All Questions

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 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, ...

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 ...

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 ...

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) ...

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 ...

$A$ 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'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 ...

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'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 $B_0$...

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 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 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 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 ...

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 ...

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, ...

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, ...

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 ...

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??

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....

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 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 ...

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'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 $\...

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 ...

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 ...

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 have a linear system whose matrix that is diagonally dominant, non-symmetric, but banded. Since the band-radius is 2 (producing only 5 variables per equation), a banded direct solver (gaussian ...

I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...

I'm trying to code a parameter estimation for a reaction-diffusion problem. Namely, knowing the distribution of tumor density $u$ at time $0$ and $T_f$ ($u^0$ and $u^f$), what are the best ...

I have a geometry and a finite element method where there are a couple internal interfaces across which I want to enforce some fairly simple jump conditions. One recommendation I've gotten has been ...

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

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...