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4
votes
2answers
1k views

Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$. Many ...
4
votes
2answers
179 views

What mapping strategy should I use when solving many large linear systems of equations?

I am working on a problem that involves solving many (thousands) of distinct linear systems of equations, each with thousands of variables. Let's assume that the size of each matrix is exactly the ...
4
votes
1answer
201 views

Choosing preconditioner for unsymmetric pressure-velocity coupled system

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
4
votes
2answers
193 views

How do the properties of a matrix affect the linear system solving

For a general matrix A, there are many properties to describe it: symmetric positive definite or indefinite, condition number, spectrum and so on. I am curious about how these properties affect the ...
4
votes
0answers
652 views

Finding roots of systems of equations with a Jacobian that is singular everywhere

Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that $$ 0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
4
votes
2answers
1k views

choice of the norm for the error of the numerical method

When I read books on finite differences they often end up using discrete $L^2$ norm for estimating the error as it naturally arises from weak formulation. I was wondering if people do that in Sobolev ...
4
votes
1answer
269 views

Points on the interface

We consider the problem $\left\{\begin{matrix} k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\ u=0 & \text{ in } \Gamma \end{matrix}\right.$ where $\Omega \subset \mathbb{R}^2$ open and ...
4
votes
1answer
2k views

Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
4
votes
2answers
459 views

Derivative-free nonlinear optimization of discrete objective function with linear constraints (simplex)

I am trying to optimize a constrained-problem with a discrete, non-linear objective function. Evaluating this function is also fairly expensive. Nevertheless, despite the above two factors, I hope, ...
4
votes
2answers
196 views

Finding dominant eigenvectors of an operator that is small but costly to evaluate

Suppose I have a symmetric linear operator $A:\mathbb{R}^k \rightarrow \mathbb{R}^k$ where $k$ is "small" (eg., $k=100$), and I want to find it's first few eigenvectors, (eg., $10$ eigenvectors). If ...
4
votes
1answer
432 views

Verlet Leap-frog Method of approximating orbits

I found the following simulation of an orbit given the semi-major axis of the ellipse, eccentricity, initial velocity, the mass of the object being orbited, and initial position of the orbiting ...
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
3answers
547 views

Backward stable projection and normalization of a vector

Given a machine precision unit vector $n$, and an arbitrary vector $v$, I want an unconditionally backward stable method to compute $$f(v) = \frac{v-nn'v}{\left|v-nn'v\right|}$$ In other words, ...
4
votes
1answer
203 views

Solving a nonlinear equation with random variable

I would like to solve an equation that looks like this UPDATE $E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$, $\theta$, is a random variable ...
3
votes
1answer
2k views

Solving Advection (Convection) - Diffusion - Reaction Partial Differential Equation in Python

I am looking for library written in Python which will enable me to solve the coupled nonlinear equations which looks like: I need the library which will enable me to couple this solver to other ...
3
votes
2answers
152 views

Generate Random Number outside Bounds:

Say I have a random number generator that generates a number within [0, RAND_MAX], and RAND_MAX < UINT_MAX. How do I ...
3
votes
2answers
3k views

C++ Library: What is the common libraries that do polynomial arithmetic?

I need to know what libraries (in C++) support polynomial arithmetic specially over a field. So I can give to it an array of coefficients of polynomial over a field and it returns the roots of ...
3
votes
1answer
884 views

Non-linear root finding when the Jacobian is almost singular

I'm trying to solve a system non linear-equations: $$ \frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0 $$ for $i = 1, \dots, 15$, using Newton's method: $$ \lambda^{k + 1} = \lambda^k ...
3
votes
3answers
778 views

Mathematical programming formulation of triangle intersection

Given variables $a_1$, $b_1$, $c_1$ and $a_2$, $b_2$, $c_2$ representing the vertices of two plane triangles, how might one specify the requirement for the two triangles to intersect as an objective ...
3
votes
0answers
82 views

Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
3
votes
1answer
155 views

Derivatives of Approximate Matrix inverses

I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below: I have a question concerning the ...
3
votes
1answer
682 views

Similarity of two CSV files (or more?)

A CSV file is characterized by a header, describing the n datarows to come. The header is a text string separated by a separator. A CSV header might look like (C1) ...
3
votes
3answers
850 views

Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
3
votes
2answers
2k views

Does the time-dependent advection-diffusion equation have an analytical solution?

The advection-diffusion problem, where $0<x<1$, $$u_t = (-au + du_x)_x$$ with Dirichlet boundary conditions, $ u(0)=1,~u(1)=0 $ , has the steady-state solution, $$ u(x) = \frac{e^{\lambda} - ...
3
votes
2answers
4k views

Octave 3D mesh, data from file

I have a big file with 3 columns: density, dimension, value. example: 10 0.3 200 10 0.4 300 20 0.3 250 20 0.4 320 ... I am trying to draw a 3d plot - ...
3
votes
2answers
2k views

What is the difference between “Newton-type” and “Newton-like” iteration?

Is there any clear classification between different iterative methods? What is the difference between Newton-type and ...
3
votes
2answers
212 views

Does length unit in FEM affect numerical condition?

I try to solve a system of coupled PDEs using FEM. Unfortunately, the originating matrix has very poor condition. After days of double checking and thinking, I suspect the following reason: Given a ...
3
votes
0answers
380 views

On solution of a class of discrete-time Lyapunov equation for systems with multiplicaitve noise

Let's consider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F_{1}...F_{p}]$ ...
3
votes
2answers
389 views

Energy Conservation in Conservation Laws with Source Terms

I'm wondering if anyone can help me understand energy conservation when using conservation law methods (i.e. Riemann solver, High-Resolution Wave-Propagation Methods) with the addition of source terms:...
3
votes
1answer
548 views

My calculated laser pulse duration is too large. Where am I wrong?

I am currently writing a small Python script to estimate the pulse duration from the optical spectrum. At the end, the idea is to observe the effects of the spectral phase on the pulse duration and ...
3
votes
2answers
183 views

Find $\min x^TAy$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

In the following problem, $A$ is a given $\mathbb{R}^{m\times n}$ matrix: \begin{align} \mbox{minimize}\quad & x^TAy \\ \mbox{subject to}\quad & 1^Tx=1^Ty=1, \\ & x\ge 0,y\ge 0. \end{...
3
votes
3answers
223 views

$H^1$-convergence rate of finite element method for Poisson equation, depending on element order

I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method $$-\nabla^2u=f$$ with $$u=...
3
votes
1answer
262 views

precision loss in non-trigonometric, periodic functions using FFTW and NaNs after marching forward in time (Fortran)

I have developed a pseudospectral solver of the Navier-Stokes equations using FFTW. I tested my formulation of right hand sides (RHS) of the NS equations against standard trigonometric functions (...
3
votes
1answer
295 views

Finite volume with cell averages vs cell totals for conservation equations

What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method? For example, consider the conservation law, $$ u_t + \...
3
votes
3answers
460 views

Numerical approximation for a known exact solution of advection-dispersion equation

My goal is to create a numerical solution of 1D-solute transport (Convective-dispersion equation, CDE) to match it's analytical solution based on experimental data. The CDE can be written as (where, C=...
3
votes
2answers
317 views

Nanoseconds vs. picoseconds in numerical quantum problems with Matlab ODEs

Hello there and thanks for taking a look at this problem. This problem is related to my previous question and I will therefore use a similar introduction from, Choice of step size using ODEs in ...
3
votes
1answer
717 views

Add User-defined/custom differential equations in OpenFoam (CFD)

I am new to OpenFoam. And I am trying to add a set (user defined) of differential equations to OpenFoam. I want to solve this user defined set of equations at each time point in addition to standard ...
3
votes
1answer
160 views

Numerical computation of the velocity in the steady Navier-Stokes equation

I've asked this question on Math.SE too. Let $d\in\left\{1,\ldots,4\right\}$ $\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open and $\partial\Lambda$ be Lipschitz $V:=\left\{u\in H_0^1(\...
3
votes
1answer
2k views

Intel MKL - Difference between mkl_intel_lp64 and mkl_gf_lp64

I am currently trying to link a program against the Intel MKL 11.0 library instead of using NetLIB or OpenBLAS. Doing this I recognized the following error which I can not explain to my self at the ...
3
votes
1answer
263 views

How can I avoid roundoff error when calculating the difference $\textrm{erfc}(a) - \textrm{erfc}(b)$?

In this excellent answer, it is recommended that one make use of the $\textrm{erfcx}$ function to avoid roundoff error in calculating dealing with $x < 25$ (approximately). So, one scales their ...
3
votes
2answers
365 views

Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE". However I have never come across a brief document that substantiates that claim, ...
3
votes
2answers
707 views

Low-rank updates in BFGS

I have read this and other threads on this site on BFGS, but I still don't have a clear understanding of what it's meant by low-rank updates. For example, I read the following in this book: The ...
3
votes
0answers
119 views

Automatically generate constraints for trajectory optimization

This is a follow up to my previous post here I'm interested in performing trajectory optimization from the problem mentioned in abov link. I want to supply the following as dynamical constraints to ...
3
votes
2answers
175 views

Optimization of known function with respect to two unknown function arguments

I have a data set, composed of points $(x_i, y_i)$ for $i=1,N$. I also have a known function $F$, which maps these points $x_i$ to $y_i$ as such $F(x_i, a(x_i),b(x_i)) = y_i$, where $a(x_i)$ and $b(...
3
votes
2answers
1k views

How Do I solve large systems given UMFPACK memory limitations?

I am trying to solve a system of equations (A x = b) for 3D heat diffusion (i.e. each equation has at most 7 terms not including the constant "b" term) using UMFPACK with boost numeric bindings to C++....
3
votes
2answers
178 views

polynomial time solvability

Consider the following optimization problem: $Min \qquad C^TX$ S.t.: $\qquad AX=0;$ $x_ix_j=x_kx_t$$\quad $for some $i\neq j\neq k\neq t$ $X=(x_1,x_2,...x_n)$ and $\quad x_j\geq 0\;\; j=1,2,......
3
votes
3answers
462 views

What are the current obstacles to reaching exascale computing?

I have heard that one of the current goals in scientific computing is to surpass the exascale level by 2018 or so... After having passed the petascale a few years ago, one would think that the ...
3
votes
2answers
437 views

Graphing electric potential of a ring of charge using MATLAB help

Here is a summary of what I am trying to do: Use MATLAB to compute the potential $V$ at any point $(x, y, z)$ in space due to a uniform ring of charge. Use a Riemann sum to compute the integral ...
3
votes
2answers
222 views

Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$ I solve the last equation using shooting method and implicit Runge-Kutta ...
3
votes
1answer
109 views

Setting up optimization problem in GEKKO

I have the following dynamical system, $\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$ $\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \label{2}$ $\eqref{1}$ represents the exact ...

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