All Questions

3
votes
0answers
53 views

Solving multiple linear programs with same constraints but different objective

I have ~30 non-negative variables and 24 equations and I want to find out the upper and lower bound for each variable. Feasible solutions are guaranteed. So for each variable, I solve two LP problem, ...
5
votes
1answer
49 views

Mass matrix and BDF time integration

I have a system of nonlinear equations on the general form: \begin{align} \mathbf{M}(\bar{y})\dot{\bar{y}} =\bar{f}(\bar{y},t) \end{align} Where $\mathbf{M}(\bar{y})$ is a matrix and $\bar{f}$ is a ...
2
votes
0answers
49 views

Why do stabilized formulations for the Navier-Stokes equation maintain the convergence rate for high order polynomial interpolation?

I have a quick questions which has been troubling me lately. When reading the FENICS Finite Element Book they assess various approaches to solver the Stokes equation. Obviously, they discuss the ...
1
vote
1answer
54 views

sum of absolute difference constraint in optimization problem

I am writing a model for an optimization problem. I need to write the following constraint: $$\sum^{N - 1}_i \lvert (a_i - a_{i+1}) \rvert \leq 2\, .$$ How to write this constraint (or linearize)? ...
4
votes
1answer
115 views

Fast matrix multiplication with matrix elements computed on-the-fly (without forming the matrix)

Is there any library or routine for high-performance matrix-matrix product, where the matrix elements are computed on-the-fly using a given function of $i$ and $j$? More specifically, in the problem ...
1
vote
1answer
79 views

Shooting method implementation

I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved. $$ \begin{aligned} \dot x_1(t)&=x_2(t)\\ \dot x_2(t)&=p_2(t)−\sqrt 2 ...
3
votes
2answers
135 views

Asymptotic error of forward Euler

I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
3
votes
0answers
29 views

Minimum axis aliged bounding box of convex polytope

I need to compute a $n-$dimensional integral with $n<10$ on a convex polytope. Since most numerical integration libraries (e.g. Cuba) expect the function to be integrated defined inside an axis ...
1
vote
0answers
91 views

Smallest circumscribed circle in spherical geometry

I work in Python 3 on astrophysics projects. I need to compute the smallest circumscribed circle of a set of points in the sky (so described by Right Ascension and Declination). I have found a code ...
1
vote
0answers
82 views

Lower bound for bilinear form in FEM

I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries. I did some research and found: $$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
3
votes
0answers
79 views

Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
3
votes
1answer
161 views

Computational complexity of Newton's method

the classical Newton's method for non-linear systems of equations is $x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems $...
3
votes
0answers
47 views

Approximate the largest simplex in N-dimensional Delaunay triangulation

I am working on determining the spatial information of a set of $M$ points in $N$-dimensional space. It is well-known that the construction of Delaunay triangulation is expensive in high dimensional ...
0
votes
1answer
38 views

Elliptic PDE: Proving that a 2nd order accurate discretization of the 2nd derivative of the unknown is 2nd order accurate for the unknown itself

For an ODE: $\frac{dy}{dt}=f(y(t),t)$ The Euler Explicit scheme reads: $y_{n+1}=y_{n}+\Delta tf_n$ and it can be easily shown with a Taylor expension that: $y_{n+1}=y_{n}+\Delta t \frac{dy}{dt}|...
1
vote
0answers
62 views

Fast calculation of $A^T B$

I need to compute a matrix-matrix product, $A^T B$, where $A$ is $n \times r$ sparse, and $B$ is $n \times q$ dense. The number of rows $n$ is far larger than both $r$ and $q$. In fact $n$ is so large ...
2
votes
0answers
85 views

Factorize laplacian in terms of first derivative matrix

I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix. The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
1
vote
0answers
36 views

Numerically solving a system of parabolic PDEs and 1st order ODEs

I'm trying to solve the following system of differential equations numerically. What are the available finite difference approaches and matlab solvers to solve such a system? Other approaches to solve ...
0
votes
0answers
58 views

Open source distributed sparse-matrix vector multiplication library

What is the current state-of-the-art open source library that has an efficient implementation distributed memory sparse matrix-vector multiplication? I need to perform repeated SpMVs and I am looking ...
-1
votes
1answer
22 views

Semi-Definite relaxation of non-linear constraint?

I am implementing an optimization problem using semi-definite approach. One of my constraints is of following form $ trace(A∗X)−(k∗trace(A∗X))+(k∗\sqrt {(trace(B∗X)} )==0$ where k is a constant, A ...
4
votes
1answer
99 views

Matrix Representation of a Discretization for a Partial Differential Equation

I want to discretize the following problem \begin{cases} \mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\\ u(...
1
vote
0answers
25 views

Why is the method of im2col with GEMM is more efficient than the method of direction implementation with SIMD in CNN

The convolutional layers are most computationally intense parts of Convolutional neural networks (CNNs).Currently the common approach to impement convolutional layers is to expand the image into a ...
6
votes
1answer
106 views

How do I integrate a function defined over an arbitrary area?

Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral $$ \int_S f(\...
1
vote
1answer
82 views

Multigrid preconditioner for conjugate gradient methods

When solving $A*x=b$ using preconditioned conjugate gradient methods one has to solve $z=K^{-1}*r$ for the preconditioning where $K$ is the preconditioner of $A$ and $r$ is the residual vector. ...
1
vote
1answer
34 views

Custom exponents for Lennard Jones in LAMMPS

I am trying to run an MD simulation using this generalized version of Lennard Jones. $$ U(r) = \left(\frac{r_0}{r}\right)^A -\frac{A}{B}\left( \frac{r_0}{r} \right)^B $$ However, I do not know how ...
0
votes
1answer
28 views

FEM 1D poisson substitution integral issue

I'm trying to solve $ \begin{cases} -u''=f \\ u(0)=0 \\ u(1)= \alpha \end{cases} $ with FEM using reference elements and local coordinates. So we have the global ...
1
vote
2answers
113 views

Method to find PDE equation coefficient satisfying mean solution?

What is the best approach to go about solving a PDE problem of the type \begin{equation} k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\ u=g\; \text{on}\; \Gamma_D\, ,\\ mean(u) = u_\text{...
2
votes
0answers
29 views

Numerical stability while modeling wave equation in staggered grid

I have modeled a simple wave equation given by: \begin{align} & \begin{cases} u_t = v_x \\ v_t = u_x \end{cases} \end{align} Boundary conditions given on interval $M=[-1,1]$ by: \begin{align} u(...
3
votes
1answer
107 views

ODE: should Euler implicit be more accurate than Euler explicit for a given computational step?

I am aware than Euler explicit is conditionally stable, and Euler implicit is unconditionally stable. And I am aware that it is probably pointless to use Euler implicit with a small computational step ...
1
vote
1answer
137 views

Libraries to deal with unstructured grids

I am dealing with a *.cgns file. This mesh format, when saved as an unstructured grid, holds nodes coordinates, nodes connectivity per element and boundary ...
1
vote
1answer
65 views

Wrong result of 'ddot' from BLAS

I am having trouble with a C/C++ program that uses the BLAS routine ddot. I am running Linux and so far LAPACK routines worked without any problems. I get a wrong ...
0
votes
0answers
22 views

Preconditioning grad(ln(u)) term

I am trying to solve a nonlinear diffusion-type problem using the finite element method which has the following terms: $-\nabla\cdot k_1\nabla u - \nabla\cdot k_2\nabla\mathrm{ln}(u) = 0$ in $\Omega$ ...
-4
votes
1answer
83 views

what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
3
votes
2answers
124 views

How to simulate basic semiconductor models using the Drift-diffusion model on Python?

I'm trying to simulate basic semiconductor models for pedagogical purposes--starting from the Drift-diffusion model. Although I don't want to use an off-the-shelf semiconductor simulator--I'll be ...
3
votes
0answers
64 views

Element Preconditioner

Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by ...
4
votes
3answers
133 views

How good are current implementations of rand() in C?

There is a lot of literature that indicates using rand() for simulations might be harmful. A couple of them are as follows: https://www.sciencedirect.com/science/...
1
vote
1answer
166 views

Double potential well with Python

I'm trying to understand the Schrödinger equation and solving it a bit better, and I'm running into some doubts while coding, even though I am adapting the code to this situation. Also I tried asking ...
1
vote
0answers
44 views

Thermal stress to displacement through finite volume method

I am attempting to solve for a displacement field where I know the thermal stresses on my discretized domain, which consists of hex cells. My first question is: is easier way to solve for the ...
1
vote
1answer
46 views

Nature of stationary points of a Lagrangian fuction

I would like to extremize a certain function $f$ with respect to a parameter $x$, under constraints $g_1(x) = 0, ..., g_m(x)=0$. In order to achieve this, I consider the Lagrangian function $L(x, \...
3
votes
2answers
126 views

Recommended visualization tools for higher order finite element solutions?

Is there any software available which can directly render higher order finite element results? In particular, 3D finite elements would be preferable. It seems gmsh has some capability in this regard ...
5
votes
1answer
103 views

Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
2
votes
0answers
28 views

Solving Compressible Euler in Primitive Variables with Galerkin P1 FEM

I have implemented a small compressible Euler solver, discretizing in primitive variables (rho, u, v, p) with standard Galerkin FEM P1 triangular elements, and mixed isotropic and anisotropic/...
1
vote
0answers
43 views

Uniformly sample a point per polytope

I want to uniformly sample a point within each of $10^5$ convex polytopes in each iteration of a solver. The polytopes in one iteration are completely different from the polytopes in another iteration....
-1
votes
1answer
67 views

Relation of Condition of a Matrix and Convergency

Can anybody explain me the relation between the condition of a Matrix and the convergency of a problem. For example how is the relation between the condition of the stiffness Matrix occuring in FEM ...
-1
votes
1answer
36 views

Modeling Diodes in Autodesk CFD

I'm extremely new to Autodesk CFD, and I'm working on a project that deals with diodes heating up and cooling down based on a fixed temperature regulated by a temperature switch. Basically, I have a ...
2
votes
1answer
86 views

Numerically solving a partial differential equation

I am trying to numerically solve the following PDE, $$\frac{\partial u^A}{\partial t} = c_1\frac{\partial^2 u^A}{\partial^2x} \,,$$ where $c_1$ is a constant. The above can be discretized using the ...
2
votes
1answer
189 views

Step-size selection for an Trapezoidal Method ODE solver (ode23t)

I was reading the documentation of the MatLab ODE solver ode23t, and I've seen that the trapezoidal rule is used. Moreover, the error is estimated by ...
3
votes
2answers
118 views

Factoring the sum of two matrices

Given \begin{equation} A_i=B+C_i \end{equation} where $A_i$,$B$ and $C_i$, $i=1,\dotsc,N$ are large square matrices, $B$ is symmetric, $C_i$ are zero matrices aside for a square block on the diagonal. ...
-1
votes
1answer
64 views

Compute affine transformation between two sets of points

Consider two sets of points $P = (P_1, ...,P_n), \ Q = (Q_1, ..., Q_m) $ included in $\mathbb{R}^3$. I'm looking to compute an optimal affine transformation that "maps" $Q$ to $P$, although the sets ...
8
votes
0answers
99 views

How to find a good preconditioner to the system $(A^T A + \lambda I) x = A^T b$?

The system in the title has a damper factor $\lambda > 0$ and the matrix $A$ is sparse and rectangular, with a structure I can exploit to solve matrix vector products very fast. My current solver, ...
1
vote
0answers
82 views

Finite differences for the one-phase Stefan problem

I am trying to code the one-phase, one-dimensional Stefan problem using finite differences in Matlab, similarly to what has already been done in Mathematica (see https://mathematica.stackexchange.com/...

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