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6
votes
2answers
4k views

Deriving the element stiffness matrix for 2D linear elasticity

I'm following the derivation from Finite Element Method using Matlab 2nd Edition, pg 311-315, which derives of the local stiffness matrix for planar isotropic linear elasticity as follows: Force ...
9
votes
1answer
788 views

Matlab Pde Toolbox: Plot solution on a line or on a submanifold

I'm using the Matlab pde toolbox to solve a certain elliptic equation in 2D. Solution is fine, although I do need to plot it along a given line, i.e. to cut a planar slice from the 3D mesh ...
4
votes
1answer
100 views

Bounded Variation Spaces

Could someone explain me (roughly) the interest of Bounded Variation (BV) Spaces for PDEs ? Is there any numerical application of those space to real problems or is it just a theoretic way to ...
2
votes
1answer
93 views

Cholesky Algorithm loop-Carried

I would like to know how to unroll loop-carried dependency inside the cholesky algorithm. What are the techniques that I should know to accomplish this work? I need to know it because I want to ...
11
votes
1answer
273 views

Are there Improved ways of computing $p \log(p)$?

Most math libraries have a number of versions of logarithm functions. Most of the time we assume them to be perfect, but actually quite a lot of them just offer a certain number of digits of precision....
2
votes
3answers
2k views

Scalar vs. vector potential for magnetostatics

When trying to solve a magneto-static boundary-value problem (BVP) ($\nabla \times \mathbf{H} = \mathbf{J}$ and $\nabla \cdot \mathbf{B} = 0$), one can use either the magnetic vector potential $\...
6
votes
2answers
5k views

Piecewise polynomial interpolation: Hermite vs Lagrange

I am a bit confused of the qualitative behavior of the two methods. Consider quadratic case, start by having points $x_i$, where I know the value and points $y_j$, where the values to be found. If I ...
4
votes
2answers
291 views

What does fundamental solutions stand for in boundary element method?

I gain some introductory knowledge from the materials I read. I feel Ok with the numerical implementation part of boundary element method when the integral equation has been formulated. But the ...
0
votes
1answer
70 views

Thermoplastic Equation solving

I was given a problem by my professor as follows Solve the System $pV=S$ $pcT=kT+BS\frac{dG}{dt}$ $\frac{dS}{dt}=\mu(V-\frac{dG}{dt})$ $\frac{dG}{dt}=f(S,T)$ Where $p$, $c$, $B$, $\mu$ are ...
8
votes
2answers
2k views

Large array in GMP

If I want to use large array say mpz_t A[100000], I got "Segmentation fault (core dumped)" during my compilation. Is there any easier way to solve this?
6
votes
2answers
2k views

Need for quad precision in scientific computing?

Even if quad precision is not directly supported by most CPUs, many Compilers (GNU, Intel) support them. Also some software packages allow to compile with quad precision, e.g. PETSc. But is there ...
2
votes
0answers
105 views

Resampling of values between body fitted and cartesian grids

Assuming there exist two 2D grids on the same domain, one of them is a cartesian one while the other is curvilinear (body fitted regular grid, with curved coordinate lines). I am looking for a way to ...
8
votes
2answers
428 views

Octree cubes to tetrahedrons

I'm trying to learn more about volume meshing and have decided to try to implement a simple volume mesher. The strategy I have chosen is to subdivide my space using an octree, refined based on some ...
4
votes
1answer
256 views

Defining electric current source excitations for surface integral equation formulations

In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields ...
3
votes
1answer
84 views

Randomly choose among N alternatives

I understand how to generate a random sequence of binary variables where 1 occurs with probability p and ...
6
votes
1answer
352 views

Ray casting algorithm for multiple disjoint polygons is still valid?

We're dealing with country borders, that is the set of multiple disjoint domains that is made of polygons. To extract the different point on the map by a given country we've been said to implement ...
4
votes
2answers
1k views

Should the discrete $L^{\infty}$ norm error increase as the mesh refines?

I'm working on an finite element code to solve the boundary value problem: $$-\frac{d}{dx}\left[ k \frac{du}{dx} \right] = f $$ $$u(0)=u(1)=0$$ The matlab code is available here. I'm testing this ...
1
vote
1answer
105 views

What kinds of maths to learn for understanding dynamical systems in cognitive science? [closed]

A current trend in cognitive science is to view the mind as a dynamical system (e.g., Continuity of Mind by Spivey, in which cognition is understood as a "continuous and often recurrent trajectory ...
4
votes
3answers
1k views

CPU benchmarks for numerical kernels

CPU benchmarks available online mostly focus on desktop apps/games and rarely on serial/parallel numerical kernels, specially sparse ones (e.g., MatMult). Some benchmarks like NAS/SciMark exists but ...
4
votes
1answer
4k views

Power Iteration on general matrices (with higher multiplicity of dominant eigenvalue)

To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$ 1) in case $A$ is ...
5
votes
1answer
62 views

Minimal point of a intersection of $n$ convex sets is always the minimal point of the intersection of two convex sets?

I have the intuition that the minimal point (in the sense of having the lowest value in one of the Euclidean space dimensions) of a intersection of $n$ convex sets is always the minimal point of the ...
4
votes
1answer
54 views

Configuration shift for determination of a true dimensionality

What would then be the way to determine a true dimensionality of a configuration of points $X\in\mathbb{R}^{n\times k}$ based on its Gram matrix $G=XX^T$? The "true" dimensionality refers to the ...
4
votes
1answer
177 views

Expectation-Maximization and local maxima

The E-M algorithm does not guarantee convergence to global maxima. Following Applet shows this nicely: http://www.cs.cmu.edu/~alad/em/ Is it possible to generate data set, that guarantees reaching ...
1
vote
2answers
159 views

Affecting the rank of a Gram matrix by configuration shift

Let certain configuration of $n$ points exist in $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$, $d<<n$. Also, let the corresponding Gram matrix be defined as $G=XX^T$. Since $X$ exists in ...
3
votes
2answers
4k views

An eigenvalue algorithm to solve constrained quadratic form minimization

I have a quadratic form $\mathbf{x}^T A \mathbf{x}$ (where $A\in \mathbb{R}^{n\times n}$ is symmetric matrix and $\mathbf{x}\in \mathbb{R}^n$) that I want to minimize given the normalization ...
3
votes
2answers
924 views

How to detect key turning points on a driven road?

I am looking for a description of algorithm which allows me to detect key turning points on the road amongs a set of all given points. I've ilustrated my problem on the below image: Green spots: ...
15
votes
1answer
793 views

What is the correct way of integrating in astronomy simulations?

I'm creating a simple astronomy simulator that should use Newtonian physics to simulate movement of planets in a system (or any objects, for that matter). All the bodies are circles in an Euclidean ...
8
votes
1answer
590 views

Using algebraic multigrid for preconditioning convection-diffusion operators

I implemented a Navier Stokes based on FEM discretization and PETSc for solving the linear system of equations. To create an efficient solution procedure, I follow the paper "Efficient ...
36
votes
6answers
2k views

Symbolic software packages for Matrix expressions?

We know that $\mathbf A$ is symmetric and positive-definite. We know that $\mathbf B$ is orthogonal: Question: is $\mathbf B \cdot\mathbf A \cdot\mathbf B^\top$ symmetric and positive-definite? ...
12
votes
4answers
436 views

Arbitrary Accuracy Scalable Rope Simulation

I am trying to simulate a rope object. The formulation I understand is an array of particles, connected by springs. These springs have very large k-values, so that the line deforms, but stretches ...
4
votes
1answer
512 views

code for surface advection (e.g. level set advection)

I have a 2D surface in 3D that I want to advect under a velocity field. More precisely, I have a surface $S$ and a velocity field $v$ and I want to advect $S$ under $v$ using the flow map of $v$, i.e....
7
votes
2answers
529 views

Numerical Green functions for a nonlinear wave equation

I am trying to put down some code to get numerically the solution of the following PDE: $$ \partial^2_t\phi-\partial^2_x\phi+\lambda\phi^3=\delta(x)\delta(t). $$ Of course, there are several ...
4
votes
1answer
2k views

Why does MATLAB's quadprog outperform MOSEK for my problem?

For a problem I am trying to solve it appears MOSEK's Quadratic Program solver is 100 times slower than MATLAB's Interior Point solver. Has anyone encountered this behavior in the past, or maybe ...
2
votes
2answers
82 views

Estimate (non-)drift in noisy data

I have a time series representing the result of a complex calculation (physical simulation). Due to round-off errors and approximation errors, there will be some "noise" on the data series. In some ...
8
votes
1answer
2k views

How to get sparse complex matrices from my code to PETSc efficiently

What is the most efficient way to get a complex sparse matrix from my Fortran code to PETSc? I understand that this is problem dependent, so I tried to give as many relevant details as possible below. ...
12
votes
1answer
490 views

Algorithms for Large Sparse Integer Matrices

I'm looking for a library that performs matrix operations on large sparse matrices w/o sacrificing numerical stability. Matrices will be 1000+ by 1000+ and values of the matrix will be between 0 and ...
9
votes
3answers
4k views

Fastest algorithm to compute the condition number of a large matrix in Matlab/Octave

From the definition of condition number it seems that a matrix inversion is needed to compute it, I'm wondering if for a generic square matrix (or better if symmetric positive definite) is possible to ...
9
votes
2answers
4k views

How to remove Rigid Body Motions in Linear Elasticity?

I want to solve $K u = b$ where $K$ is my stiffness matrix. However some constraints may be missing an therefore some rigid body motion may be still present in the system (due to eigenvalue zero). ...
4
votes
3answers
521 views

Quality Measures for Various Pseudo-Random Number Generators

According to this paper, Ideally, a pseudorandom number generator would produce a stream of numbers that: are uniformly distributed, are uncorrelated, never repeats itself, ...
2
votes
2answers
1k views

Quadratic program With Linear Constraint vs. Eigen Decomposition Time Complexity-Comparison. Which is faster?

Say I had the choice of choosing one out of the following two optimization problems which I could use to solve my problem. Which choice is the fastest? How much of a trade-off would it be-as in - Is ...
4
votes
2answers
261 views

Does right-hand side influence convergence rate of a Krlylov supspace method?

Consider general system $Ax=b$. Does convergence of the Krylov subspace methods depend on actual vector $b$ assuming initial guess is zero? I mean such factors as locality of the source (with the ...
2
votes
1answer
213 views

How are collisions detected in simulations?

I want to understand how does the simulation identifies and models when two bodies are colliding with each other. For example car crash simulation. The car moves towards the wall or another crash. ...
6
votes
2answers
421 views

FEM toolbox for discretization of higher order PDEs

Is there any (open source) FEM toolbox that allows the direct discretization of higher order PDEs without the need to split them up into systems of second order?
2
votes
2answers
127 views

efficient mean of solving constrained OLS problems?

I was wondering whether there was a efficient procedure for solving constrained quadratic approximations of the form: $$\underset{k\in \mathbb{R}}{\min}\;||x_i-kx_0||_2$$ for fixed values of $x_0,...
3
votes
2answers
141 views

Effect of boundary condition on the local error

Any error analysis is based on the Taylor expansions. So, if I take a finite difference scheme, I can calculate the value of the function at any point using the known value at another node via Taylor. ...
5
votes
1answer
58 views

correct complexity notation

I have written an algorithm where the 2 input arguments are a file and a list of values. I would like to say the algorithm complexity is: ...
2
votes
2answers
529 views

Ill-conditioned Gram Matrix Assembly

I'm trying to find the best approximation to the function $e^x$ in the finite dimensional polynomial space $P_4$ with respect to the standard basis vectors $B=\{1,x,x^2,x^3,x^4\}$ with inner product $$...
5
votes
4answers
520 views

Where to find Leszek Demkowicz's finite element codes or alternatives?

I know that long back Dr. Leszek Demkowicz finite elements codes(1Dhp,2Dhp,3Dhp) were available in his website. I'm finding it difficult to locate it now. Is there any alternatives available to these ...
4
votes
1answer
2k views

Finite Volume Method vs. Finite Element Method for Eulerian and Lagrangian Reference Frames

As far as I can tell, finite volume methods are primarily for eulerian reference frames (such as in fluid dynamics simulations) while finite element methods are easily ameanable to lagrangian ...
8
votes
5answers
365 views

Iterative solution to a nonlinear equation

I appologize in advance if this question is silly. I need to compute the root of \begin{equation} u -f(u) =0 \end{equation} Where $u$ is a real vector and $f(u)$ is a real-vector valued function. ...

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