All Questions

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 $\...

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 ...

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 ...

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 ...

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 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 ...

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

I have calculated the first derivative of following equation using Euler method (first order), Three point Finite Difference method (second order) and Four point Finite Difference method (third order)....

I am struggling with an equation that represents the Weak form of Galerkin method: $ \phi^{T}F(\textbf{u})\sim \int_{\Omega}^{ } \phi.f_{0}(\mathit{u},\nabla \mathit{u}) + \nabla\phi:f_{1}(\mathit{u},...

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 ...

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} + ...

I am trying to learn the Lattice-Boltzmann method and was looking for some good beginner resources explaining the method. I have been looking at some codes online, but have been having trouble ...

I have a quadratic eigenvalue problem of the form: $$(A_2 s^2 + A_1 s + A_0)\hat{v} = 0$$ where $s$ is the eigenvalue. The matrices $A_i$ contain derivatives up to order six, and I have six boundary ...

In a diverging pipe section like the following, the pipe of radius $r$ splits into two pipes of radius $r/2$. Consider a solute transported by convection from node 1. $$\frac{\partial C}{\partial t}...

I need help computing the following integral: $$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$ in this integral $\vec{r}$ ...

I have the correct plot for specific heat capacity when I am using the formula which is $C_V$ = differentiation of entropy with respect to temperature. However, When I try to calculate $C_V$, by using ...

I am trying to implement a fem code on tet10 elements. I follow the lecture notes for tet10 implementation given in http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch10.d/AFEM.Ch10.pdf ...

I need to calculate the inverse of a dense matrix, with some elements taking values as high as 1e9 and some around 1e2. What would be the best method to do it? Note: I am more concerned about the ...

Background & Problem formulation I'm trying to write a simple program in C++ that performs adaptive numerical integration of vector valued integrands (in one variable), i.e. $$\int_a^b \bar{f}(...

I am dealing with a highly nonlinear system of two PDEs. I already have a code to solve the system in case of Dirichlet boundary conditions. The explicit system is: $$ \begin{eqnarray*} \partial_{t}u ...

The journal Association for Computing Machinery Transactions on Mathematical Software (ACM TOMS) publishes many articles on numerical algorithms that include software implementations. According to ...

can you give me a link for a good and simple explanation of the Shortley-Weller finite-difference scheme? I tried to google it but all I get is (inaccessible) academic publications. I also tried ...

I've seen a lot of publications in Computational Physics journals use different metrics for the performance of their code. Especially for GPGPU code, there seems to be a great variety of timing ...

In his talk "What is a good linear finite element?", Shewchuk states that small dihedral angles in linear tetrahedra elements cause ill-conditioning of the stiffness matrix. Do small dihedral angles ...

I'm part of a Computational Science course and come from a non-programming background, so please forgive me my ignorance. I'm working on a set of code in C to numerically solve the Navier Stokes ...

Let us say that I have a function like so: def f(x): return g(x)*h(x) Now, g(x) and ...

While trying to find cell-centered gradients in finite volume method computation of incompressible fluid flow I get over-determined linear system. This is a well known "cell based least-square" ...

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$ where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...

I am working on estimating the position and orientation (pose) of a model (rigid object) from its silhouette in an image. For this, I have constructed an error measure between the model in its pose ...

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...

I have a small FEM implementation program. And I want to add imposing multifreedom constraints (MFC) feature to it. The theory of master-slave method is given here (page 10 for general case). ...

In general, is there a partial trace algorithm (ideally for systems of any size) that can be coded using basic matrix operations found in software like Mathematica or Maple? All of the methods I'm ...

I have difficulties in understanding the role of the weight function $w(x)$ that occurs in the solution of PDEs via the Galerkin approach. Consider a linear differential equation of the form $$ \...

If I have a psd, symmetric matrix $\mathbf{A}$ and I need to do LU decomps on $\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$ (where $\mathbf{D_i}$ is a diagonal psd matrix, where $\mathbf{D_i}$ changes ...

Assume the optimal value of a primal problem is bounded. Is the following statement true? If the primal problem is bounded, then its dual problem is bounded as well.

I have a Hermite Cubic Finite Element Space on a computer in the form of Matlab m-files. More specifically, I can evaluate four "shape functions" $N_1, N_2, N_3,$ and $N_4$, for which the following ...

I have to calculate a huge differential equation. With my laptop, it's going to be computed for several days. Is there a free (I need just for 3 days) fast server for scientific calculations? My ...

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and $0<x&...

Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last? I mean, not like a trig ...

I have a function that is not only infinitely differentiable, but it is also very cheap to calculate any of those derivatives. It looks like: $f(\boldsymbol{C}, \boldsymbol{x})=\sum_{i} C_{i} \prod_{...

My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...

I happened to know that there are advanced established techniques for non-smooth convex optimization in research. For example, these two papers: Nesterov, "Smooth minimization of non-smooth functions"...

I have a linear system of complex numbers. I am using LAPACK' zgesv (actually I am using intel MKL LAPACKE, but I am assuming the algorithm is the same). No assumption can be made about the system. I ...

Please can you help me with my problem? On Wikipedia, in article Discrete sine transform, this is written (chapter Computation): "Although the direct application of these formulas would require O(N2) ...

Consider an instance of non-convexoptimization problem: It seems that this problem is NP-complete. How can I find a suitable reduction for this?

I want to compute an expression of the form: $$L = \ln\sum_i e^{x_i}$$ Suppose that there are many small terms, say $e^{x_i} \approx \epsilon$. If there are $N_\epsilon$ such terms, their ...

Suppose that I have a large (on the order of 10^6 unknowns) 3D scalar Poisson system which I would like to precondition. The boundary conditions have been treated so that the system is SPD. I.e., $$\...

I am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...