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1500 questions
14
votes
1 answer
What spatial discretizations work for incompressible flow with anisotropic boundary meshes?
High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this regime because the inf-sup constant degrades as…
Jed Brown
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14
votes
3 answers
Why aren't Krylov subspace methods popular in the Machine Learning community compared to Gradient Descent?
Historically, iterative methods for solving relatively simple-structured systems $Ax=b$ with $A$ being a $4\times 4$ matrix or to find the eigenvalues of that matrix assuming in both problems that $A$ is non-singular were not quite popular and were…
SPARSE
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14
votes
2 answers
How do I find the minimum-area ellipse that encloses a set of points?
I have a set of points that resembles more of an ellipse than a circle. I implemented the optimization formulation below and the solution gives a circle. I tried with various initial values, still to no avail. Have I overlooked or mistaken…
physicsnovice
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14
votes
3 answers
Why would you need frameworks like MPI when you can multi-task using threads?
MPI is an interface which enables us to create multiple processes to be run on a single machine or on a cluster of machines, and enables message passing or in short sorts of communication between processes.
I am sure they are other lots of…
Shivam Jindal
- 143
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14
votes
3 answers
What are the symptoms of ill-conditioning when using direct methods?
Suppose we have a linear system and we know nothing about its conditioning and have no preliminary information about the solution. We blindly apply Gaussian elimination and obtain some solution $x$. Is it possible to determine whether this solution…
faleichik
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14
votes
2 answers
What are new c++20 features that are relevant to scientific computation?
In my research department we plan a small seminar on the new c++20 language standard. There are exhaustive lists online presenting the new features of the language standard, some of which will be of interest to this community, some will be…
MPIchael
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14
votes
1 answer
Conserving Energy in Physics Simulation with imperfect Numerical Solver
I am creating a C++ Physics Simulation where I need to move an rigid body through an acting force field.
Problem: simulation does not conserve energy.
Quesiton: abstractly, how is conservation of energy handled in modern day physics simulations?
My…
akarshkumar0101
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14
votes
1 answer
Understanding the Wolfe Conditions for an Inexact line search
According to Nocedal & Wright's Book Numerical Optimization (2006), the Wolfe's conditions for an inexact line search are, for a descent direction $p$,
Sufficient Decrease: $f(x+\alpha p)\le f(x)+c_1\alpha_k\nabla f(x)^T p$
Curvature Condition:…
Paul
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14
votes
2 answers
Impose the compatibility conditions for mixed finite elements method in Stokes equation
$\newcommand{\v}[1]{\boldsymbol{#1}}$
Suppose we have following Stokes flow model equation:
$$
\tag{1}
\left\{
\begin{aligned}
-\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f}
\\
\mathrm{div} \v{u} &= 0
\end{aligned}
\right.$$
where the viscosity…
Shuhao Cao
- 2,552
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14
votes
3 answers
Boundary conditions for the advection equation discretized by a finite difference method
I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs.
The books and notes which I currently have access to all say similar things:
The general rules governing…
Simon M
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14
votes
3 answers
Rule of thumb for sparse vs dense matrix storage
Suppose I know the expected sparsity of a matrix (i.e. the number of non-zeros / total possible number of non-zeros). Is there a rule of thumb (perhaps approximate) for deciding whether to use sparse matrix storage (specifically, compressed row…
josh_eime
- 153
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14
votes
1 answer
Fast Automatic Differentiation for numpy?
I would like to use automatic differentiation to calculate gradients to function written in numpy.
I've come across a number of packages, including
autograd
tangent
chainer
But none of them seem to support things like numba and numexpr, which I'd…
user357269
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14
votes
5 answers
Examples of PDE computations using parallelism in both space and time
In the numerical solution of initial boundary value PDEs, it is very common to employ parallelism in space. It is much less common to employ some form of parallelism in the time discretization, and that parallelism is usually much more limited. …
David Ketcheson
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14
votes
5 answers
Scientific computing vs numerical analysis
I'm a double major in computer science and mathematics. I love both subjects. I'm thinking in taking a graduate career, perhaps in scientific computing. What's the real difference between scientific computing and numerical analysis? Are they studied…
blade
- 243
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14
votes
6 answers
Approximate spectrum of a large matrix
I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard.
I am willing to settle for an approximation. Are there approximation methods to do this?
While I hope for a general answer…
MRocklin
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