Questions tagged [reference-request]
This tag is for requests for books, papers, and citations.
264
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Could you recommend some books on FEM that explain various data-structures in FEM?
I want to understand the data structure of elements, elements around elements, and so on, and various other data structures in FEM, could you please recommend some books?
- 21
6
votes
1
answer
194
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Continuous vs discontinuous space-time FEM
What are some reasons for approximating a (e.g. parabolic) PDE using the space-time method, with continuous finite elements in time, vs discontinuous finite elements in time?
Are there e.g. ...
- 131
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votes
0
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How to get an "optimal point" for refinement in FEM adaptive mesh refinement?
Consider the following 1D problem
\begin{align*}
\begin{cases}
\displaystyle
-\frac{d^2u}{dx^2} = f(x), \hspace{0.5cm} x\in (a,b) \\[4mm]
u(a) = u_{a}, \ \ u(b) = u_{b}
\end{cases}
\end{align*}
I ...
- 51
1
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1
answer
109
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Non-Uniform Grids: Approximation Quality: First Order Finite Difference vs. First Order Finite Volume
Consider the advection/transport equation in 1D with constant velocity $a(x) \equiv 1$
$$u_t(t,x) + u_x(t,x) = 0$$
on a, say, periodic domain.
On uniform grids $$ \{x_i\}_{i = 1, \dots, N}, \quad x_{i ...
- 819
6
votes
0
answers
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Matrix-free FEM references
I've seen that many people are using matrix-free fem codes in my community (mechanical engineering). I have to admit that I googled a bit and I didn't manage to find a good reference for the subject. ...
- 211
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votes
0
answers
55
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Solving L1 minimization problems in Eigen
I have an $m\times n$ dense matrix $\mathbf{M}$ and wish to solve $\mathbf{M}\mathbf{x} = \mathbf{b}$ via any of the "L1" methods using Eigen. By this I mean I'm happy with using basis ...
- 2,055
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70
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Book recommendation on multiphysics
I want to learn multiphysics such as fluid-structure interaction where the simulation is performed for heat transfer, fluid dynamics, and electrodynamics.
Could you please recommend some books about ...
- 21
1
vote
1
answer
92
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Automatic Differentiation using foward mode on matrices
Whilst googling I see reverse mode automatic differentiation (AD) tends to be used when optimising neural networks.
Would it not be better to use forward mode and treat your input as a single variable,...
- 13
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votes
0
answers
9
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Non-Temporal Weighted Graph Datasets
I am searching for datasets to evaluate an algorithm designed for tasks such as node-classification (edge-prediction, etc.) on weighted and potentially directed graphs.
The Stanford Network Analysis ...
- 1
1
vote
2
answers
109
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Nonlinear Hyperbolic PDEs: Known solutions
I would like to collect some test-problems for nonlinear hyperbolic PDEs (Euler Equations, Shallow Water Equations, Ideal MHD, Acoustic Perturbation, ...) for which analytical solutions are known.
A ...
- 819
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Kolmogorov n-width
Could someone please point me to an understandable definition of the Kolmogorov n-width? I'm having a hard time figuring out what is the output of the definition - is it an integer?
Edit: I realize ...
- 654
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3
answers
163
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Reference request: Philosophy of Computational Science
Do you suggest any references (papers, monographs, books) about the philosophy of computational science?
Recently, I found out about the following two:
Winsberg, E. (2009). Computer simulation and ...
- 8,209
4
votes
1
answer
96
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Stabilized Many Stage Runge-Kutta methods instead of Local/Multirate Time Stepping
Locally refined meshes are often inevitable for accurate, yet feasible computations.
In the context of time-dependent PDEs, however, this comes at the cost that (due to the CFL condition) reducing the ...
- 819
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0
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52
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Finding optimal values from multiple parameter estimation runs
I've performed a parameter estimation repeat (i.e. 1000 parallel runs with the same
initial values of parameters). I am trying to estimate ~20 parameters using measurements from experiments.
After ...
- 459
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118
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What are the prerequisites and resources to self-learn the Boundary Element Method for Contact Mechanics problems?
What are the prerequisites to learning BEM?
In which order is it advisable to learn BEM and FEM - either one before the other, or does it not matter?
What are some good resources to self-learn BEM?
P....
- 21
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vote
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73
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A priori estimates in finite elements for inhomogeneous heat equation
Consider the problem
$$\partial_t u-\Delta u = f\\
u(\Sigma_1)=f_D\\
\partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$
where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be ...
- 53
5
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177
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Dense least-squares with millions of variables
Suppose $X$ is a dense $m\times n$ data matrix and we seek to find $w$ by approximately iterating least-squares filter equation:
$$w = w - \mu (X'X)^{-1}X'(Xw-b)$$
What are known approaches for $10^9&...
- 1,799
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answers
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Unconstrained convex optimization: correlation between dimensionality and Lipschitz constant
The author of the SIAM News article "Optimization Theory and Perspectives on the Field of Machine Learning" mentions:
... For unconstrained convex optimization, GD (gradient descent) ...
- 8,542
4
votes
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A priori FEM estimates without $H^2$ regularity
In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's ...
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63
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Reference request for finite elements theory
Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how:
without needing isoparametric ...
- 131
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0
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71
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Implementation of integration schemes for ordinary differential equations in Python and peformance comparison
I look for a book/manual where I can find implementations of different integration schemes for ordinary differential equations (like 4-th order Runge-Kutta) in Python with Numba.
To be more specific, ...
- 171
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127
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About using SOCP solvers to solve QCQP
I have noticed that some commercial solvers transform QCQPs into SOCPs and use SOCP algorithms to solve the resulting problem. I am wondering if there is a benefit to this approach over using a pure ...
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66
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Requesting for Finite Difference Methods reference in Portuguese or English
Crossposted on Mathematics SE
I have been assigned a group project for an introductory Linear Algebra subject on Finite Difference Methods and sparse matrices. Our professor advised we use Gilbert ...
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141
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Mesh refinement in the Finite Element Method
I need some good references on how to implement programmatically the hp-refinement of meshes in the Finite Element Method in two/three-dimension. I've searched the web a lot and read many articles and ...
- 194
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votes
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145
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Fast way of computing the action of a matrix power on a vector
For integer $k>0$, it is well-known that one can use binary exponentiation to evaluate the matrix power $\mathbf A^k$, where $\mathbf A$ is an $n\times n$ matrix.
However, it is not clear to me if ...
- 61
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80
views
How to Impose nonhomogeneous Neumann Boundary Condition in the DG Formulation
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
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votes
2
answers
558
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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
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answer
146
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DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
0
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2
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294
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Searching for recent code source for "Parallel scientific computing in C++ and MPI "
I am learning C++ scientific computing with "Parallel scientific computing in C++ and MPI A Seamless Approach to Parallel Algorithms and their Implementation" since it kept coming up a lot ...
- 88
1
vote
1
answer
88
views
Discrete model of cell - cell communication
I am trying to understand how cell to cell communication is studied using a discrete modelling framework. Could someone please suggest suitable references or libraries which already have ...
- 459
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vote
2
answers
59
views
Two-dimensional ordering issue – alternate sort order ascending/descending to reduce fluctuations - trivial?
I have a solution in search of a problem that some of you could perhaps help me with.
Let $L$ be a list of elements. Each element has two inherent properties/attributes ($a$, $b$) that can each be ...
- 23
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0
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108
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Finite Difference Approximation for the Laplacian in 2D that produces a nonsymmetric matrix
Consider the following PDE
\begin{align}
-\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\
u &= 0 \ \ \text{en} \ \partial ((0,1)\times (0,1)) \label{P2}
\end{align}
if we ...
- 83
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Finding the source of numerical instability in a electrostatic problem solved by conformal mapping
I'm using conformal mapping to solve a 2D electrostatic problem (calculating the potential $u(x,y)$ in the plane). Let $C_1$ and $C_2$ be two circles at an electric potential $U_1$ and $U_2$, ...
2
votes
1
answer
468
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Tensor product representation for the 9-point finite difference approximations for the Poisson equation
If we use 5-point finite difference approximations in a uniform rectangular grid to solve the Poisson PDE
\begin{align}
-\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\
u &= ...
- 83
2
votes
3
answers
198
views
Stable finite elements for the mixed form of the elasticity equations
The mixed form of the elasticity equations is to find the unique critical point of the Hellinger-Reissner functional
$$J(u, \sigma) = \int_\Omega\left\{\frac{1}{2}A\sigma : \sigma + u\cdot\left(\nabla\...
- 9,013
2
votes
2
answers
209
views
Scientific computing code development hands on introduction
I have a background in Computational Mechanics but my knowledge remains very user-oriented. What I mean by that is that I have a fairely good knowledge about how to use a commercial engineering ...
- 550
6
votes
4
answers
965
views
How important is learning hardware/architecture for scientific computing?
Apologies if this is a bit of a soft, unclear, or opinion-based question. I'm a relatively new PhD student in a (computational) quantum chemistry group. My group develops and maintains a few software ...
- 327
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votes
1
answer
188
views
Help me choose a book on the numerical integration of PDEs
For ODEs I have these books:
Griffiths, David, Higham, Desmond J., Numerical Methods for Ordinary Differential Equations, 2010
Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, ...
- 119
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46
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HLLC Riemann solver with shock test 2 - extension to low densities
I am currently using the HLLC solver to solve a 1-D system of Euler equations with very satisfactory results.
However, there are cases where my solution produces low-density, high velocity states, ...
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1
answer
74
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What is the best cooling and flippling schedule in simulated annealing?
I've noticed that some heuristics for it on my problem which work surprisingly well. I guess it ought to be systematically studied although I cannot find guides or overviews for it.
- 193
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108
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A priori error estimates - finite element method - mixed boundary conditions
Consider the problem
$$ \left\{\begin{array} {rcl}
-\Delta u & = 0 & \text{ in } \Omega \\
u & = 0 & \text{ on } \Gamma_D \\
\frac{\partial u}{\partial n} &= g &\text{ on } \...
- 965
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votes
1
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496
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Reference request: C++ and numerical analysis book
I'm a master student with a good Numerical analysis background. I'm going to do a master thesis in the same subject, but I need to use C++ since my advisor loves it, and I also believe it's the best ...
- 211
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vote
1
answer
44
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Introductory reference on bit-twiddling? [closed]
I'm looking for introductory resources--web pages, book chapters, articles--that introduce the C language with an immediate focus on bit-twiddling functions (e.g. bitwise XOR, AND, shifts), and of ...
- 121
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votes
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118
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What is this QR-factorization-based preconditioning called?
I have recently started to delve into someone else's code, and there is a part in there I don't quite understand. The authors of the code use some form of pre-conditioning to speed up the optimization....
- 203
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83
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Book recommendation on numerical methods for solving Integro-Differential equations
I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
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FEM Python book
Is there any book or site available with Finite element Method for partial differential equations with python code apart from Fenics?
user37062
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2
answers
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Efficient ODE steppers with query of $f$ and $\nabla f$ is efficient
Assume we have an IVP $y'(t) = f(t,y)$, and that $\partial_t f$ and $\nabla f$ are cheap to compute.
Assume further that more derivatives are not cheap to compute, or inaccessible for some reason, ...
- 2,055
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votes
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51
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Finding a CFD paper with extra degree of freedom variable in mass conservation
I am trying to find a paper that I saw about a year ago. I am not sure of the actual date of the paper.
I believe it was a finite difference CFD paper. The interesting part of the paper was the ...
2
votes
1
answer
258
views
Diagonalization of Hermitian matrices vs Unitary matrices
What are the general algorithms used for diagonalization of large Hermitian matrices and Unitary matrices? ($>5000 \times 5000$)
LAPACK seems to diagonalize Hermitian matrices almost 20 times as ...
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votes
2
answers
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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 ...
- 1,799