All Questions
1,732 questions
0
votes
0
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98
views
Interpolation polynomials with constraints
Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
- 502
4
votes
0
answers
46
views
Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
1
vote
0
answers
60
views
Galerkin’s Method for hyperbolic PDEs: proving convergence without using compactness
Lawrence Evan's PDE book prove the existence of solution to the following problem where $L$ is an elliptic operator:
$$
\begin{cases}
u_{tt} = -Lu+f,\\
u|_{t=0} = u_0,\\
u|_{\partial U} = 0
\end{cases}...
- 1,755
2
votes
1
answer
158
views
Are the coefficients in the stationary phase approximation computed explicitly somewhere
In Stein's "Harmonic analysis" book, page 334, one can find
the asymptotic expansion
An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
- 852
0
votes
0
answers
67
views
Concentration of bilinear forms
This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
- 6,962
0
votes
0
answers
39
views
Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101
0
votes
1
answer
58
views
Integration algorithm and analytic property
This question is the continuation of the previous one.
In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
- 81
1
vote
0
answers
92
views
Computing smallest singular value of a matrix with explicit error control?
Many good algorithms are out there computing truncated SVD: What is the time complexity of truncated SVD?.
I am trying to implement some codes to find the smallest singular value of a big matrix $A$. ...
- 1,755
1
vote
0
answers
44
views
Error bounds for a Romberg-style improvement of a non-linear approximation
I have a (possibly non-linear) functional $F$, which I want to numerically approximate by a (typically non-linear) $\widehat{F}_h$. For a suitable class of functions, I have asymptotic error behaviour
...
- 3,065
1
vote
0
answers
126
views
Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
- 81
14
votes
3
answers
2k
views
Guaranteed correct digits of elementary expressions
Let $E$ be the set of elementary expressions, defined as the smallest set of mathematical expressions such that $1 \in E$, and if $a,b \in E$ then $a + b$, $a - b$, $ab$, $a/b$, $\exp(a)$, $\log(a)$ ...
- 361
0
votes
0
answers
30
views
Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
- 23
1
vote
1
answer
309
views
Numerical estimation of partial derivatives of convolved functions when closed forms do not exist
Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
- 879
0
votes
0
answers
34
views
Ensuring symmetry in mixed derivatives using RBF-FD method
I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a bivariate function $f(x, y)$ at a set of ...
- 1
-1
votes
1
answer
214
views
Best approximation of the modulus function
While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
- 149
4
votes
3
answers
682
views
Approximation for complex variables
Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...
- 149
2
votes
1
answer
213
views
Is matrix B obtained from matrix A?
Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
- 49
0
votes
0
answers
36
views
Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
- 393
0
votes
0
answers
56
views
Godunov splitting convergence research
The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
- 101
2
votes
2
answers
272
views
Numerical integration method that doesn't involve derivative in the error bound
Consider the integral $\int_a^b f(t)dt$.
There are many numerical integration methods, like trapezoidal rule, Simpson's rule, Gaussian quadrature, but all they involve derivative in the error bound.
...
- 49
0
votes
0
answers
36
views
ILPs with square constraint matrix
Given the Integer Linear Programming ($\text{ILP}$) problem
\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to}& \mathbf{A}^T x \ge b \\
\text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
- 13.2k
0
votes
0
answers
87
views
How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
- 1,759
1
vote
0
answers
49
views
Computing geodesic length of Euclidean lines in the manifold of positive definite matrices
I am working with the manifold of positive definite matrices $PD(n)$ equipped with the affine-invariant Riemannian metric (AIRM) $g_P(V,W):=tr(P^{-1}VP^{-1}W)$, where $P \in PD(n)$ and $V,W \in T_P PD(...
- 518
0
votes
0
answers
26
views
Monotony of enforced subtour merging
Is it true that for a symmetric TSP instance in the sequence of edges generated by successively:
calculating the optimal 2-factor
adding cardinality constraints on the edgesets of the 2-factor's ...
- 13.2k
1
vote
0
answers
183
views
Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart
I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...
- 33
0
votes
0
answers
171
views
Solve NP-hard type problems with linear programming
I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.
I ask this ...
3
votes
1
answer
308
views
Root finding algorithm for an analytic function
Given an analytic function $f(x)$. What is the best algorithm to find roots on the interval $[a,b]$ inside the radius of convergence> What is its complexity with respect to the length of input of ...
- 81
0
votes
0
answers
64
views
Alternatives to McCormick Envelope
I have an optimization problem for which I have the optimal solution obtained by the ILP.
However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
- 3
2
votes
0
answers
119
views
Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
4
votes
0
answers
214
views
Computational complexity of zeros of an analytic function
The work of Friedman and Ko, page 342, Corollary 4.3.1
states that all zeros of analytic polynomial time computable function are polynomial time computable, but for me that is not clear how it could ...
- 81
0
votes
1
answer
169
views
How to integrate an indicator function/constraint into the cost function of a linear program?
I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...
- 3
2
votes
0
answers
68
views
Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
- 167
0
votes
1
answer
40
views
How to handle the evaluation of functions on staggered ghost nodes?
I have a convection-diffusion-reaction steady state PDE in the form
$$
\frac{\partial C}{\partial x} = \frac{1}{u_0(x)}\left(\frac{\partial}{\partial z} \left( \mathcal{D}(z) \frac{\partial C}{\...
- 111
1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
- 6,155
3
votes
0
answers
130
views
Series acceleration for $\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta$, $\beta\ll 1$
The probability mass of the Conway-Maxwell-Poisson variable $K$ is given by
$$
\mathsf P(K=k)=\frac{1}{Z(H,\beta)}\left(\frac{H^k}{k!}\right)^\beta
$$
where
$$
Z(H,\beta)=\sum_{k=0}^\infty\left(\frac{...
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 129
1
vote
0
answers
61
views
Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues
I am looking for a reference justifying the following statement.
Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$
$$
...
- 5,380
0
votes
1
answer
28
views
Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
- 13.2k
3
votes
1
answer
273
views
Inflection point calculation for cubic Bézier curve encounters division by zero
I've been working on finding the inflection points of a cubic Bezier curve using the method described in a paper Hain, Venkat, Racherla, and Langan - Fast, Precise Flattening of Cubic Bézier Segment ...
- 133
2
votes
1
answer
217
views
How to do LU factorization efficiently based on the factorized result added with a low-rank matrix?
Suppose a square $n\times n$, dense matrix $A^{\text{old}}$ has been factorized into $L^{\text{old}}$ and $U^{\text{old}}$ components by performing a LU decomposition $A^{\text{old}} = L^{\text{old}}U^...
- 21
2
votes
0
answers
79
views
Proof of the convergence of the Rayleigh-Ritz Method?
In this article The convergence of the Rayleigh-Ritz Method in quantum chemistry by Bruno Klahn & Werner A. Bingel they have at page 11
Let $H_B$ be that Hilbert space which can be obtained as the ...
4
votes
1
answer
210
views
Is Stokes equation a saddle point problem or a minimum problem?
Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are
\begin{equation}
\begin{cases}
- \Delta u + \nabla p = f \text{ ...
- 697
1
vote
0
answers
41
views
The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
- 47
1
vote
0
answers
145
views
Complexity of calculating the expectation of $\operatorname{Tr} h(A)$, $A$ is a random matrix
$A$ is a $d_1\times d_1$ random matrix. Given $\{g_i\}~(1\leq i\leq n)$ iid Gaussian variables, $f_{ij}(g_1,g_2,...,g_n)~(1\leq i,j\leq d_1)$ are degree-$d_2$ polynomials. And $f_{ij}\equiv f_{ji}~(\...
- 91
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
0
votes
1
answer
114
views
Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
1
vote
1
answer
186
views
Practical calculation of Canterbury approximants
I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
- 3,065
0
votes
2
answers
131
views
Reshaping data vector into a matrix for deconvolution using a circulant matrix
Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
- 879
1
vote
0
answers
128
views
Resolving singularities in numerical integration
I am now trying to compute numerically the following integral.
$$
\begin{split}
L_1^s(\hat{\phi}_s)(r,\zeta,\theta_\zeta) &=\frac{1}{\sqrt{2}\pi}
\int\limits_{0}^{2\pi} d\varphi
\int\limits_0^{\pi}...
- 11