Questions tagged [reference-request]

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

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Solving systems of the form $y_i=UW x_i$ for $U,W$

I'm looking for pointers/examples of solving system of equations $y_i=f_W(f_U(x_i))$ for $W,U$ where $f_M(x) \approx M x$ $U,W$ are updated simultaneously $i\in (0, 10^{12})$ Simplest example is ...
Yaroslav Bulatov's user avatar
2 votes
1 answer
294 views

Literature request covering Chebyshev's pseudospectral collocation method

I would like to request some literature recommendations covering Chebyshev's pseudospectral collocation method for solving space-time PDEs. It would be nice if it even contained some example problems ...
Nikola Ristic's user avatar
0 votes
0 answers
44 views

Which numerical method can I use to solve this system of hyperbolic PDEs?

Backround The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
Nikola Ristic's user avatar
2 votes
1 answer
94 views

references for optimization in the context of parameter identification with finite elements

i am performing parameter identification for a non-linear partial differential equation (elasticity) that I solve with finite elements. My optimization problem is a non-linear least squares data-...
Simon's user avatar
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1 vote
0 answers
43 views

Solution to the Liouville-Gibbs equation

What would be the approach to numerically solve for $\rho(x,t)$ the following equation with some initial conditions $$\frac{\partial\rho}{\partial t} +\sum_{i=1}^n\left(\frac{\partial(\rho g_i)}{\...
homocomputeris's user avatar
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 ...
Yaroslav Bulatov's user avatar
1 vote
0 answers
90 views

Closed formula to diagonalize discretized (perhaps randomized) Laplacians

I was wondering whether there is a closed formula for the eigenvalues and eigenvectors of the discretized Laplacian in (edit) $[0,1]^n$ with a uniform grid, using what I imagine is a $2n+1$ stencil. ...
Aner's user avatar
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6 votes
0 answers
117 views

References on the theory of Petrov-Galerkin methods for more "basic" problems

In my reading on various aspects of FEM, Petrov-Galerkin methods often arise in the study of solutions of convection-dominated systems, such as Hughes' work on Navier-Stokes, or systems where optimal ...
whpowell96's user avatar
  • 2,259
1 vote
0 answers
70 views

Has the arithmetic for exotic (unsigned float with positive exponent) number format been solved?

The data type is a doubly unsigned float. This is where the value and exponent are both strictly positive. The range of this number should include $0$ and $[1, ~2^{\text{exponent}})$, skipping all ...
Jeff's user avatar
  • 133
1 vote
0 answers
86 views

Resource to learn assembly code

I'm a PhD student in mechanical engineering and I have to perform a lot of simulations for my project. In my lab we use several well-known libraries, from FEM to machine learning. As we're doing ...
FEGirl's user avatar
  • 281
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0 answers
48 views

Efficient cutting of mesh edges

I am looking for efficient algorithms to cut a mesh along edges. I have a (half-edge) mesh and a list of inner edges that I want to cut, such that both are new boundary edges. At each vertex there can ...
allo's user avatar
  • 607
1 vote
1 answer
135 views

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?
吴yuer's user avatar
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5 votes
2 answers
305 views

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. ...
Lilla's user avatar
  • 259
5 votes
0 answers
91 views

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 ...
Warren's user avatar
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1 vote
1 answer
162 views

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 ...
Dan Doe's user avatar
  • 1,033
7 votes
0 answers
271 views

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. ...
FEGirl's user avatar
  • 281
1 vote
1 answer
97 views

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,...
Gideon Ilung's user avatar
1 vote
2 answers
123 views

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 ...
Dan Doe's user avatar
  • 1,033
1 vote
1 answer
432 views

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 ...
NNN's user avatar
  • 610
7 votes
3 answers
169 views

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 ...
nicoguaro's user avatar
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4 votes
1 answer
102 views

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 ...
Dan Doe's user avatar
  • 1,033
0 votes
0 answers
57 views

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 ...
Natasha's user avatar
  • 421
1 vote
1 answer
133 views

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....
SNIreaPER's user avatar
1 vote
1 answer
84 views

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 ...
Lilla's user avatar
  • 259
5 votes
0 answers
195 views

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&...
Yaroslav Bulatov's user avatar
2 votes
0 answers
38 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) ...
Anton Menshov's user avatar
  • 8,652
4 votes
1 answer
119 views

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 ...
Lilla's user avatar
  • 259
0 votes
0 answers
66 views

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 ...
Lilla's user avatar
  • 259
1 vote
0 answers
81 views

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, ...
Artem Alexandrov's user avatar
1 vote
1 answer
219 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 ...
Abdullah Ali Sivas's user avatar
3 votes
1 answer
71 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 ...
ahb's user avatar
  • 33
0 votes
1 answer
182 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 ...
Masa's user avatar
  • 194
6 votes
1 answer
208 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 ...
むすんでひらいて's user avatar
1 vote
1 answer
89 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) ...
TheComander's user avatar
10 votes
2 answers
677 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-...
bob_bill's user avatar
2 votes
1 answer
177 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) ...
TheComander's user avatar
0 votes
2 answers
389 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 ...
coyote's user avatar
  • 188
1 vote
1 answer
92 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 ...
Natasha's user avatar
  • 421
1 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 ...
Rien's user avatar
  • 23
1 vote
0 answers
119 views

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 ...
TheComander's user avatar
1 vote
1 answer
116 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 $U_2$, ...
Pedro H. N. Vieira's user avatar
2 votes
1 answer
740 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 &= ...
TheComander's user avatar
3 votes
3 answers
209 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\...
Daniel Shapero's user avatar
2 votes
2 answers
250 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 ...
Dude's user avatar
  • 570
6 votes
4 answers
1k 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 ...
tmph's user avatar
  • 337
0 votes
1 answer
235 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, ...
DeltaIV's user avatar
  • 119
0 votes
0 answers
52 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, ...
AtmosphericPrisonEscape's user avatar
1 vote
1 answer
107 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.
Moonwalker's user avatar
2 votes
0 answers
119 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 } \...
Beni Bogosel's user avatar
  • 1,005
2 votes
1 answer
605 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 ...
FEGirl's user avatar
  • 281

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