Questions tagged [reference-request]

This tag is for requests for books, papers, and citations.

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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 ...
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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&...
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1 vote
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21 views

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) ...
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4 votes
1 answer
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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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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 ...
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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, ...
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1 vote
1 answer
70 views

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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3 votes
1 answer
63 views

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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122 views

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 ...
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6 votes
1 answer
113 views

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 ...
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1 vote
1 answer
71 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) ...
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10 votes
2 answers
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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-...
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2 votes
1 answer
115 views

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) ...
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0 votes
2 answers
169 views

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 ...
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1 answer
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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 ...
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1 vote
2 answers
58 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 ...
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0 answers
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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 ...
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1 vote
1 answer
104 views

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 $U_2$, ...
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2 votes
1 answer
303 views

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 &= ...
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1 vote
3 answers
153 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 - (\nabla\cdot\sigma)\...
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2 votes
2 answers
197 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 ...
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6 votes
4 answers
938 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 ...
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  • 327
0 votes
1 answer
167 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, ...
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  • 119
0 votes
0 answers
40 views

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 vote
1 answer
59 views

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.
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2 votes
0 answers
96 views

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 } \...
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2 votes
1 answer
410 views

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 ...
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1 vote
1 answer
43 views

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 ...
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3 votes
0 answers
81 views

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....
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1 vote
0 answers
69 views

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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4 votes
1 answer
1k views

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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1 vote
2 answers
75 views

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, ...
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0 votes
0 answers
50 views

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 ...
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2 votes
1 answer
217 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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7 votes
2 answers
2k 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 ...
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1 vote
0 answers
65 views

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 ...
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  • 533
5 votes
1 answer
1k views

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 ...
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0 votes
1 answer
152 views

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 ...
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5 votes
0 answers
43 views

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 ...
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1 vote
0 answers
57 views

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 ...
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2 votes
1 answer
158 views

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?
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1 vote
0 answers
42 views

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 ...
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  • 163
3 votes
1 answer
139 views

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 ...
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0 votes
1 answer
121 views

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 ...
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  • 533
0 votes
0 answers
74 views

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-...
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1 vote
1 answer
494 views

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 ...
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2 votes
1 answer
605 views

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 ...
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  • 163
1 vote
1 answer
185 views

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 ...
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3 votes
2 answers
672 views

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 ...
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1 vote
3 answers
326 views

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 ...
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