Questions tagged [reference-request]
This tag is for requests for books, papers, and citations.
273
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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 ...
- 227
2
votes
1
answer
89
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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-...
- 123
1
vote
0
answers
42
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)}{\...
- 395
12
votes
2
answers
2k
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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 ...
- 2,273
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86
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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.
...
- 131
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111
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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 ...
- 2,054
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0
answers
68
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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 ...
- 133
1
vote
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83
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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 ...
- 281
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0
answers
47
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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 ...
- 607
1
vote
1
answer
130
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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?
- 151
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votes
2
answers
293
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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. ...
- 157
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0
answers
90
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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
vote
1
answer
142
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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 ...
- 904
7
votes
0
answers
252
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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. ...
- 281
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votes
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109
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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 ...
- 2,145
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76
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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 ...
- 151
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vote
1
answer
96
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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
12
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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
119
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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 ...
- 904
1
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1
answer
298
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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 ...
- 668
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votes
3
answers
168
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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 ...
- 8,370
4
votes
1
answer
101
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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 ...
- 904
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votes
0
answers
55
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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 ...
- 421
1
vote
1
answer
132
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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....
- 73
1
vote
1
answer
77
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 ...
- 243
5
votes
0
answers
189
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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&...
- 2,273
1
vote
0
answers
22
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) ...
- 8,602
4
votes
1
answer
113
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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 ...
- 243
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66
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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 ...
- 157
1
vote
0
answers
81
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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, ...
- 181
1
vote
1
answer
206
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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 ...
- 2,411
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1
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70
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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 ...
- 33
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votes
1
answer
175
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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
6
votes
1
answer
189
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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
1
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1
answer
88
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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) ...
- 83
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votes
2
answers
644
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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
2
votes
1
answer
172
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) ...
- 83
0
votes
2
answers
371
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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
91
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 ...
- 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 ...
- 23
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0
answers
119
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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
1
vote
1
answer
116
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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
685
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 &= ...
- 83
3
votes
3
answers
204
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
$$J(u, \sigma) = \int_\Omega\left\{\frac{1}{2}A\sigma : \sigma + u\cdot\left(\nabla\...
- 10.1k
2
votes
2
answers
231
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Scientific computing code development hands on introduction
I have a background in Computational Mechanics but my knowledge remains very user-oriented. What I mean by that is that I have a fairely good knowledge about how to use a commercial engineering ...
- 570
6
votes
4
answers
1k
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How important is learning hardware/architecture for scientific computing?
Apologies if this is a bit of a soft, unclear, or opinion-based question. I'm a relatively new PhD student in a (computational) quantum chemistry group. My group develops and maintains a few software ...
- 337
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votes
1
answer
215
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Help me choose a book on the numerical integration of PDEs
For ODEs I have these books:
Griffiths, David, Higham, Desmond J., Numerical Methods for Ordinary Differential Equations, 2010
Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, ...
- 119
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votes
0
answers
51
views
HLLC Riemann solver with shock test 2 - extension to low densities
I am currently using the HLLC solver to solve a 1-D system of Euler equations with very satisfactory results.
However, there are cases where my solution produces low-density, high velocity states, ...
1
vote
1
answer
96
views
What is the best cooling and flippling schedule in simulated annealing?
I've noticed that some heuristics for it on my problem which work surprisingly well. I guess it ought to be systematically studied although I cannot find guides or overviews for it.
- 193
2
votes
0
answers
117
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A priori error estimates - finite element method - mixed boundary conditions
Consider the problem
$$ \left\{\begin{array} {rcl}
-\Delta u & = 0 & \text{ in } \Omega \\
u & = 0 & \text{ on } \Gamma_D \\
\frac{\partial u}{\partial n} &= g &\text{ on } \...
- 985