Questions tagged [reference-request]
This tag is for requests for books, papers, and citations.
3
votes
1answer
99 views
Derivation of backward differentiation formulas(BDF)
I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations.
From the description given here, I could follow the steps till equation (5).
I am finding ...
3
votes
0answers
31 views
Guide for finite-difference schemes for Hamilton-Jacobi-Bellman Equations
I need to solve a simple, low-dimensional Hamilton-Jacobi-Bellman equation.
Is there a simple guide for doing this numerically using finite-difference schemes? I found a few research articles ...
1
vote
0answers
140 views
Good C++ optimization library for BFGS
To implement maximum likelihood estimators, I am looking for a good C++ optimization library that plays nicely with Eigen's matrix objects. Eigen has some capability of interfacing of its own but if ...
1
vote
0answers
35 views
Has there been a comparison bewteen SIMPLE/SIMPLER and JFNK for steady CFD?
I'm looking for a comparison between the Jacobian-Free Newton-Krylov (JFNK) method performance compared to the conventional CFD nonlinear solution methodologies like SIMPLE.
Does anyone know if such ...
2
votes
2answers
74 views
Elliptic equation with finite volume and unstructured high order geometry
I have found that in unstructured mesh, discretizing the laplacian operator with finite volumes requires special care, as given in An Introduction to Computational Fluid Dynamics: The Finite Volume ...
2
votes
1answer
73 views
Term for the typical “linear in the larger dimension, quadratic in the smaller” cost for linear algebra
Many dense linear algebra decompositions (QR, SVD...) on an $m\times n$ matrix have cost
$$
O(\max(m,n)\min(m,n)^2)
$$
when implemented in practice on a computer. Is there a colloquial name or a more ...
2
votes
0answers
30 views
Sensitivity Lagrangian solution general case
I have asked this question already on maths and mathoverflow.
Just a question about a literature reference. I am writing a paper for engineers.
Usually for the Lagrange multiplier problem
$$
\...
6
votes
2answers
73 views
What good are hard-sphere event-driven molecular dynamics simulations in the face of chaos?
Simple hard-sphere dynamical systems can exhibit chaotic dynamics. Due to finite-precision arithmetic when implemented on a computer, the presence of chaos implies that for a given set of initial data,...
12
votes
1answer
261 views
For noisy or fine-structured data, are there better quadratures than the midpoint rule?
Only the first two sections of this long question are essential. The others are just for illustration.
Background
Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
0
votes
1answer
64 views
References in Scientific Computation for a Computer Science Undergrad? [duplicate]
TLDR: What references and pre-requisites are necessary for a Computer Science Undergraduate to get ready for a Masters and Career in Scientific Computation / Computational X
For a Computer Science ...
1
vote
1answer
15 views
Implementation of a ratio with a well-defined limit
I'm implementing a function $f(x_1, x_2, \dots x_n)$ given by the ratio of two closed form equations.
$$ f(x_1, x_2, \dots, x_n) = \frac{g(x_1, x_2, \dots, x_n)}{h(x_1, x_2, \dots, x_n)} $$
...
0
votes
2answers
516 views
Physics Simulation in C++
OK, I know a bit of C++ (very basic syntax), and I want to do physics simulation in C++, like stuff like (also the things mentioned here):
Ripples and waves over a 2-d surface
Vibrating string/...
6
votes
1answer
220 views
How to directly compute the inverse of an ill-conditioned dense matrix
I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try?
...
3
votes
1answer
73 views
Computation of the heat kernel from Brownian motion
This question is rather simple but I have some difficulties to find code. Let us suppose that I wrote a routine, in a given language, that computes the evolution of a particle doing Brownian motion in ...
2
votes
0answers
146 views
Analysis and numerical methods for PDEs arising in industrial problems
Assume some basic knowledge of numerical methods for PDEs, acquired through A. Quarteroni's Numerical models for differential problems.
I'm looking for a reference to get started on the analysis of ...
3
votes
1answer
214 views
Radiation boundary condition (heat transfer)
I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM:
$-k \frac {\...
1
vote
0answers
111 views
Reference request: free/open source books on intro numerical PDE
I'm interested in finding a free online textbook that is an introductory textbook on numerical PDE for engineers. It should cover the basics of finite differences, finite elements, and Fourier series ...
0
votes
1answer
115 views
pde-constrained optimization
I'm trying to solve a problem where I have a initial and final distribution of tumor, and my goal is to find the best map of parameters (diffusion and reaction terms) for a reaction-diffusion equation,...
9
votes
1answer
205 views
Resources on mesh generation for finite element methods
I know that this is not really apart of the rules as this is a recommendation question and these don't really have an answer per say. But, like this forum posting: https://stackoverflow.com/questions/...
1
vote
2answers
222 views
C++ template design pattern for groups (algebra)
Having both programmed my share of c++ and studied some beginners group theory some year ago, I got curious about this...
Is there any particularly popular template based (object oriented) design ...
7
votes
0answers
107 views
Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?
I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
10
votes
1answer
240 views
First appearance of the phrase “inverse crime”
In research on inverse problems, it's common to construct a synthetic data set from a known set of parameters and then test whether the inversion technique can reconstruct those parameters. In doing ...
1
vote
0answers
30 views
Computing Direct Scattering Transform
I'm working on the Nonlinear Schrodinger equation (NLSE) in 1d:
$$i\psi _t (t,x)+ \psi _{xx} + |\psi|^2\psi = 0 \, ,$$ for $t\geq 0$ and $x\in \mathbb{R}$.
This equation is integrable, and so ...
1
vote
1answer
125 views
Comparison between FEM libraries & languages
Is there any modern online resources which compares the most popular finite element method libraries/packages/languages?
1
vote
1answer
117 views
Leveraging scipy for matrix free finite elements
This will be a very general question.
I have a 3D finite element code in Python which I would like to extend to handle "large" problems (~10^8 unknowns in the global system). Right now I am using the ...
3
votes
2answers
126 views
Sharing a matrix with the math community
What would currently be the best way to share a matrix with the math community?
(I'm aware of Matrix Market but it seems the last update was in 2007...)
4
votes
1answer
170 views
Resources for solving fluid-structure interaction problems
I would like to get started solving Fluid-Structure interaction problems.
I already have some experience with Finite Elements, including my own MATLAB and Julia software packages for developing ...
2
votes
2answers
128 views
Preconditioner for scalar laplacian system
Suppose that I have a large (on the order of 10^6 unknowns) 3D scalar Poisson system which I would like to precondition. The boundary conditions have been treated so that the system is SPD. I.e.,
$$\...
0
votes
1answer
83 views
Newbie help on FEM contact problem
I am modelling the time-evolution of a soft material with a tetrahedral mesh. I use an FEM method to compute the forces on each node, and then numerically integrate the positions and velocity of the ...
1
vote
0answers
78 views
Discrete operator textbooks
This will be a vague question.
When I was writing a finite element matrix assembly routine, a colleague noticed that I had a bug in my code because the sparsity pattern of the one of the blocks didn'...
1
vote
0answers
79 views
Kahan Summation for Three-Term Recurrences
Kahan summation applies to summation problems, but not to three-term recurrence relations. However, a three-term recurrence shares many of the features of a summation-albeit with a rescaling step at ...
4
votes
0answers
146 views
Rhie--Chow interpolation on PDE level
The Rhie--Chow [1] interpolation seems to be a standard tool in the Finite-Volume discretization of incompressible flows.
It is commonly defined on the discrete level [2].
In the lecture notes [3] --...
6
votes
2answers
415 views
Reference request for computational fluid dynamics
I have good background of finite element methods and continuum mechanics and I am familiar with fluid mechanics. My aim is to understand the required theory and to write my own simple codes using ...
0
votes
2answers
89 views
A program to simulate cellular automaton model
I have worked on mathematical modeling based on differential equations, and now I want to simulate a cellular automaton based on a ...
1
vote
0answers
13 views
Optimizing estimator of composed functions when function is known
Note: This was cross-posted from comp-sci, as I didn't know this community existed!
I have a problem which I'm looking to see if there is literature on:
Consider three types of actors, a Director, ...
9
votes
1answer
272 views
$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)
I know that the piecewise linear finite element approximation $u_h$ of
$$
\Delta u(x)=f(x)\quad\text{in }U\\
u(x)=0\quad\text{on }\partial U
$$
satisfies
$$
\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)}
$...
1
vote
2answers
2k views
BLAS libraries for Octave or Matlab, preferrably with GPU support?
I just searched around a bit for BLAS implementations and was amazed by the sheer amount of libraries around. Does someone know of a benchmark or otherwise rating of the various libraries?
How easy ...
5
votes
0answers
225 views
How do I implement a working Lanczos biorthogonalization method for Krylov subspace?
I am trying to implement Lanczos biorthogonalization algorithm to construct a Krylov subspace basis. Given a matrix $A$, and vectors $v,w:(v,w)=1$, it produces three matrices $V,W,T$ such that $V^T W =...
0
votes
1answer
87 views
Solving Poisson equation while suffering from the curse of dimensionality
I have a heat transfer equation in a cube in $R^{100}$: $[0,1]\times[0,1]\times[0,1]\dots$:
$$
\nabla^2 \varphi = f,
$$
with boundary conditions set in a form that in the number of points $p_i$, ...
1
vote
2answers
166 views
Flow past square cylinder benchmark in 2D. Famous papers to compare
I'm looking for a benchmark for the flow past square cylinder problem in 2D to compare my results. I have found some papers, but it seems that there are different results in different papers. I want ...
2
votes
0answers
42 views
A test suite of large systems of nonlinear equations
I am looking for a kind of modern test set of large nonlinear problems. The only option I managed to find so far is rather dated: http://folk.uib.no/ssu029/Pdf_file/Testproblems/testprobRheinboldt03....
1
vote
1answer
50 views
Literature on comparing Simplex and Interior-Point-Methods (or combining both of them)
Do you know some interesting literature concerning the comparison of Simplex and Interior-Point-Methods referring to linear optimization?
I also read about the possibility of combining both of them ...
3
votes
0answers
142 views
Numerical methods for calculating the inverse CDF when closed form approximation not available
I need to calculate the inverse CDF for a probability distribution, however there is no closed form approximation available in the literature.
The distribution I am working with is the Normal Inverse ...
3
votes
1answer
220 views
Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?
Are there numerical methods of solving the following fully nonlinear time-dependent PDE:
$$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$
for $\textbf{...
8
votes
1answer
637 views
Integer vs float multiplication performance, modern CPUs
Are there benchmarks for how many multiplications of various integer types compared to floating point types can be achieved per second on modern CPUs?
I'm trying to get some hint if it would be ...
0
votes
1answer
69 views
Pseudo random numbers
I am learning how to use pseudo random number generators but the instructor just told us how they work without explaining why they work.
For example, can one prove that the numbers generated by LCG ...
2
votes
2answers
186 views
Why do Newton-Krylov iterations stagnate in this problem? [closed]
Consider this integro-differential heat equation taken from SciPy documentation page:
$
\nabla^2 P = \alpha \left(\iint_\Omega \cosh(P)dx dy \right)^2
$
which was found in this question.
In the ...
4
votes
1answer
399 views
Intro to DG Finite Element methods
I wrote a number of 1D/2D FE and FD programs as a bachelor student, but the main problem I continually came into contact with was gradient shocks related to convection/diffusion problems in convection-...
9
votes
1answer
208 views
When is it advantageous to iterate integrals numerically?
If there is an $(n+1)$-dimensional integral of the form
$$ \int_{[0,1]^{n+1}} f(x, y)\,\mathrm{d}^n x \,\mathrm{d}y,$$
normally one would evaluate this using a multi-dimensional integration library ...
1
vote
1answer
2k views
Introduction to Lattice Boltzmann methods [closed]
I am trying to learn the Lattice-Boltzmann method and was looking for some good beginner resources explaining the method. I have been looking at some codes online, but have been having trouble ...
