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6
votes
3answers
1k views

Eigenvectors with the Power Iteration

To compute the eigenvector corresponding to dominant eigenvalue of a symmetric matrix $A\in\mathbb{R}^{n\times n}$, one used Power Iteration, i.e., given some random initialization, $u_1\in\mathbb{R}^...
3
votes
1answer
678 views

SQP optimization algorithm tuning advice

I am using a stable version of SQP algorithm from a lib. Parameters setting is left to the developer, althought default values are at hands. I launch solver on very simple optimization problems s.a. ...
0
votes
0answers
51 views

Lagrangian Duality [duplicate]

Possible Duplicate: Linear programming boundedness Consider the following LP: $\max$ $\sum_{i=1}^N b_i \pi_i$ s.t. $\;\;$ $\pi_i-\pi_j\leq 1 $ $\quad$ for each $(i,j) \in \tilde{A}$ $\...
13
votes
2answers
1k views

Alternatives to von neumann stability analysis for finite difference methods

I'm working on solving the coupled one-dimensional poroelasticity equations (biot's model), given as: $$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$ $$\...
4
votes
2answers
2k views

Extracting a 2D matrix from a multidimensional array in C

In my c language program, I have to store multiple dense $m\times m$ matrices corresponding to gridpoints $x_i$ with $i=1,...,n$. I decided to create a three dimensional array $A\in R^{n\times m\...
3
votes
1answer
123 views

Single nodes after mpmetis partitioning

I was checking partitioning capabilities of Metis (mpmetis) when I noticed, that it leaves two single nodes. I have marked them in red Have you seen something similar or maybe it is my mistake? The ...
0
votes
0answers
402 views

How to move particles in a variational Monte Carlo simulation?

I'm attempting to implement some code that moves particles and then calculated the acceptance/rejection of the move, but I'm stuck in a rut. Here is my Python code: ...
2
votes
1answer
410 views

Convergence of step-length in a globally-convergent newton line search method with non-degenerate Jacobian

I'm working on a problem given in Nocedal & Wright's Numerical Optimization 2nd Edition, pg 303 Exercise 11.7: Consider a line-search newton method in which the step length $\alpha_k$ is ...
7
votes
1answer
783 views

Polynomial Fitting from Chebyshev Coefficients

I have been reading Numerical recipes about how to create a power series approximation to a function once you have a Chebyshev approximation to the function. However it is still very unclear to me how ...
5
votes
4answers
3k views

Simulated Annealing proof of convergence

I implemented downhill simplex simulated annealing algorithm. Algorithm is very hard to tune, w.r.t. parameters including cooling schedule, starting temperature... My first question is about ...
4
votes
2answers
404 views

Effects of memory speed/architecture on Pardiso scaling

I am using a program that utilizes the PARDISO solver as part of the Intel math kernel library. I am currently in the process of deciding on a new computer to run the simulations on. I have a ...
5
votes
2answers
3k views

Relation and difference between combinatorial optimization, discrete optimization and integer programming

I wonder what relation and difference are between combinatorial optimization and discrete optimization? Thanks! Originally by reading Wikipedia, I thought discrete optimization consists of ...
10
votes
3answers
1k views

Relative comparison of floating point numbers

I have a numerical function f(x, y) returning a double floating point number that implements some formula and I want to check that it is correct against analytic ...
3
votes
1answer
73 views

semiboundedness of the operator and it is affect on stability

I remember seeing in the book by Kreiss "Time-dependent partial differential equations and their numerical solution" that if some elliptic differential operator satisfies $$(Lu,u)\leq K(u.u)$$ for the ...
2
votes
2answers
143 views

What are good examples of problems which are stiff due to very long interval of integration?

There is a class of stiff initial value problems for ODEs that have small Lipschitz constants, slowly-changing solutions, but very long interval of integration. The only practical example of such a ...
6
votes
5answers
2k views

What other method can be used to solve differential equations except ode functions in Matlab?

Recently I am taking advantage of the ode45, as well as ode23s, ode15s solvers in ...
10
votes
4answers
3k views

Fast and accurate double precision implementation of incomplete gamma function

What is the state of the art way of implementing double precision special functions? I need the following integral: $$ F_m(t) = \int_0^1 u^{2m} e^{-tu^2} d u = {\gamma(m+{1\over 2}, t)\...
6
votes
3answers
1k views

Convergence of fixed point iterations of a non-linear matrix system

I'm working on modeling two phase immiscible flow in a porous medium. When I setup the system of equations, I obtain a non-linear system of equations that can be expressed in the form: $A(x)x=b$ ...
11
votes
2answers
7k views

CVXOPT VS. OpenOpt

CVXOPT: http://abel.ee.ucla.edu/cvxopt/index.html OpenOpt: http://openopt.org/Welcome What's the relation between them? What are the advantages/disadvantages of them, respectively? BTW, is there any ...
0
votes
3answers
181 views

Equivalence of linear systems, solving one instead of the other

This question is related to recently posted one, but I guess it deserves a separate attention. Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{...
14
votes
1answer
287 views

How do low rank modifications affect Krylov method convergence?

Say I have a linear system $A x = b$, which converges quickly using a suitable Krylov method (such as CG or GMRES) for all $b$. If $B$ is a matrix with low rank $r$, will the same Krylov method on ...
2
votes
3answers
10k views

Dealing with non-monotonically increasing data

How do you deal with data that you need to be monotonically increasing in order to work with interpolation libs and other functions, when it is in fact not monotonically increasing?
3
votes
1answer
234 views

Looking for parMetis visualizer?

Is there any visualizer for parMetis (mpmetis), which can visualize FEM mesh grids after partitioning?
1
vote
0answers
498 views

How to solve advection equation using semi-lagrangian method?

I am working on something that involves solving an advection equation $\partial{x}/\partial{t}+\vec{u}\cdot\nabla{x}=0$ in 3D. I discretized the space into 3d cartesian grid and used the Semi-...
8
votes
2answers
7k views

Simultaneous maximization of two functions without available derivatives

I have two variables k and t as functions of two other variables p1 and ...
3
votes
1answer
353 views

Parallelization of LSE solvers using CUDA

I want to know methods which are fully parallelizable on CUDA architecture. I have implemented the Jacobi and Conjugate Gradient methods and now Im thinking about the Bi-Conjugate gradient method. I ...
25
votes
6answers
7k views

Visualizing very large link graphs

I am looking for a tool to visualize very large directional link graphs. I currently have ~2million nodes with ~10million edges. I have tried a few different things, but most take hours to even do ...
8
votes
0answers
440 views

What's a good numerical/optimization software package for solving the 2-D optimal stopping problem?

I am looking for a numerical software package to help me solve the 2-dimensional "free boundary" PDEs that arise in optimal stopping problems. In one dimension a standard optimal stopping problem in ...
3
votes
1answer
148 views

How to add perturbation to a base state in physical grid space?

For study 3D instabilities problem i.e a base state (velocity field and temperature field) and an additional perturbation. The method must be independent of the type of instabilities considered but ...
8
votes
1answer
2k views

Open-source, thread-safe implementation of convex optimization solvers in C/C++?

Is there an open-source, thread-safe implementation of convex optimization solvers in C/C++? Some libraries such as NLopt, Ipopt, OPT++ don't meet my requirements. OPT++ and Ipopt aren't thread-safe,...
2
votes
1answer
187 views

Recovering coordinates by eigendecomposition without double-centering

Suppose an Euclidean distance $D\in\mathbb{R}^{n\times n}$ matrix between a set of $n$ objects is given. To obtain inner-products (which will be further be used to recover coordinates), entries of $D$ ...
8
votes
2answers
426 views

Octree cubes to tetrahedrons

I'm trying to learn more about volume meshing and have decided to try to implement a simple volume mesher. The strategy I have chosen is to subdivide my space using an octree, refined based on some ...
6
votes
1answer
12k views

Concave polygon 'hull' finding

I implemented an algorithm to find the alpha shape of a set of points. The alpha shape is a concave hull for a set of points, whose shape depends on a parameter alpha deciding which points make up the ...
6
votes
1answer
1k views

Appropriate space for weak solutions to an elliptical pde with mixed inhomogeneous boundary conditions

I'm working with the following mixed inhomogeneous boundary value problem: $\nabla(\kappa\nabla u)=f$ in $\Omega$ with $\partial\Omega = \Omega_1 \bigcup\Omega_2$ such that $u=g$ on $\partial\...
2
votes
2answers
317 views

Numerical solution of fractional integro-diffrential equ. using collocation method?

problem comes from "Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan" $D^...
1
vote
1answer
87 views

neural networks: multilayer on-off perceptrons

This article says that all any multilayer perceptron with a linear on-off functions for all the neurons can be reduced to a two-layered perceptron. Now, consider a two input/one output perceptron. ...
15
votes
3answers
1k views

Numerical methods for discontinuous r.s. ODEs

what are state of art methods for numerical solution of ODEs with discontinuous right side? I'm mostly interested piecewise-smooth right side functions, e.g. sign. I'm trying to solve the equation of ...
2
votes
1answer
160 views

FEM simulation of a material being stretched

I'm trying to teach myself FEM. The problem I have in mind is to completely investigate a plastic (e.g rubber band) being stretched (therefore, boundary conditions move as you stretch). I imagine this ...
3
votes
2answers
106 views

How can you explain the following bound on the inner product?

I am reading a paper on stability of CG, and I came across the following statement: \begin{equation} \frac{\|A\|\,\|p\|^2}{\langle p,Ap\rangle} \leq \kappa(A) \end{equation} where $\kappa(\cdot)$ is ...
1
vote
1answer
54 views

Finding most informative feature subsets given dataset, clustering algorithm and gold standard partition

I have an $n \times m$ matrix of data $\mathbf{D}$ as well as a $k$-partition $P$ of $n$ indices each representing a row in a dataset. Assuming an arbitrary clustering algorithm $A$, I would like to ...
10
votes
2answers
409 views

Task-based shared-memory parallel libraries in Scientific Computing

In recent years, several libraries/software projects have appeared that offer some form or another of general-purpose data-driven shared-memory parallelism. The main idea is that instead of writing ...
1
vote
2answers
2k views

Gram-Schmidt method to identify linearly dependent vectors

A method to orthogonalize a set of vectors (vectors of unit length that are mutually orthogonal) is the Gram-Schmidt process: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Note that the ...
13
votes
1answer
7k views

Understanding the Wolfe Conditions for an Inexact line search

According to Nocedal & Wright's Book Numerical Optimization (2006), the Wolfe's conditions for an inexact line search are, for a descent direction $p$, Sufficient Decrease: $f(x+\alpha p)\le f(x)+...
1
vote
1answer
120 views

Adaptive Linear Algebra Libraries

After reading the first answer here about how the best way to find the most performant sparse solver is to try almost everything, I began to wonder if there was any past work on libraries or research ...
2
votes
3answers
986 views

Convergence of the gradient descent and linear vs non-linear fixed point iteration

Suppose a system $$Ax=b$$ is given, with $A\in\mathbb{R}^{n\times n}$ being a symmetric positive-definite matrix, and some non-zero $b\in\mathbb{R}^n$. The gradient method with optimum step length can ...
33
votes
3answers
14k views

How to choose a method for solving linear equations

To my knowledge, there are 4 ways to solving a system of linear equations (correct me if there are more): If the system matrix is a full-rank square matrix, you can use Cramer’s Rule; Compute the ...
13
votes
2answers
942 views

Impose the compatibility conditions for mixed finite elements method in Stokes equation

$\newcommand{\v}[1]{\boldsymbol{#1}}$ Suppose we have following Stokes flow model equation: $$ \tag{1} \left\{ \begin{aligned} -\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f} \\ \mathrm{div} \...
3
votes
1answer
221 views

Using same memory space for global and local Petsc DA Vectors

I have constructed a local and global distributed array in Petsc 3.2 using: DMCreateGlobalVector(da, &v); DMCreateLocalVector(da, &lv); In order to ...
8
votes
1answer
1k views

Fourier transform for Neumann boundary condition

I need to solve system of two coupled partial differential equations numerically. \begin{align} \frac{\partial x_1}{\partial t} &= c_1\nabla ^2 x_1 + f_1(x_1,x_2)\\ \frac{\partial x_2}{\partial ...
10
votes
3answers
351 views

Ways of visualizing event data in search of performance issues

I'm trying to optimize an MPI application with a highly asynchronous communication pattern. Each rank has a list of things to compute, and sends messages as necessary if the inputs or outputs reside ...

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