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Numerical algorithms for problems in analysis and algebra, scientific computation

1 vote

How to handle the evaluation of functions on staggered ghost nodes?

Evaluation out of bounds is related to boundary conditions. Doesn't the boundary treatment for $C$ already indicate a way to handle the coefficients? If not, I would use some way that extends the coef …
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3 votes
Accepted

Adding constraints as penalty with $\| \cdot \|_0$ norm

The claim in the paper is false. Since the problem is not convex, the claim does not follow from general results. However, there are some results in this direction in quite general cases: If $x^*$ i …
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2 votes
Accepted

Iterative method for $p$-Laplacian

I know this method under the name lagged diffusivity. I learned it from the paper Vogel, Curtis R., and Mary E. Oman. "Iterative methods for total variation denoising." SIAM Journal on Scientific …
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1 vote

Orthogonal system of functions ordered by norm of second derivative

The optimization problem for $f_3$ is $$ \min_g \int_{-1}^1 |g''(x)|^2dx \quad \text{s.t.}\quad \int_{-1}^1 g(x)dx = 0,\ \int_{-1}^1 x g(x)dx = 0,\ \int_{-1}^1 |g(x)|^2dx = 1. $$ Using Lagrange Multip …
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0 votes

Iterative matrix inversion with $L^\infty$ norm

As a shameless plug for my own work: As you want to use the $\infty$-norm as a stopping criterion, you may be interested in homotopy methods. For the sake of completeness, assume that you want to find …
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1 vote

Linear convergence rate of proximal point algorithm

I am not aware of results on the linear rate of this variant of the proximal point method. Let me note that convergence is usually shown by the following observation: Since $C$ is a bijection, you may …
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0 votes

Approximating a function with sums of powers

You may want to restrict $x$ to be positive and then expressions of the type you are dealing with called "posynomials" if the $c_k$ are also positive. Posynomials are convex and they are indeed used i …
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5 votes

Real world example of use of Monte Carlo method for high dimensional integrals

Some buzzwords that should lead to some non-textbook examples: In the fields of uncertainty quantification, statistical inverse problems or Bayesian inference one wants, for example, compute condition …
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1 vote

Reference Request: Variational Problem

A bit long for a comment. Let's clean up the formulation a bit: First, the domain $[-1,1]^2$ of definition does play any role, and hence, we assume that all respective quantities are functions on s …
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5 votes

Numerical approximation to the Wasserstein metric?

Yes, there are. First note that the Wasserstein metric is, after discretization, the solution of a linear program (LP) that can be fed into any LP solver. Moreover, there are specialized algorithms, …
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7 votes

What is an extragradient method?

I can confirm that there is no agreement of what "extragradient method" really means. I know the interpretation by Christian Clason but I also know the one that is linked in Carlo Beenakkers answer. L …
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1 vote

Selecting Rays for Simulated Radon Transform

Sorry, not an answer, but too long for a comment. If your question is motivated by practical applications of the Radon transform such as computerized tomography in medical imaging or non-destructive …
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3 votes

Discrete gradient on point clouds

Certainly, there is not a standard method - and be aware that the calculation will be sensitive to noise. A straightforward of calculating the gradient would be: Take some point $x$ and choose a numb …
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4 votes

Roots of modified polynomials

I guess, Chapter 2, §1 of "Perturbation Theory of Linear Operators" by Kato will answer your question.
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2 votes

dense lattices in high dimensions

You face the curse of dimensionality. Besides the pretty old but simple and robust Monte Carlo integration and its relatives there are also methods based on sparse grids. For an overview see E. No …
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