Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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4 votes
4 answers
2k views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", &...
10 votes
3 answers
995 views

Geodesic triangles in finite element method

I've been working on a new method for 2-dimensional finite element method (FEM) on Riemannian manifolds that involves using geodesic triangles instead of approximating them in an embedded form using &...
0 votes
0 answers
103 views

Can the best constants in harmonic analysis be approximated in principle?

Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
2 votes
0 answers
246 views

Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]

Here it is an Advection-Diffusion equation in 2D: $$ \frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
2 votes
1 answer
275 views

Possible pathological properties of positive definite matrix

Suppose $A$ is a positive definite matrix such that$$ I \preceq A \preceq 1.01I.$$ Is it possible that $\sum\limits_{i=1}^n A_{1i}$ can be arbitrarily large?
0 votes
2 answers
301 views

Solving sparse linear least squares or a positive definite 5-band matrix system fast

I want to quickly solve the following linear least-squares problem $$\min_{x \in \mathbb{R}^n} \left\| A x - b \right\|_2^2$$ with a special sparse structure where each row in $A$ has only up to $4$ ...
7 votes
2 answers
8k views

Conditions for convergence of Euler's method

It is known that a sufficient and necessary condition for $$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$ to have a unique solution is $f$ Lipschitz in $y$ and continuous in $t$. However, ...
0 votes
0 answers
39 views

Rigorous definition of space and time order of accuracy of numerical PDEs

Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
2 votes
0 answers
845 views

Problem using finite difference to solve a initial value problem

I tried to use 'finite difference' method to solve an Initial Value Problem (IVP). For the two boundaries I used periodical condition and for the differential operators I used 4th degree center ...
3 votes
0 answers
88 views

Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$

Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$. Now let $p\ge1$, ...
0 votes
0 answers
39 views

Approximate piecewise linear function with finite incontinuities with polynomial on discrete points

I wish to use single polynomial to approximate a piecewise linear discontinuous function. I wish to minimize the $L^\infty$ norm between such function and original function at every discrete point of ...
2 votes
1 answer
113 views

Finite difference approximation

I'm trying to find formulas for the finite difference approximation "Five-points-stencil" of the first derivative for non-constant grid spacing. It's needed for the outermost left and right ...
32 votes
4 answers
2k views

"Wild" solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional assumptions. In ...
0 votes
0 answers
82 views

Lagrange's interpolating polynomial

Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does ...
1 vote
1 answer
189 views

What's a good approximation for the first derivative at the endpoints of given datapoints for a cubic spline interpolation?

I'm using a cubic spline interpolation for given data points. The boundary condition for the spline is that $f'(a)$ and $f'(b)$ are given (I'm using a finite difference formula $\frac{y_1-y_0}{x_1-x_0}...
0 votes
0 answers
158 views

Numerical approaches to functional equations

I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y . ...
1 vote
0 answers
62 views

Find a vector in the null space of a large dense matrix, where elements in the matrix are not directly accessible

I am working with Conjugate Gradient method to solve for 𝐴𝑥=𝑏, where 𝐴 is an extremely large PSD and Singular matrix. I cannot directly access the elements of 𝐴. The only thing I can do is ...
2 votes
1 answer
213 views

Linear system with sum of Kronecker products

Here and here, specific ways to address the equation in $x$, for $N=2$, are given: $$\sum_{i=1}^N (A_i\otimes B_i)x=c$$ Is anything know about the case $N>2$? I am looking in fact for an efficient ...
2 votes
0 answers
48 views

Efficiently determining surface intersections along a line segment

Background In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
1 vote
1 answer
49 views

Implementable numerical scheme for the equation $a=\text{Erf}\big(z/\sqrt{2N_{a}}\big)$

Let $z>0$ be fixed and $A$ be the set of non-increasing functions from $\mathbb R_+$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|_\infty$. Define by $F$ the operator on $A$ by \begin{equation*} F(...
1 vote
1 answer
103 views

Numerical solution to some functional equation

Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as $$ p_a(t,x):=\frac{1}{\sqrt{2\pi N_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N_a(t)}\right)-\exp\left(-\frac{(x+z)^...
1 vote
0 answers
18 views

Optimal Truncation of LDL-factorization to improve conditioning

Suppose I factored real symmetric quasi-definite $ A_0= L_0 \cdot D_0 \cdot L_0^T$ and the factorization exists, with $D$ diagonal and $L$ unit lower-triangular; and suppose $L$ and $D$ are badly ...
1 vote
0 answers
33 views

Slope assertion in Cholesky on digital computers

For a real symmetric positive definite linear system $$ A \cdot x = b, $$ solved using Choelsky with forward- and backward-substitution, we know it for the numerical approximation $\tilde{x}$ to $x$ ...
2 votes
0 answers
267 views

open problem in numerical analysis [closed]

I am interested in open and current issues in numerical analysis, there are good references in this respect. Thanks for your response
3 votes
2 answers
190 views

Practical symmetric equivalent to QR factorization updates

As we know, the QR-factorization $Q\cdot R=A$ of any real symmetric $n \times n$ matrix $A$ with full rank is unconditionally numerically stable. Further, when A is rank-1-updated, the factorization ...
1 vote
1 answer
53 views

Does norm of discrepancy decrease monotonously in CGLS/CGNR

I am the author of the package for tomographic reconstruction https://github.com/kulvait/KCT_cbct I have implemented CGLS/CGNR , algorithm which applies conjugate gradients on normal equation $$ A^\...
4 votes
0 answers
430 views

What is the computational complexity of Arnoldi algorithm for diagonalization?

What is the space and time computational complexity of finding $k$ eigenvalues of an $N\times N$ matrix using the iterative Arnoldi algorithm? I know that exact diagonalization scales like $O(N^3)$, ...
6 votes
3 answers
2k views

Estimating the variance of a discrete normal distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
3 votes
2 answers
229 views

Exact simulation of a large sample histogram

Say I want to create a histogram of $N$ random points from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins,...
6 votes
2 answers
458 views

Optimal polynomial approximation of rational function $\frac{1}{1-x}$

I've been working on the following polynomial approximation problem. I want to find the optimal Chebyshev approximation of the rational function $\frac{1}{1-x}$ on the real interval $x\in[-\rho, \rho]$...
2 votes
1 answer
81 views

Pressure integrated by parts in finite element method

Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first ...
2 votes
1 answer
95 views

Maximizing a skew-symmetric 4D cross product

How do I find two orthonormal 4D vectors, $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$, which maximize this function: $-19x_1y_0 - 33x_2y_0 + 11x_3y_0 + 19x_0y_1 - 21x_2y_1 - 33x_3y_1 + 33x_0y_2 + ...
5 votes
3 answers
464 views

Optimisation under constraint of Wasserstein distance

Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...
3 votes
1 answer
151 views

Approximation in Bochner spaces

Is there any result like the Bramble-Hilbert lemma for Bochner spaces? More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)...
1 vote
1 answer
96 views

Numerical methods for systems of trilinear polynomials

I have some large system of particular non-linear polynomial equations: each equation mentions at most three variables no variable appears with a degree larger than 1. I'm not an expert in this area ...
7 votes
3 answers
888 views

How to numerically compute $x \ln x$ and related functions near $0$?

I was recently trying to find a numerical solution to a thermodynamics problem and the expression $x\ln x$ appeared in one of the computations. I did not have to find its value very near $0$, so the ...
5 votes
1 answer
891 views

What makes a geometric construction more or less stable?

I'm not entirely sure if this is "research level" math or not, but I asked on Math.SE and it was suggested I try asking here, so hopefully it's of interest to this community. (Original question on M....
1 vote
1 answer
130 views

Finding minimax approximation of a permutation equivariant polynomial

Is there any known method to approximate a given permutation-equivariant smooth function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ as multivariable polynomial function $p: \mathbb{R}^{n} \to \mathbb{R}^{...
35 votes
9 answers
14k views

What is... a grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The ...
14 votes
14 answers
5k views

Basic software libraries for numerical analysis using modern programming languages?

I'm looking for a software library with a scope similar to "numerical recipes", but implemented in a modern programming language. "Modern" in this context means to me: object oriented (not C or ...
12 votes
2 answers
13k views

Closest point on Bézier spline

Given a two-dimensional cubic Bézier spline defined by 4 control-points as described in the Wikipedia entry, is there a way to solve analytically for the parameter along the curve (ranging from 0 to 1)...
4 votes
0 answers
98 views

Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes

Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e., $$ \|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
1 vote
0 answers
45 views

Error bounds for Sobolev space norm approximation on a finite grid

Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space, \begin{multline} f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
3 votes
0 answers
61 views

Enhanced dissipation for Kolmogorov flow

My problem is $$\frac{\partial u}{\partial t}+\ sin(y)\frac{\partial u}{\partial x}=\nu(\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2})$$ with periodic boundary conditions and ...
18 votes
2 answers
10k views

Meaning of $\Subset$ notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
3 votes
4 answers
3k views

Fast multiplication of constant symmetric positive-definite matrix and vector.

Consider the matrix $H=H^T$, $H>0$, $H \in R^{n \times n}$, and the vector $v \in R^n$. In a numerical algorithm, I need to compute the product $b = Hv$. Right now I am following the naive approach:...
2 votes
0 answers
82 views

Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications

What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
1 vote
0 answers
53 views

Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary

Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...
9 votes
2 answers
887 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
7 votes
2 answers
1k views

Mathematics of sustainable development and energy sobriety in the classroom

Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...

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