Questions tagged [reference-request]
This tag is for requests for books, papers, and citations.
272
questions
33
votes
9
answers
6k
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Modern resources for learning FEM
I need to get started using Finite Element Methods. I am about to start reading Numerical solutions of partial differential equations by the finite element method by Claes Johnson, but it's dated 1987....
- 1,709
30
votes
11
answers
10k
views
Robust algorithm for $2 \times 2$ SVD
What is a simple algorithm for computing the SVD of $2 \times 2$ matrices?
Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
- 966
28
votes
9
answers
4k
views
Modern C++ in scientific computing?
I am looking for books or articles, or blog-posts, or any published material in general, that address specifically the uses of C++ modern features (move semantics, the STL, iterators, lazy evaluation, ...
- 1,709
25
votes
3
answers
16k
views
Recommendation for Finite Difference Method in Scientific Python
For a project I am working on (in hyperbolic PDEs) I would like to get some rough handle on the behavior by looking at some numerics. I am, however, not a very good programmer.
Can you recommend ...
- 677
24
votes
4
answers
5k
views
When do orthogonal transformations outperform Gaussian elimination?
As we know, orthogonal transformations methods (Givens rotations and Housholder reflections) for systems of linear equations are more expensive than Gaussian elimination, but theoretically have nicer ...
- 1,832
19
votes
3
answers
1k
views
Is it well known that some optimization problems are equivalent to time-stepping?
Given a desired state $y_0$ and a regularization parameter $\beta \in \mathbb R$, consider the problem of finding a state $y$ and a control $u$ to minimize a functional
\begin{equation}
\frac{1}{2} \...
- 677
17
votes
4
answers
742
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 ...
- 1,832
17
votes
2
answers
4k
views
Drawbacks of Newton-Raphson approximation with approximate numerical derivative
Suppose I have some function $f$ and I want to find $x$ such that $f(x)\approx 0$. I might use the Newton-Raphson method. But this requires that I know the derivative function $f'(x)$. An analytic ...
- 273
16
votes
4
answers
2k
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 ...
- 3,344
16
votes
1
answer
538
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 ...
- 3,344
15
votes
3
answers
384
views
Citable references for software best practices
I'm currently writing up my PhD thesis. I spent a significant fraction of my PhD cleaning up and extending existing scientific code, applying software engineering best practices which were previously ...
- 273
14
votes
4
answers
4k
views
Looking for Runge-Kutta 8th order in C/C++
I would like to use Runge-Kutta 8th order method (89) in a celestial mechanics / astrodynamics application, written in C++, using a Windows machine. Therefore I wonder if anyone knows a good library / ...
- 277
14
votes
4
answers
35k
views
Difference between Nodes and CPUs when running software on a cluster?
I'm looking into moving some computations of mine to a data center to get more computation power. In the context of this process, I am getting confused by the differentiation of a computation node and ...
- 984
13
votes
3
answers
1k
views
Blaze linear algebra library?
The paper "Expression Templates Revisited: A Performance Analysis of Current Methodologies" in SIAM Journal of Scientific Computing references the "Blaze" linear algebra library. I haven't heard of it ...
- 805
13
votes
3
answers
1k
views
Finite elements on manifold
I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere.
Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy (...
- 133
13
votes
2
answers
919
views
Automatic generation of integration points and weights for triangles and tetrahedra
Usually one would consult a paper or book to find integration points and weights for unit triangle and tetrahedra. I am looking for a method to automatically compute such points and weights. The ...
user530
12
votes
2
answers
1k
views
Higher precision floating-point arithmetic in numerical PDE
I have the impression, from very different resources and talks with researches, that there is a growing demand for high precision computations in numerical partial differential equations. Here, high ...
- 3,590
12
votes
2
answers
2k
views
Why are systems with clustered eigenvalues easy to solve?
I came across the following slide by Theo Diamandis & Zachary Frangella on what makes the linear system $Ax=b$ easy to solve using the conjugate gradient method.
Transcription:
CG converges ...
- 2,253
12
votes
2
answers
415
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, ...
- 2,022
11
votes
5
answers
5k
views
Memory efficient implementations of partial Singular Value Decompositions (SVD)
For model reduction, I want to compute the left singular vectors associated to the - say 20 - largest singular values of a matrix $A \in \mathbb R^{N,k}$, where $N\approx 10^6$ and $k\approx 10^3$. ...
- 3,418
11
votes
3
answers
1k
views
Which linear algebra texts should I read before learning numerical linear algebra?
Assuming one wishes to study numerical linear algebra in depth (and follow journals on numerical linear algebra and matrix theory), which would be a better course/better book to take up at first:
...
- 3,344
11
votes
1
answer
2k
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 ...
- 18.5k
11
votes
3
answers
3k
views
Volume of 3D convex hull of small point sets all on the hull
I have a question that is similar to this one asked before except in 3D, and I only need the volume, not the actual shape of the hull.
More precisely, I'm given a small set of points (say, 10-15) in ...
- 4,480
11
votes
1
answer
4k
views
weighted SVD problem?
Given two matrices $A$ and $B$, I'd like to find vectors $x$ and $y$, such that,
$$ \min \sum_{ij} (A_{ij} - x_i y_j B_{ij})^2. $$
In matrix form, I'm trying to minimize the Frobenius norm of $A - \...
- 870
10
votes
3
answers
2k
views
Benchmarks for Gröbner bases and polynomial system solution
In the recent question Solving system of 7 nonlinear algebraic equations symbolically, Brian Borchers experimentally confirmed that Maple can solve a polynomial system that Matlab/Mupad cannot handle. ...
- 11.1k
10
votes
4
answers
437
views
Reference request: Rigorous analysis of algorithms for PDE and ODE
I'm interested in suggestions for book references on the subject of numerical PDE and ODE, in particular, a rigorous analysis of such methods in a manner written for professional mathematicians. It ...
10
votes
1
answer
1k
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
$$
\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)}
$...
- 799
10
votes
2
answers
640
views
FEM for vector valued problems: reference request
I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-...
- 67
10
votes
2
answers
812
views
Finite difference scheme for "wave equation", method of characteristics
Consider the following problem
$$ W_{uv} = F $$
where the forcing term can depend on $u,v$ (see Edit 1 below for the formulation), and $W$ and its first derivatives. This is a 1+1 dimensional wave ...
- 677
10
votes
1
answer
534
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/...
- 489
9
votes
4
answers
409
views
Fast explicit solution for $\mathbf{A}\mathbf{x} = \mathbf{b}$, $ \mathbf{b} \in \mathbf{R}^3$, low condition number
I am looking for a fast (dare I say optimal?) explicit solution the 3x3 linear real problem, $\mathbf{A}\mathbf{x} = \mathbf{b}$, $\mathbf{A} \in \mathbf{R}^{3 \times 3}, \mathbf{b} \in \mathbf{R}^{3}$...
- 792
9
votes
2
answers
1k
views
Initially Bracketing Minimum for Line Search
Leafing through a few textbooks, I've noticed that the problem of initially bracketing a minimum during a line search tends be an afterthought (at least in my undergraduate texts). Are there well-...
9
votes
3
answers
1k
views
Construction of $C^1$/$H^2$-conforming finite element basis for triangular or tetrahedral mesh
In the paper Hierarchical Conforming Finite Element Methods for the Biharmonic Equation, P. Oswald claimed Clough-Tocher type elements has $C^1$-continuity while being a cubic polynomial on each ...
- 2,542
9
votes
2
answers
4k
views
When is it easy to invert a sparse matrix?
(Crossposted on cstheory.SE)
When is it easy to invert a sparse matrix? Specifically, I'm wondering about the cases in which matrix inversion has similar cost to sparse matrix multiplication, hence ...
- 2,253
9
votes
1
answer
2k
views
What are the strategies for local Adaptive Mesh Refinement (local AMR) on unstructured meshes?
I am interested in local AMR on unstructured meshes. Currently, I'm working with the OpenFOAM library - it supports completely unstructured local AMR:
cell refinement criteria determine a list of ...
- 1,916
9
votes
1
answer
163
views
Numerically stable algorithms for computing remainder of polynomials
Let $f, g \in \mathbb{R}[x]$ and $\deg f > \deg g$. I am looking for asymptotically fast and numerically stable algorithms for computing $f \bmod g$. In the applications intended, both $f, g$ are ...
user182
9
votes
1
answer
472
views
Is there some good mailing list for `Computational Science`?
I am wondering whether there is some very good mailing list or google groups for Computational Science, where we can discuss questions instead of only asking and replying questions.
In fact, I am ...
- 367
9
votes
0
answers
445
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 ...
8
votes
5
answers
1k
views
Some good reading on polygon algorithms
What are some good resources (books, articles, sites) about polygon intersection and union algorithms?
- 81
8
votes
3
answers
282
views
Reference Request for Profiling High Performance Computing Codes
I write codes in Fortran and C for various matrix algorithms. However, when I profile my codes using VTune, I usually run into some terminology that I cannot fully appreciate. Is there a good resource ...
- 3,344
8
votes
2
answers
716
views
Helmholtz and Biharmonic equation examples with exact solution
I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare my numerical solutions with it.
I was able to find quite a few ...
- 1,403
8
votes
3
answers
1k
views
Good introduction to numerical methods for magnetohydrodynamics (MHD)
I very recently started to read up about magnetohydrodynamics (MHD). While I have experience in the fluid part (both theory and numerics), my knowledge about the magneto part is very limited.
At the ...
- 1,238
8
votes
1
answer
250
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,$$
normally one would evaluate this using a multi-dimensional integration library ...
- 11.4k
8
votes
2
answers
425
views
The effect of decoupling a coupled system of PDEs
I asked a somewhat similar question previously but perhaps it might have been too specific for anyone to really answer. Here is a bit more general of a question that I am struggling with. Consider the ...
- 937
8
votes
1
answer
160
views
F(x) = 0 vs. ||F(x)||^2->min
In many areas of application, one needs to solve a nonlinear system of equations
$$
F(x) = 0.
$$
Sometimes, the formulation
$$
\|F(x)\|^2 \to\min
$$
is used. Clearly, every solution $\hat{x}$ of $F(x)=...
- 3,118
8
votes
1
answer
494
views
Good tutorials on how to use Butcher tables?
I tried to go to the primary sources in order to understand how to use Butcher tables to simplify the algebra I need to do when using Taylor series to find the order of accuracy of a scheme, for ...
- 669
8
votes
0
answers
4k
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 ...
- 183
7
votes
4
answers
792
views
Introductory book on computational physics [duplicate]
I'm currently working on my MS in CS and have developed an interest in astrophysics. Luckily one of my professors is a astrophysicist and is currently doing research through computational physics and ...
- 115
7
votes
2
answers
670
views
Recommendations for a usable, fast GPL-compatible derivative-free numerical optimization library that can be interfaced to C++
I am dealing with optimization of functions for which I do not have derivatives available, and the optimization is not constrained. I am searching for a high quality GNU Public License-compatible ...
- 1,916
7
votes
3
answers
265
views
Wanting to learn about matrix solvers
Edit: I was advised to replace the question with a more specific one.
Coming from a very theoretical background, I'm pretty ignorant about what practical matrix solvers exist. (I have been, and will ...
- 173