Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,224
questions
4
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4
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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
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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
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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
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Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
2
votes
1
answer
275
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Possible pathological properties of positive definite matrix
Suppose $A$ is a positive definite matrix such that$$ I \preceq A \preceq 1.01I.$$ Is it possible that $\sum\limits_{i=1}^n A_{1i}$ can be arbitrarily large?
0
votes
2
answers
301
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Solving sparse linear least squares or a positive definite 5-band matrix system fast
I want to quickly solve the following linear least-squares problem
$$\min_{x \in \mathbb{R}^n} \left\| A x - b \right\|_2^2$$
with a special sparse structure where each row in $A$ has only up to $4$ ...
7
votes
2
answers
8k
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Conditions for convergence of Euler's method
It is known that a sufficient and necessary condition for
$$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$
to have a unique solution is $f$ Lipschitz in $y$ and continuous in $t$. However, ...
0
votes
0
answers
39
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Rigorous definition of space and time order of accuracy of numerical PDEs
Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
2
votes
0
answers
845
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Problem using finite difference to solve a initial value problem
I tried to use 'finite difference' method to solve an Initial Value Problem (IVP). For the two boundaries I used periodical condition and for the differential operators I used 4th degree center ...
3
votes
0
answers
88
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Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$
Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\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
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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
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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
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"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
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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
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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
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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
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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$$
Is anything know about the case $N>2$?
I am looking in fact for an efficient ...
2
votes
0
answers
48
views
Efficiently determining surface intersections along a line segment
Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
1
vote
1
answer
49
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Implementable numerical scheme for the equation $a=\text{Erf}\big(z/\sqrt{2N_{a}}\big)$
Let $z>0$ be fixed and $A$ be the set of non-increasing functions from $\mathbb R_+$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|_\infty$. Define by $F$ the operator on $A$ by
\begin{equation*}
F(...
1
vote
1
answer
103
views
Numerical solution to some functional equation
Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as
$$
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...