Numerical algorithms for problems in analysis and algebra, scientific computation

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3
votes
2answers
535 views

Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...
0
votes
2answers
73 views

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

I want to quickly solve linear least squares problem for $x \in \mathbb{R}^n$ $$min_x \left\| A x - b \right\|_2^2$$ with a special sparse structure where each row in $A$ has only up to 4 ...
0
votes
1answer
85 views

Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...
2
votes
0answers
63 views

Scale vector in scaled pivoting (numerical methods)

I'm teaching students about several numeric methods, including scaled pivoting. There's a small section in this subject that I could never find a clear explanation to, either as intuition, or a more ...
1
vote
1answer
187 views

Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?

I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ...
4
votes
1answer
112 views

Numerical equality testing

I am working on developing an online homework system. One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe $\sin(2x)$)...
2
votes
0answers
77 views

Finding an explicit constant in finite element error estimates

Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ("...
2
votes
1answer
337 views

How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$. I could not a find a good way of computing the Teichmuller flow on this ...
5
votes
0answers
182 views

Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
9
votes
3answers
555 views

Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
3
votes
0answers
34 views

Solving a Certain Constrained Isoperimetric Approximation-Problem

This question is related to my question Differential Geometric Aspects of Rubber Bands, where I asked for a mathematical model of contracting rubber bands. In contrast to my former question, the ...
2
votes
0answers
61 views

Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e., $G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $ ...
2
votes
0answers
178 views

Generalized arithmetic progressions contained in Bohr sets

Recall that a generalized arithmetic progression of dimension $d$ is, by definition, a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...
11
votes
2answers
410 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
1
vote
0answers
42 views

LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
4
votes
0answers
86 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in S^{D-...
2
votes
1answer
210 views

The bubble function

In the finite element method and more precisely the MINI element method in two dimensions, they use a function called the "bubble function" which is related to a triangle K of the space meshing and is ...
0
votes
1answer
106 views

Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...
9
votes
1answer
271 views

What happens to continuous spectrum upon discretization?

Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...
2
votes
1answer
208 views

Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
3
votes
0answers
103 views

Estimating overshoot in spline interpolation

Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...
0
votes
0answers
122 views

inflow/outflow Boundary Conditions for flow in pipe

I have a question about boundary condition of solving Navier-Stokes equation through pipe. When I simulate the flow in pipe using periodic boundary condition, it works good. But when I tried to change ...
3
votes
2answers
184 views

What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more? I think that over the years numerical folks (who are faced with ...
5
votes
0answers
117 views

Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup: Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function. For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} f(\...
10
votes
4answers
628 views

Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function. How do rounding errors affect the results? I'm looking for references on this issue, ...
2
votes
1answer
120 views

cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline} F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, x_{3k+1},x_{3k+2},x_{3k+3})\in\...
6
votes
4answers
449 views

Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers $[1,n]...
4
votes
2answers
142 views

Accuracy of the formulas for angles between almost colinear vectors

Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities: In ...
5
votes
1answer
158 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 ...
0
votes
0answers
65 views

Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$. Suppose we have diagonalized using $LMR=D$. I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
3
votes
2answers
327 views

Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the integrand....
1
vote
1answer
135 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method (...
0
votes
1answer
174 views

Generating random variables from the Cantor Distribution [closed]

I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In ...
0
votes
0answers
50 views

Comparing Calculation Error in Divergent Numerical Methods

I'm not an expert in numerical methods, but I'm doing a simulation based on non-linear differential equations (General Relativity), there solutions has singularities, thus at some points numerical ...
0
votes
1answer
86 views

What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...
3
votes
2answers
343 views

Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form: \begin{equation} dX_t = f(...
-1
votes
1answer
739 views

BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better? The BDF2 method requires the ...
9
votes
2answers
294 views

Approximation theory on the disc

Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...
0
votes
0answers
59 views

Shooting method for non-oscillatory solutions

I am having some trouble with using the shooting method : I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I ...
4
votes
2answers
279 views

Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous function $f$ such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is an interpolation polynomial ...
0
votes
1answer
138 views

A kind of Discrete Fourier Transform

Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]_\mathbb{N}$ by $$DFT_z(k)=\frac{1}{N} \sum_{j=0}^{N-1} z_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th ...
0
votes
0answers
61 views

Exact diagonalization of tridiagonal centrosymmetric matrices

It is said that one can diagonalize a tridiagonal matrix using the analytical Lanczos method http://arxiv.org/abs/cond-mat/9712283v1. In some references in it, they always say that the starting point ...
1
vote
1answer
61 views

Convergent algorithm for dividing a body into two regions of equal volume

Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ ...
2
votes
1answer
274 views

Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something. Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$, i.e. $\gamma_0\sim 14.134...$. 1) what is ...
0
votes
0answers
79 views

Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
3
votes
0answers
113 views

error estimate of linear interpolation in high dimension

Consider convex functions $f,g$ on $[0,1]^d$. Let $x_1,\cdots,x_n$ be $n\geq d+1$ fixed point in $[0,1]^d$ that is equally 'distributed' in the sense that $$c_1\leq \frac{n\mathrm{vol}(K)}{\mathrm{...
4
votes
1answer
211 views

Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
-2
votes
2answers
88 views

Systems of ODEs that fulfill a matrix relationship at steady state [closed]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$ with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
4
votes
0answers
86 views

Estimating polynomial approximation error in high dimension

Question Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...
0
votes
0answers
37 views

Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...