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6
votes
0answers
122 views

What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
6
votes
0answers
217 views

Linear vs Non Linear inverse problems: Does non-linearity help?

This is not a typical question with a deterministic answer. If this is not the right place, feel free to close it. For the past one year I have been working on various kinds of inverse problem. Most ...
6
votes
0answers
392 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula $$ (A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u} $$ results in small errors in relation to the standard matrix inverse operation after each application, ...
6
votes
0answers
419 views

Implementing a Hill-Type Muscle Model

I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work. Both papers link to the paper by Geyer and Herr, which describes this model: However, the paper on this ...
6
votes
0answers
111 views

How does an unpivoted QR fail to reveal rank?

An unpivoted QR factorization produces a triangular factor $R$. A rank-revealing QR factorization is typically done with column pivoting. My question is, how does an unpivoted QR factorization fail to ...
6
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0answers
97 views

Reference request for numerical variational method

I have a variational problem where the unknown function is a periodic path $\gamma:[0,1)\to\mathbb{R}^2$, and the functional is $$ \int_0^1\left( \tfrac12\|\dot\gamma(s)\|^2 + \mathcal{F}[\gamma]\...
6
votes
0answers
168 views

Find the solution of linear equation using Wiedemann/ Krylov method

Let given $M =$ 1 0 1 0 1 1 1 1 1 and $b =$ 1 0 1 How to find the solution $x_3$ where $x=${$...
6
votes
0answers
160 views

Strategy for solving a non-trivial differential equation

I would like to numerically solve an equation of a type as shown below. Does anyone of you have an idea how to approach such a problem? Any links to literature or for further reading would be greatly ...
6
votes
0answers
249 views

Integrating highly oscillatory functions

I have a logarithmic grid, upon which i have two functions that are similar to this one (this is only the last 100 points): These are essentially very similar to a Sin function at this point. I need ...
6
votes
0answers
252 views

A simple question about 1D finite element derivatives

For 1D derivative we have \begin{equation} F(x) = \frac{\partial f(x)}{\partial x} \end{equation} \begin{equation} f(x)=\sum_{i}f_ie_i(x) \end{equation} \begin{equation} F(x)...
6
votes
0answers
603 views

Stochastic Galerkin projection approach for using generalized polynomial chaos expansion (GPCE) in solving PDE

I want to know if there is any way to define the test and trial function in the way that I want instead of using the default functions. So if I want define the polynomial and basis and coefficient, ...
6
votes
0answers
2k views

What is the best OpenFOAM RAS turbulence model for a motorbike/vespa problem?

I am learning OpenFOAM as a hobby and using my Vespa racing as the topic to apply it to. The objective is to produce modifications that improve the top speed (as well as getting some values such as ...
6
votes
0answers
416 views

How to do FEM in sector elements?

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 ...
5
votes
0answers
160 views

Solving $AXB + X\odot C = D$ matrix equation

Can anyone see a way to solve this equation efficiently? $$AXB + X\odot C = D$$ I tried a straightforward solution that involved vectorizing $X$ but that turned out too expensive for my application -...
5
votes
0answers
66 views

Minimum of quadratic assignment (QAP) with convex objective

Suppose $A,B\succeq0$ and $C\in\mathbb R^{n\times n}$. I am hoping to solve an instance of the following optimization problem: $$ \min_{\textrm{permutation matrices }P} \mathrm{tr}(BP^\top AP+C^\top ...
5
votes
0answers
75 views

Sensitivity of BFGS to the accuracy of the gradient

I am studying how to speed-up the BFGS method using quantum computing techniques. I have used a method of speeding up the gradient of the function, but it sacrifices the precision value of the ...
5
votes
0answers
47 views

Evaluate Nth root of a rational to a correctly rounded float

Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible. I want to evaluate an expression in the ...
5
votes
0answers
94 views

How to solve a 4th order nonnegative LASSO problem?

I need to solve the following 4th order nonnegative LASSO problem: $$ \min_{x \geq 0} \quad || |Ax|^2 - b ||^2 + \lambda ||x||_1 $$ where $|\cdot|^2$ denotes element-wise squared. $A$ is small size (e....
5
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0answers
81 views

Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
5
votes
0answers
99 views

Best way to numerically compute elliptic integrals of the third kind with complex arguments?

I need to compute elliptic integrals of the third kind with complex arguments, preferably in C++. Is there code out there to do this? I have discovered the Arb library, but that does much more than I ...
5
votes
0answers
74 views

Is there a numerically stable way to take $\epsilon \rightarrow 0$ in integrals of the form $\int \frac{f(x)dx}{x+i\epsilon}$?

The Sokhotski-Plemelj theorem states, $$\lim_{\epsilon\rightarrow 0^+}\int_a^b\frac{f(x)dx}{x+i\epsilon} = \mathcal P \int_a^b \frac{f(x)dx}{x} - i\pi f(0). $$ Is there a numerically stable way to ...
5
votes
0answers
174 views

Any way to avoid catastrophic cancellation when computing the discriminant of a quadratic function?

Homework disclaimer... The task: We are using the following algorithm to solve the quadratic equation $x^2+2px+q=0$: $x_1=|p|+\sqrt{p^2-q}\mathtt{;}$ $\mathtt{if}\,p>0\,\mathtt{then}\,...
5
votes
0answers
88 views

How can Navier--Stokes equations have asymmetric solutions such as Karman vortex streets

The Navier--Stokes equations are axially symmetric, so with symmetric boundary conditions, how can features such as Karman vortex streets develop? I understand that in reality symmetry does never ...
5
votes
0answers
271 views

Cressman interpolation and objective analysis

I have read this question and answer – Interpolation of scattered data to a regular grid in python and I am doing something similar as I have temperature values of the atmosphere at different heights ...
5
votes
0answers
176 views

conjugate gradient for Newton's method with non positive definite Hessian matrix

I want to minimize a non-linear function $f(x)$ using Newton's method. At each optimization step, I compute a descent direction $d$ to update $x$ using a second-order approximation of $f(x)$: $$ \...
5
votes
0answers
68 views

Discrete sine and cosine transform for mixed derivatives

Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's ...
5
votes
0answers
65 views

Padua-type pointset for functions singular on line $x=y$

The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a ...
5
votes
0answers
150 views

Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
5
votes
0answers
131 views

Roots of transcendental equation involving bessel functions

I need an efficient way to numerically find the first $n$ positive roots $\lambda_n$ of the transcendental equation $$ \dfrac{J_0 (\lambda_n r) Y_1 (\lambda_n) - J_1 (\lambda_n) Y_0 (\lambda_n r)}{...
5
votes
0answers
79 views

Minimize interesting objective function with knowledge of gradient nonlinearity?

I plan on using a Quasi-Newton method (L-BFGS) to minimize a non-linear objective function. $$ f: \mathbb{R}^n \rightarrow \mathbb{R}$$ The gradient is kind of interesting: as the values of the ...
5
votes
1answer
471 views

Finite volume a posteriori error estimation

I'm wondering what alternatives there are to a grid convergence study to judge solution accuracy for a given grid resolution when doing steady-state RANS simulations on an automatically generated ...
5
votes
0answers
149 views

How to optimally choose points for multivariable Hermite interpolation?

I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input. I would like to create interpolation polynomial for it. In one-dimensional case ...
5
votes
0answers
142 views

Rank-one Update to a Rank Revealing QR (RRQR) Factorization?

Suppose we are given an RRQR factorization for some matrix $A \in \mathbb{R}^{m \times n}$, $A\Pi = QR$ where $m > n$. Is there a cheap way to update $A' = A + uv^{\top}$ given this factorization?...
5
votes
0answers
447 views

Sparse matrix format and sparse-matrix sparse-matrix multiplication

I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices). Mainly I have to multiply such ...
5
votes
0answers
302 views

Optimization on the manifold of stochastic matrices

So I have an optimization problem of the form $$\text{maximize}\hspace{3mm}f(A):{\bf R}^{K\times K}\rightarrow{\bf R}$$ $$\text{subject to}\hspace{19mm}A^T{\bf 1}=\bf{1}$$ $$\hspace{33mm}A\geq 0$$ ...
5
votes
0answers
124 views

Are there any benefits of computable analysis to numerical algorithms

Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis). When I heard of the existence of computable analysis I ...
5
votes
0answers
140 views

Difference of convex functions optimization problem in R

I am seeking of any already written R package which could help in an optimization technique which is called Difference of convex functions. This technique is sketched here and could be very useful ...
5
votes
0answers
118 views

Load balancing/partitioning with unknown weights

For a grid-based numerical simulation, I am looking for a load balancing/partitioning algorithm that not only distributes my grid elements, but also determines (approximates) their respective weights. ...
5
votes
0answers
155 views

Is it generally unstable to use in multidimensional simulations finite difference schemes with higher orders than 2?

I'm part of a team trying to generalize a 1D Advection-Diffussion-Reaction code we inherited by extending it to 2D by using dimensional splitting, i.e. solving advection and diffusion for x and y ...
5
votes
0answers
183 views

Comparison between Voronoi and Delaunay 2D quadrature methods

This question is a search for further answers from a question on maths.stackexchange.com. I've inherited some numerical quadrature code that is designed to integrate sparse 2D data. The quadrature is ...
5
votes
0answers
150 views

Elliptical problem with Robin BC

Working in finite differences, I am using a transformation on the temperature variable $\Theta = \int_{T0}^T \kappa(T)dT$ to linearize the steady-state heat equation into a Poisson equation $-\...
5
votes
0answers
83 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
5
votes
0answers
180 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
5
votes
0answers
92 views

Benchmarks or generic configurations for optimal flow control

I am about to test my algorithms for solving optimal control problems of type: Find an input $u$, such that for a time interval $(0,T]$ the cost functional $$J(v,u) = \mathcal M(v(T)) + \int_0^T\...
5
votes
0answers
480 views

frozen coefficient vs. constant coefficient

This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of ...
5
votes
0answers
182 views

Finding quadrature weights for a given set of points? How to select points such that all weights are positive?

Currently, I fit a Finite Element solution of a PDE on a spectral basis. The matrices ($R^{25000\times 2000}$) of the corresponding system of linear equations are highly ill-conditioned ($\kappa \...
5
votes
0answers
480 views

How does GAMG in OpenFOAM really work?

I use OpenFOAM for CFD simulations. A very popular preconditioner is GAMG which needs a low number of iterations per a time step in SIMPLE or PISO solvers that are used to simulate the fluid flow. I ...
5
votes
0answers
195 views

Discontinuous Galerkin for flow through porous media

I am struggling with DG methods for 2 phase flow through porous media. I managed to get the global pressure, total flux equations to work with an unconditionally stable mixed FE DG formulation as ...
5
votes
0answers
113 views

How to choose a stable PML for pseudo-spectral method with strongly varying velocity

My friend was working on this, and he asked me about the stability of PML while applying on pseudo-spectral method, I believe his concern was how to choose the difference(if the difference should be ...
5
votes
0answers
117 views

Wavelets frame for $L^2[0,\infty)$

I need a wavelet frame for $L^2[0,\infty)$. Moreover, the wavelet should be twice differentiable and with continuous second order derivatives. Hopefully, the wavelet should have compact support (...

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