Numerical algorithms for problems in analysis and algebra, scientific computation

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31 views

Low degree polynomial approximation to a square pulse

Let $\epsilon,\kappa>0$ be fixed and if we seek a polynomial $W(x)$ that satisfies: $$W(x) \in (1-\epsilon,1+\epsilon)\mbox{ if }x\in \big(-\frac{s+\kappa}{2},\frac{s+\kappa}{2}\big)$$ $$W(x) \in ...
4
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1answer
512 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
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0answers
37 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 $f$ function such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is interpolation polynomial on ...
3
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1answer
73 views

Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...
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1answer
183 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 ...
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2answers
540 views

Sampling from Sine Kernel and Airy Kernel

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples ...
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1answer
117 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 ...
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3answers
210 views

What is an extragradient method?

I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia ...
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0answers
28 views

Differentiability of piecewise function [closed]

I am trying to model the feedback response of some natural system in a system of ordinary differential equations (ODE), and then solve the system via Runge-Kutta method. $a(t)$ is a controled ...
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114 views

$Ax=b$ in a function space (again)

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, Let ${\mathbb{C}}^n$ ...
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22 views

Multiple integral of the resolvent kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...
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30 views

finite elements for a nonlinear heat equation [closed]

I'm looking to compute the (real-valued) solution $u(x,t)$ of the following equation $$ k(x,u) \partial_t u - \Delta u = f \text{ on } \Omega \times [0,T]$$ where $k$ is a continuous but nonlinear ...
2
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1answer
86 views

Special Function, Series Expansion, or Simpler Form of a Certain Infinite Product?

$\prod _{n=1}^{\infty } \left(1+a (c+n)^b\right)$ where a > 0, b < -1, and c >= 0 Is there a special function, series expansion, or other simpler (or maybe just interesting) representation of ...
1
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1answer
26 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 $\partial \Omega$ ...
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0answers
46 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}} \\ ...
4
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1answer
144 views

Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem. Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that $$ \| D_1 A ...
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3answers
192 views

What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states: $$ i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...
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0answers
97 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 ...
5
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2answers
193 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
4
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1answer
152 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 ...
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0answers
36 views

Bound on change of function given bound on Hessian

Suppose I have very some smooth function $F(x)$, and let $x_0 = \text{argmin}_x F(x)$. I would like to bound $F(x) - F(x_0)$ from above, in terms of the gradient $\nabla f(x)$ and the Hessian matrix ...
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2answers
48 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, ...
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2answers
1k views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
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0answers
43 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 ...
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30 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 ...
2
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0answers
63 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...
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1answer
37 views

Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability: One factor would be the condition number of the approximation operator. The other factor ...
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1answer
118 views

books on very large scale linear optimization

Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific ...
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2answers
145 views

iterative solution better than analytic solution? [closed]

My supervisor and I were discussing a specific optimisation problem this afternoon. To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference ...
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0answers
51 views

Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...
7
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1answer
163 views

Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
8
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1answer
234 views

Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation. I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
6
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1answer
136 views

Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am ...
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1answer
214 views

Improving Newton's Inequalities using the Taylor Theorem

Newton's inequalities say that if $f(x) = \sum \binom{n}{k} a_k x^k$ is a polynomial with all real roots then $ a_k^2 > a_{k-1}a_{k+1}$. The proof this result uses that if $f(x)$ has all real ...
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0answers
37 views

General reparameterization of a b-spline

Say I have a bspline function (or curve) of order $k_1$, defined over some knot vector $\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$ Do you know of a process of finding ...
3
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1answer
59 views

The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...
7
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1answer
154 views

Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
7
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4answers
221 views

Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here. I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...
2
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0answers
34 views

Conditions for convergence of Euler's method

It is know that a sufficient and necessary condition for $$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$ assumes a unique solution of $f$ is Lipschitz in $y$ and continuous in $t$. ...
5
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4answers
683 views

Quanitative de Moivre–Laplace theorem (reference request)

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution: $${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi ...
1
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1answer
192 views

Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
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0answers
60 views

Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
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40 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
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787 views

Eigenvalue Problems with Linear Constraints

The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...
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1answer
61 views

Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
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1answer
44 views

Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...
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0answers
35 views

Numerical method for self-consistency of one-dimensional probability density function

I have an integral equation for self-consistency of one-dimensional probability density function, like this $$\rho_x(x) = \frac{1}{|a|}\int \int \rho_x\left(\frac{s-b}{a}\right) \rho_P(p) ...
2
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1answer
37 views

IVP accuracy - scheme accuracy Vs. derivative accuracy?

General Question: If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ...
7
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3answers
289 views

Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+): $$ \int_{-1}^1\textrm{d}t ...
0
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0answers
48 views

Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation, $i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $ ...