i,3)+scenario(i,4)+scenario(i,5);end 

for i = 1:52    vout(52+i) = scenario2(i,2)+scenario2(i,3)+scenario2(i,4)+scenario2(i,5);end

% contraintes d'inégalités linéaires%  A*X  <= bA = [];b = [];

% contraintes d'égalités linéaires%  Aeq*X  = BeqAeq = [];beq = [];

% initialisation du vecteur des variables%x0 = [1;2;0.5;0.5]; 

%  mes limites inférieure et supérieures : 

mat1 = zeros(1,468);

for i=1:52    mat1(1,8+9*(i-1)) = -10000;end

lb = mat1;

mat2 = zeros(1,468);for i = 1:52      mat2(1,1+9*(i-1))= 20000;    mat2(1,2+9*(i-1))= 20000;    mat2(1,3+9*(i-1))= 20000;    mat2(1,4+9*(i-1))= 20000;    mat2(1,5+9*(i-1))= 1;    mat2(1,6+9*(i-1))= 1;    mat2(1,7+9*(i-1))= 1;    mat2(1,8+9*(i-1))= 10000;    mat2(1,9+9*(i-1))= 30000;end   ub = mat2;

for j=1:52    x0(1+9*(j-1))=100;       x0(2+9*(j-1))=100;      x0(3+9*(j-1))=100;      x0(4+9*(j-1))=100;       x0(5+9*(j-1))=100;    x0(6+9*(j-1))=100;     x0(7+9*(j-1))=100;      x0(8+9*(j-1))=100;        x0(9+9*(j-1))=100;end

for i = 0:0.02:1        w1 = i;    w2 = 1-i;    % -----X = FMINCON(FUN,    X0,A,B,Aeq,Beq,LB,UB,NONLCON,OPTIONS)

options=optimset('Algorithm','interior-point','MaxFunEvals',10000000);

[x,fval] = fmincon(@(x) multiobj(x),x0,A,b,Aeq,beq,lb,ub,@(x) mycon3(x,vout),options,w1,w2);

end

someone could help me with the problem, Matlab tells me Failure in initial user-supplied objective function evaluation. FMINCON cannot continue ?

I have currently used the Krylov and Multifrontal methods in order to solve disperse linear systems. Next figure shows the characteristic curves number of nodes vs CPU time of the Krylov and Multifrontal methods. Does anyone know about direct or iterative methods for decrementing the computation time in the solution of large sparse linear systems? Which one?

I am trying to solve a large system of linear equations AX=B, where A is a Jacobean matrix, non-symmetric, very large (about 18000*18000), sparse matrix. Which method is the fastest method for it?

Jacobean matrix is ordered in a way that each of the four parts of the Jacobean matrix is block diagonal.

I want to minimize the error function, which I defined as the sum of the square distances between data points and a given numerical solution. My error function then solely depends on the ODE parameters. Which iterative optimization method should I use in order to minimize the error function in the parametric space? My ODEs are nonlinear. I would like a solution based on an open source software such as Octave.

Dear all,

I am running simulations in OpenFOAM with icoFoam. But after a few time steps, the Courant number increases to a very large value. I wonder if anyone know the possible reasons.

BTW: I can assure that the settings in OpenFOAM 4.1.0 are correct, because I succeeded in the simulation of one type of structures and have got reasonable results. The meshCheck is also OK.

I have been using OpenFOAM for several months and this is the first time I met this kind of problem.

Any suggestion is appreciated. Thank you in advance.

Best Regards,

Bill

Simulink cannot solve the algebraic loop containing 'file name/fractional order derivative' at time 0.1 using the TrustRegion-based algorithm due to one of the following reasons: the model is ill-defined i.e., the system equations do not have a solution; or the nonlinear equation solver failed to converge due to numerical issues.

To rule out solver convergence as the cause of this error

What are the applications of implicit iterative algorithms? Moreover, is there any nonlinear nonexpansive mapping example satisfying some result (any research paper in literature with implicit iteration) having implicit iteration process?

I am working on a CFD analysis of a 80m2 greenhouse used for drying seeds. The space has 8 shelves stacked at 4 different heights and openable windows for excess moisture and heat to be removed. I use both '2-k-e' and 'constant effective velocity' solvers and used them separately as well as in succession to generate results. I vary Relaxation Factor on x/y/z velocities, mass and temperature and I also vary Termination Residuals to direct the solver towards convergence.

My question is to anybody with experience of none-symmetrical CFD work at scales close to room/building level; what is a reasonable number of iterations before ditching the simulation as non-convergent? I have had convergence at up to 100,000 iterations when my PC was running over the weekend, so my inclination is to wait, but this particular CFD task leads to a lot of unstable, osculating patterns on Normalised Residuals (see attached) and I wonder if it is worth giving up as soon as the residuals begins to follow similar patterns and ‘repeat’ themselves?

Choosing the initial values of a parameter at beginning of procedure like estimation of parameters is crucial. AS far as I know, good starting point leads to fast converging, for example, the good starting point,  the fast converging in a calculation of hessian matrix.

now, How do I choose an initial value for the parameter?

thanks

Let be [D] a (nxn) ill-conditioned matrix and [G] its inverse. If the inverse exists then it can be efficiently implemented by the linear solving of [D]{gj}={bj} with the LU method using partial pivoting where {gj}: is a column vector of [G] and {bj}: is a column of the identity matrix [I] for j=1 to n and (bj(i)=0 for i=1 to n such that bj(j)=1). The resulting matrix can be checked with the satisfying condition [D][G]=[I].

I was just trying to solve system of transcendental equations using 'Bisection Method'. Iterative procedure is used to check the convergence. In lower order modes, I am unable to obtain the solutions whereas the same code is able to give solutions for arbitrary higher order modes? In those lower order modes, I think, solutions are stuck in LOCAL minima. How can I overcome local minima in 'Iterative Procedure'

if we use iteration methods like successive over relaxation, we should use a suitable value of the weight factor and this value will be updated through iteration to force iteration variables to get rapid convergence. my question is how to update this value? what is mathematics we use to do that?

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, strictly diagonally dominant $M$-matrix. So I establish a stationary iteration scheme as follows,

$$Sx^{(k+1)} = Tx^{(k)} + b.$$

According to numerical results, it seems that this iterative scheme is always convergent. Is it rational from theoretical analysis? So, can we prove the convergence of this iterative scheme? i.e.,

show $\rho(S^{-1}T) < 1$, where $\rho(\cdot)$ is the spectral radius.

Please also refer to the URL for details.

Hi all,

In point (Gauss-Seidel) method of iterative solution, simple algorithm is applied at each point where the function value has to be determined in successive manner, which essentially means that all point methods are a explicit methods. In fact Gauss Seidel method is explicit.

So, How point explicit G-S works ?

whereas in line method of iteration (called implicit method), a row of many grids are solved simultaneously with the known values (existing values) from neighbouring grids. So this method becomes implicit method?

I need a clarification on this form an numerical simulation expert. Please someone answer.

Waiting for a reply.

Thanks

Vivek 

I am student of MS Mathematics . currently I am working on this topic  (Nonlinear Complementarity Problem) want to know numerical solution of  such algorithms those have been done on Numeric iterative method .

Consider the problem of fitting a curve of the form

y(x)=a0+a1*exp(-x/b1)+a2*exp(-x/b2)+...+an*exp(-x/bn)

to noisy (x,y) data.  If we have a good initial guess for a0,a1,b1,a2,b2,..., then an iterative method can be used to find a local minimum of the least squares fit to the data.  But is there a good way to generate a good initial guess automatically?

A routine for this purpose, exp2fit.m (see attachment), was contributed to the Matlab File Exchange in 2008 by someone named Per Lundqvist.  It contains some unexplained formulas for generating initial guesses for n=1 and n=2 exponential terms, based on integrals of the data.

-Can anyone explain these formulas?

-Can the approach be extended to the sum of 3 exponentials (and beyond)?

-Are there better alternatives?

If there is no extranous fixed points for any iterative methods, what does mean?

Such methods, possible to diverge at any points in complex plane. ?

If there is many extranous fixed points, then what it means?

For example, one optimal fourth order modified newton's method gives 30 extranous fixed points for solving p(z)=z^3-1. for this results what say's.

Your valuable answer and comments  lead me to good work

Thanks in advance

The method should not be based on optimization technique which takes long time for convergence. Thanks in advance.

Hello everybody,

I have implemented a numerical model which consists--among others--of a Finite Difference scheme that is used to solve the quasi-stationary magnetic diffusion equation in three dimensions; i.e., a modified version of the Laplacian operator in Cartesian coordinates. For solving the generated system of linear equations (A)x = b I use the SOR algorithm. The model ran well so far, no matter what input I chose. I should mention that the system matrix A consists of complex-valued elements.

However, for some input parameters the SOR solver does not deliver useful results, or more specifically, it does not converge to the given level of accuracy at all. The system matrix has 8e6x8e6 entries which shouldn't be a problem because, for most input parameters, the SOR algorithm delivers physically correct results. It has to have something to do with the entries of the system matrix per se.

For solving the equation on the Cartesian grid I use the 7-point-stencil. If the unmodified version of the Laplacian operator is solved, the weighing factor at the actual node (i,j,k) is -6. In my modified version, it is -6 + i*x, where x denotes the imaginary part. I even tried SPQR from the SuiteSparse package provided by Tim Davis but since this is not an iterative method MATLAB throws an "out of memory" error (I can use up to 192 GiB (!)). It would really be great if someone encountered a similar problem and wants to share their solution. If interested, I could upload the system matrix as well as the boundary condition vector.

Thank you very much in anticipation!

Best regards,

Chris Volkmar

I found some example about iterative learning control but they are not enough for understanding it.

Currently I am using Gauss Siedel Iterative method to solve CFD equations. I am facing instability problem i.e. iterations keep on increasing without change in rms value for a given variable such as temperature. 

I want to know which are some stable iterative methods so that one can implement to get rid of diagonal dominance criteria to be satisfied?

I am interested in the numerical solution of convection-diffusion problems where the convection dominates. In the iterative solution with Gauss-Seidel, instabilities can occur for large Peclet numbers. Is the "downwind numbering" as in the paper by Bey & Wittum a possible solution for this problem on structured grids and with variable coefficients (advection speeds)? Unfortunately, I don't have access to the paper.

I want to know about the H-optimality. I have found in " A duality theorem and an iterative method of finding h-optimal experimental designs" by V. Melas and S.M.Ermakov.

I'm doing a relocation using hypoDD software. When I use LSQR for iteration method, everything works fine. But then I try to use SVD to enhance the quality from subset of events and the process stopped.

I have attached the command window when I run hypoDD using SVD. According to the error command, I have try to increase MAXEVE0 in hypoDD.inc into a considerable amount (about a millions).

I also try to remove/purge hypoDD in .bin, and changing MAXEVE0 parameter before reinstalling hypoDD. Those attempts still fail to solve the issue.

Can anyone give me some advices?

I want to tune natural frequencies of first few modes of vibration of a shell. Which are the best iterative methods for finite element  model updating of above mentioned problem.

Any suggestions?

Is there a simple condition for global convergence (i.e. regardless of initial guess) of the Newton-Raphson method to solve f(x) = 0, in both the scalar and vector cases? Particularly in the case in which f is not doubly differentiable?

I have experience on iterative methods like Jacobi, SOR and AOR and preconditioning as well.

Using some iteration method, we can solve a nonlinear system of equation. I know lots of old methods like, Newton Raphson, Gauss newton, marquardt, levenberg marquardt etc. I want to know about some upgraded methods.