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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 ...
15
votes
4answers
590 views

What are some applications which require interval arithmetic?

I have a very basic notion about interval arithmetic (IA), but it seems to be a very interesting branch of computational science both theoretically and practically. It is clear that the obvious ...
15
votes
4answers
1k views

How to reorder variables to produce a banded matrix of minimum bandwidth?

I'm trying to solve a 2D Poisson equation by finite differences. In the process, I obtain a sparse matrix with only $5$ variables in each equation. For example, if the variables were $U$, then the ...
15
votes
3answers
2k views

Why isn't my Matrix-Vector Multiplication Scaling?

Sorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices....
15
votes
1answer
428 views

Puzzling remark about stability region of fifth-order Runge-Kutta method

I came across a puzzling remark in the paper P. J. van der Houwen, The development of Runge-Kutta methods for partial differential equations, Appl. Num. Math. 20:261, 1996 On lines 8ff on page 264, ...
15
votes
2answers
2k views

Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems. Fisher's equation (a nonlinear reaction-diffusion PDE), $$ u_t = du_{xx} + \beta u ...
15
votes
4answers
1k views

Testing numerical optimization methods: Rosenbrock vs. real test functions

There seem to be two main kinds of test function for no-derivative optimizers: one-liners like the Rosenbrock function ff., with start points sets of real data points, with an interpolator Is it ...
15
votes
6answers
18k views

Constraints involving $\max$ in a linear program?

Suppose $$\begin{align*} \min A &\mathrm{vec}(U) \\ &\text{subject to } U_{i,j} \leq \max\{U_{i,k}, U_{k,j}\}, \quad i,j,k = 1, \ldots, n \end{align*}$$ where $U$ is a symmetric $n\times ...
15
votes
3answers
762 views

What is a scalable preconditioner for high-frequency Helmholtz?

Standard multigrid and domain decomposition methods do not work, but I have large 3D problems and direct solvers are not an option. What methods should I try? How are my choices affected by the ...
15
votes
5answers
4k views

Parallel optimization algorithms for a problem with very expensive objective function

I am optimizing a function of 10-20 variables. The bad news is that each function evaluation is expensive, approx 30 min of serial computation. The good news is that I have a cluster with a few dozen ...
15
votes
4answers
1k views

Book reference for Numerical Analysis

I've had a glimpse of Numerical Analysis (majorly, Numerical Methods like root finding, quadratic equations and other preliminary stuff) in my Calculus class but now, I find myself wanting more ...
15
votes
2answers
385 views

SciComp Modeling Jobs

The meta seemed to suggest that career advice is ok . . . so here goes. I have a couple of close friends in the ML and mathematical modeling fields just finishing PhD's and starting out on the job ...
15
votes
4answers
3k views

Selecting most scattered points from a set of points

Is there any (efficient) algorithm to select subset of $M$ points from a set of $N$ points ($M < N$) such that they "cover" most area (over all possible subsets of size $M$)? I assume the points ...
15
votes
1answer
2k views

Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate ...
15
votes
1answer
4k views

How to derive the Weak Formulation of a Partial Differential Equation for Finite Element Method?

I have taken a basic introduction to Finite Element Method, which did not emphasize a sophisticated understanding of a 'weak formulation'. I understand that with the galerkin method, we multiply both ...
15
votes
2answers
4k views

Stopping criteria for iterative linear solvers applied to nearly singular systems

Consider $Ax=b$ with $A$ nearly singular which means there is an eigenvalue $\lambda_0$ of $A$ that is very small. The usual stop criterion of an iterative method is based on the residual $r_n:=b-Ax_n$...
15
votes
2answers
1k views

Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
15
votes
3answers
471 views

Scientific Programming Contests

I regularly compete in so called "Programming Contests", where you solve difficult algorithmic problems with your own code and problem solving skills during a limited time-frame. For referential ...
15
votes
2answers
1k views

What are the efficient, accurate algorithms for evaluation of hypergeometric functions?

I'm curious to know what good numerical algorithms exist for evaluation of the generalized hypergeometric function (or series), defined as $${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{k=0}^{\...
15
votes
1answer
742 views

What is the correct way of integrating in astronomy simulations?

I'm creating a simple astronomy simulator that should use Newtonian physics to simulate movement of planets in a system (or any objects, for that matter). All the bodies are circles in an Euclidean ...
15
votes
3answers
5k views

How is B3LYP implemented in Gaussin 0*, GAMESS-US, Molpro, … etc?

Specifically I want to extend work involving B3LYP started with Gaussian 03 but continued with GAMESS-US. The energies provided by the default B3LYP methods are not the same. There is a discussion ...
15
votes
3answers
62k views

How to determine the amount of FLOPs my computer is capable of

I would like to determine the theoretical number of FLOPs (Floating Point Operations) that my computer can do. Can someone please help me with this. (I would like to compare my computer to some ...
15
votes
3answers
6k views

Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
15
votes
2answers
413 views

Is there any way to do “double preconditioning”

Question: Suppose that you have two different (factored) preconditioners for a symmetric positive definite matrix $A$: $$A \approx B^TB$$ and $$A \approx C^TC,$$ where the inverses of the factors $B, ...
15
votes
2answers
982 views

Open source implementation of rational approximation to a function

I am looking for some open source implementation (any of Python, C, C++, Fortran is fine) of rational approximation to a function. Something along the article [1]. I give it a function and it gives me ...
15
votes
3answers
6k views

What are the advantages and disadvantages of the particle decomposition and domain decomposition parallelization algorithms?

I am running molecular dynamics (MD) simulations using several software packages, like Gromacs and DL_POLY. Gromacs now supports both the particle decomposition and domain decomposition algorithms. ...
15
votes
3answers
2k views

I/O Strategies for computational problems with large data sets?

My research group focuses on molecular dynamics, which obviously can generate gigabytes of data as part of a single trajectory which must then be analyzed. Several of the problems we're concerned ...
15
votes
1answer
2k views

Intuitive motivation for BFGS update

I am teaching a numerical analysis survey class and am seeking motivation for the BFGS method for students with limited background/intuition in optimization! While I don't have time to prove ...
15
votes
1answer
557 views

What is the current state of polynomial preconditioners?

I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ...
15
votes
1answer
3k views

How can I estimate the condition number of a large sparse matrix using PETSc?

I have a PETSc Mat and would like to estimate its condition number.
15
votes
1answer
1k views

Convergence rate of FFT Poisson solver

What is the theoretical convergence rate for an FFT Poison solver? I am solving a Poisson equation: $$\nabla^2 V_H(x, y, z) = -4\pi n(x, y, z)$$ with $$n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + (...
15
votes
1answer
1k views

can I trust this numerical triple integral from Matlab?

Computational Science people: I originally posted this question at Math Stack Exchange and someone commented that I might get "much better" answers here: I am a novice at numerical methods and ...
15
votes
2answers
16k views

Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
15
votes
1answer
434 views

How effective is the 'tendrils of knowledge' approach to Comp. Sci?

I was reading this on Math SE. The basic question is : Assume that someone wishes to study something advanced; one way to do this would be to start off from basics and build up. But the "bigger ...
15
votes
1answer
1k views

How to numerically calculate residues?

I need to calculate the following integral: $$ {1\over 2\pi i} \int_C f(E) \, d E $$ $$ f(E) = {\rm Tr}\,\left(({\bf h} + E)\,{\bf G}(E) \right) $$ Where $\bf h$ is a matrix (one particle kinetic and ...
15
votes
1answer
273 views

How to to easily reproduce published results in my own articles using my own code

I wrote a program/library which I used to obtain results in an article. (Here it is, but my question is general.) I have tests that I run regularly using ctest (it ...
15
votes
1answer
2k views

Visualizing discontinuous Galerkin/finite element data

I would like to visualize simulation results, obtained using the discontinuous Galerkin (DG) approach, within ParaView. Similarly to finite volume methods, the problem domain is divided into cube-...
15
votes
1answer
208 views

Usefulness of elements with mesh-dependent stability

After doing some mathematics related to the stability of elements in 3D Stokes problem I was slightly shocked to realize that $P_2-P_1$ is not stable for an arbitrary tetrahedral mesh. More precisely, ...
15
votes
1answer
691 views

Are there any open source inverse-based multilevel ILU implementations?

I am very impressed with the serial performance of multilevel inverse-based ILU preconditioners, particularly for heterogeneous Helmholtz, but I am surprised to not be able to find any open source ...
14
votes
5answers
12k views

Minimizing the Sum of Absolute Deviation ($ {L}_{1} $ Distance)

I have a data set $x_{1}, x_{2}, \ldots, x_{k}$ and want to find the parameter $m$ such that it minimizes the sum $$\sum_{i=1}^{k}\big|m-x_i\big|.$$ that is $$\min_{m}\sum_{i=1}^{k}\big|m-x_i\big|.$$...
14
votes
5answers
693 views

Repeated nearest neighbor calculation for millions of data points too slow

I have a dataset running into millions of data points in 3D. For the calculation I am doing, I need to calculate neighbor (range search) to each data point in a radius, try to fit a function, ...
14
votes
3answers
4k views

What are the symptoms of ill-conditioning when using direct methods?

Suppose we have a linear system and we know nothing about its conditioning and have no preliminary information about the solution. We blindly apply Gaussian elimination and obtain some solution $x$. ...
14
votes
6answers
974 views

Approximate spectrum of a large matrix

I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard. I am willing to settle for an approximation. Are there approximation ...
14
votes
6answers
2k views

Are there open-source scientific libraries which use modern Fortran with OOP?

I've spent the last couple of months on coding a Fortran program for solving a particular PDE system (describes fluid flow/combustion). I tryed to use latest-standard Fortran and the new OOP ...
14
votes
4answers
348 views

How to deal with too much data?

Our plasma dynamics simulations often produce too much information. During the simulations we record various physical properties on a grid (x,y,z,t) that is as large as (8192x1024x1024x1500), for at ...
14
votes
3answers
1k views

Why is my parallel solver slower than my sequential solver?

I was playing around with PETSc and noticed that when I run my program with more than one process via MPI it seems to run even slower! How can I check to see what is going on?
14
votes
2answers
7k views

What is counterpoise correction?

What is counterpoise correction exactly ? Can you explain when it is needed and why ?
14
votes
3answers
1k views

How to express this complicated expression using numpy slices

I wish to implement the following expression in Python: $$ x_i = \sum_{j=1}^{i-1}k_{i-j,j}a_{i-j}a_j, $$ where $x$ and $y$ are numpy arrays of size $n$, and $k$ is a numpy array of size $n\times n$. ...
14
votes
3answers
649 views

PDEs in Many Dimensions

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...
14
votes
2answers
2k views

FeniCS: Visualizing high order elements

I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...

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