Questions tagged [reference-request]
This tag is for requests for books, papers, and citations.
249
questions
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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 ...
5
votes
0
answers
97
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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
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0
answers
21
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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) ...
4
votes
1
answer
81
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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 ...
0
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0
answers
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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 ...
1
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0
answers
59
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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, ...
1
vote
1
answer
70
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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 ...
3
votes
1
answer
63
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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 ...
0
votes
1
answer
122
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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 ...
6
votes
1
answer
113
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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 ...
1
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1
answer
71
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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) ...
10
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2
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503
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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-...
2
votes
1
answer
115
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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) ...
0
votes
2
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169
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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 ...
1
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1
answer
85
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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 ...
1
vote
2
answers
58
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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 ...
1
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0
answers
93
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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 ...
1
vote
1
answer
104
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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
303
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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 &= ...
1
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3
answers
153
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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 - (\nabla\cdot\sigma)\...
2
votes
2
answers
197
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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 ...
6
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4
answers
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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 ...
0
votes
1
answer
167
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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, ...
0
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0
answers
40
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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, ...
1
vote
1
answer
59
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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.
2
votes
0
answers
96
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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 } \...
2
votes
1
answer
410
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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 ...
1
vote
1
answer
43
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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 ...
3
votes
0
answers
81
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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....
1
vote
0
answers
69
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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 ...
4
votes
1
answer
1k
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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?
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2
answers
75
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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, ...
0
votes
0
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50
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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
217
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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 ...
7
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2
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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
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0
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65
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computing time scale and steady state concentration in microfluidic channels
I have been performing convection-diffusion transport studies on microfluidic channels like the following
The inlet concentration is specific and I obtain the time-dependent concentration profiles of ...
5
votes
1
answer
1k
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4th order tensor rotation - sources to refer
I am trying to model a linear elastic material in Abaqus using a UMAT. For my application, I need to rotate the 6x6 compliance matrix for a given set of eigenvectors (or a rotation matrix). I came ...
0
votes
1
answer
152
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Red flags for numerical computing?
I've learnt the hard way that you should avoid:
computing small numbers as the difference of two large numbers
evaluating chaotic functions with imprecise inputs.
Are there any other red flags a ...
5
votes
0
answers
43
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Origin of phrase `computational microscope'
I have heard the term 'computational microscope' used to describe the practice of molecular simulation (in the context e.g. computational chemistry, materials science) and its use as a numerical tool ...
1
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0
answers
57
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Assume $AX = C$. How to determine which entry of $BX - D$ is non-negative?
Let $A,B$ be $n \times n$ matrices and $C,D$ be $n \times 1$ matrices. Moreover, all entries of $A,B,C,D$ are non-negative. Assume that there is a unique matrix $X$ that solves $AX = C$.
My goal is ...
2
votes
1
answer
158
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Applications of Julia in Chemistry and Molecular Physics?
I was wondering if there are any Theoretical & Computational Chemistry (MM, QM) codes or publications out there that are based primarily on the Julia programming language?
1
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0
answers
42
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Advice for a topic in a seminar
I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence ...
3
votes
1
answer
139
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Adaptive Lagrangian-Eulerian methods and practical benchmark results
Does anyone know of any published study that talks about the practical aspects of running Adaptive Lagrangian-Eulerian techniques for solid and/or fluid mechanics problems?
I'm looking for things ...
0
votes
1
answer
121
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Avalability of SNOPT optimization solver
I'd like to know if SNOPT solver is available free of cost for academic research in any of the optimization software packages.
I came across a few softwares that have SNOPT, but those require a ...
0
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0
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74
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How to solve odd-order differential equations in FEM? Petrov-Galerkin?
I've recently learned about using weighted residuals with the Galerkin method to numerically approximate even-order differential equations (for linear elements, I'm still a beginner). It seems for odd-...
1
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1
answer
494
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Simple particle-in-cell examples
I am studying about the 1D EM-PIC (Electro Magnetics using particle-in-cell) simulation. I want to have a simultaneous time-integration of the electric/magnetic fields plus the motion of free charges ...
2
votes
1
answer
605
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C++ book recommendation- Scientific computing and C++
I'm a master's student in Math interested in Numerical Analysis. I know there are lots of questions like that on this site, but I think this is the best place to ask. So, I'm looking for an ...
1
vote
1
answer
185
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Is C++ and Object-Oriented Numeric Computing for Scientists and Engineers by Daoqi Yang still relevant?
I'm looking to learn C++ primarily from a scientific computation perspective. The approach of the textbook seems ideal to me as it covers C++ from first principles with an emphasis on numerical ...
3
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2
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672
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Minimum number of elements (mesh size) for electromagnetic simulation
Does someone have a reference for the minimum number of elements (or maximum mesh size) for electromagnetic simulations where a mathematical or numerical explanation is given?
I have found several ...
1
vote
3
answers
326
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Morley element implementation reference
I am looking for a detailed reference on the implementation of the Morley element for FEM, specifically for the biharmonic equation. By detailed, I mean that it should discuss the problems associated ...