All Questions

$\newcommand{\v}[1]{\boldsymbol{#1}}$ 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} \...

Can anybody help me to find the books on numerical solutions(finite difference and Crank–Nicolson methods) of Poisson and diffusion equations including examples on irregular geometry, such as a domain ...

Question What is the most efficient algorithm for finding a row of a matrix which matches a given row? This is the same as a table lookup based on multiple criteria. Context Finite Element Matrices ...

I am interested in globally maximizing a function of many ($\approx 30$) real parameters (a result of a complex simulation). However, the function in question is relatively expensive to evaluate, ...

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...

I'm trying to optimize a flow distributor in a tank such that the velocity and temperature distribution across any cross-section is relatively uniform. There are many parameters I can adjust to the ...

Thomas algorithm can be used to solve a tridiagonal matrix: $$ \begin{bmatrix} {b_ 1} & {c_ 1} & { } & { } & { 0 } \\ {a_ 2} & {b_ 2} & {c_ 2} & { } & { }...

I was just implementing the upwind scheme for a linear transport equation $u_t + cu_x = 0$ where $c=0.5$ and I saw that the solution was indeed advected but over time it starts to diffuse. Can anyone ...

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...

I am working on a project that I need to add a regularization into the NNLS algorithm. Is there a way to add the Tikhonov regularization into the NNLS implementation of scipy [1]? [2] talks about it, ...

The OpenCL programming paradigm promises to be a royalty free opens standard for heterogenous computing. Should we invest our time in developing software based on OpenCL? Pros/cons?

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...

I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...

I'm just getting tucked into fortran 95 for some quantum mechanics simulations. Honestly, I've been spoiled by Octave so I've taken matrix exponentiation for granted. Given a (small, $n\leq 36$) skew-...

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...

There are many different linear solvers, some which work best for diagonally dominant matrices, some for symmetric, some for positive definite ones, some for banded matrices, etc... There are direct ...

I am a Mechanical Engineering grad student, currently working on a project which will be scaled up in the new future to require quite some processing power. I am using C++ for the code that I have, ...

$A$ is a $n \times n$ symmetric positive definite (SPD) sparse matrix. $G$ is a sparse diagonal matrix. $n$ is large ($n$ >10000) and the number of nonzeros in the $G$ is usually 100 ~ 1000. $A$ has ...

Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann ...

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...

I have an inhomogeneous linear system $$ Ax=b $$ where $A$ is a real $n\times n$ matrix with $n\leq 4$. The nullspace of $A$ is guaranteed to be of zero dimension so the equation has a unique ...

I'm trying to implement the Broyden-Fletcher-Goldfarb-Shanno method to find the minimum of a function. I need two initial guesses $x_{-1}$ & $x_0$ and an initial Hessian Matrix approximation $B_0$...

Indefinite systems of matrices appear for example in the discretization of saddle point problems by mixed finite elements. The system matrix can then be put in the form $$\begin{pmatrix} A & B^t \...

A lot of adaptive FEM libraries use more advanced mesh data structures to handle adding/removing nodes, edges, triangles, tetrahedra, etc. For example, the p4est library uses octree data structures ...

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...

I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could ...

I am solving the Dirichlet problem $$ \begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases} $$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...

I know the method of constructing solutions. For example I have a BVP: $$ u_{xx} + u = 0 $$ subjected to: $$ u_{x}(0) = f_1, $$ $$ u_{x}(1) = f_2 $$ If I want to check the correctness of my ...

I'm given a stream of acceleration data with timestamps. The sampling is non-uniform. Apart from Euler, is there a way to integrate the acceleration into velocity? Something more accurate or of ...

How would you represent functions of type $[-1, 1]^n \to \mathbb R \;$ for moderate $n$? How would you integrate them? For small $n$ (1-2) such functions can be represented as histograms, vectors in ...

Disclaimer In the process of typing up this question, I determine its solution. Since I went through the trouble of typing up the question in its entirety, I will post its answer as well. It may ...

I'd like to place as many random points as possible in a 2D square $S=[0,1]x[0,1]$ such that the euclidean distance $d$ between any two points $d$ is greater than a given value $b$ (b is small). I'm ...

This is originally a problem in programming, but since almost no one on Stackoverflow know how to solve this I went here instead; https://stackoverflow.com/questions/23003612/javascript-angular-...

What are theoretical hurdles in applying Galerikin method on, say, first order time dependent ODE? Is there no way we can form an inner product??

So this semester, I'm taking a Finite Element Method course at my graduate school. We started out making codes for 1D bars and came all the way to 8 node solid elements. However, I seem to have run ...

I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate). Let there be a equally ...

We have the advection equation $u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R}$ with initial condition $u(0,x)=u_0(x)$. Suppose that we have the following sheme: I want to find the CFL ...

I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it? thanks for ...

Sorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices....

Is there any preprocessor directives that could be used to use the polylog function? Or is it included in cmath? If so, do you call it by Li or by polylog? EDIT: What I really am trying to do is give ...

I wish to model a fishing rod (or a rope) by joining short segments. (The segments may have equal (short) length but each segment should be assigned its own individual mass.) One segment will ...

I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} -...

I want to find a solution for $xA=0$, where $A$ is a square matrix. I know that most of the LAPACK routines solve for $Ax=b$. So I take $A^T$ as a, and set $b=0$. I have an additional restriction of $\...

Let $p$ be a polynomial with floating-point coefficients and let $a$ be a floating-point value. Is there a method for accurately evaluating the sign of $p(a)$ in floating-point arithmetic? I don't ...

Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ...

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...