arXiv subjects

Jesse van Rhijn

Publications and source records attributed to Jesse van Rhijn.

10 recordsLinked to original sources

arXiv2026

Non-periodic Fourier propagation algorithms for partial differential equations

Spectral methods for partial differential equations (PDEs) with non-periodic boundary conditions arising in computational physics often use polynomial expansions on non-uniform grids. Here, we implement a Fourier method that employs fast trigonometric expansions on a uniform grid with non-periodic boundaries using fast discrete sine transforms (DST) or/and discrete cosine transforms (DCT) to solve parabolic PDEs. We implement this method in two ways: either using a Fourier spectral derivative or a Fourier interaction picture. Both methods can treat vector fields with a combination of Dirichlet and/or Neumann boundary conditions in one or more space dimensions. As examples, we use them to solve a variety of computational physics PDEs with analytical solutions, including the Peregrine solitary wave solution. For the 1D heat equation problem, our method with an interaction picture is accurate up to machine precision. Soluble examples of stochastic partial differential equation (SPDE) with non-periodic boundaries in one and two space dimensions, with physics and interdisciplinary applications are also treated. We compare the results obtained from these algorithms with publicly available solvers that use polynomial spectral methods, and study their relative performance and error scaling. Polynomial methods with non-uniform spatial grids have lower spatial discretization errors when the solutions change slowly in space, typically with large spatial grids. For problems with rapid spatial variation, Fourier methods can outperform polynomial expansions, owing to their smaller maximum space interval, and are generally faster due to the computational efficiency of discrete Fourier transform methods. We verified this by making a complexity analysis in which we studied the total error at the optimum combination of time and space steps for a given resource use.

math.NA↗

arXiv2025

Complexity of Local Search for Euclidean Clustering Problems

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k = 2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $\mathcal{X} \subseteq \mathbb{R}^d$; a problem which is equivalent to Min Sum 2-Clustering.

cs.CC↗

arXiv2025

Convergence and Running Time of Time-dependent Ant Colony Algorithms

Ant Colony Optimization (ACO) is a well-known method inspired by the foraging behavior of ants and is extensively used to solve combinatorial optimization problems. In this paper, we first consider a general framework based on the concept of a construction graph - a graph associated with an instance of the optimization problem under study, where feasible solutions are represented by walks. We analyze the running time of this ACO variant, known as the Graph-based Ant System with time-dependent evaporation rate (GBAS/tdev), and prove that the algorithm's solution converges to the optimal solution of the problem with probability 1 for a slightly stronger evaporation rate function than was previously known. We then consider two time-dependent adaptations of Attiratanasunthron and Fakcharoenphol's $n$-ANT algorithm: $n$-ANT with time-dependent evaporation rate ($n$-ANT/tdev) and $n$-ANT with time-dependent lower pheromone bound ($n$-ANT/tdlb). We analyze both variants on the single destination shortest path problem (SDSP). Our results show that $n$-ANT/tdev has a super-polynomial time lower bound on the SDSP. In contrast, we show that $n$-ANT/tdlb achieves a polynomial time upper bound on this problem.

cs.DS↗

arXiv2024

Counting Locally Optimal Tours in the TSP

We show that the problem of counting the number of 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is $O(1.2098^n \sqrt{n!})$. Based on numerical experiments, we conjecture that the true bound is at most $O(\sqrt{n!})$, which is approximately the square root of the total number of tours.

cs.DS↗

arXiv2024

Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering

We analyze the running time of the Hartigan-Wong method, an old algorithm for the $k$-means clustering problem. First, we construct an instance on the line on which the method can take $2^{Ω(n)}$ steps to converge, demonstrating that the Hartigan-Wong method has exponential worst-case running time even when $k$-means is easy to solve. As this is in contrast to the empirical performance of the algorithm, we also analyze the running time in the framework of smoothed analysis. In particular, given an instance of $n$ points in $d$ dimensions, we prove that the expected number of iterations needed for the Hartigan-Wong method to terminate is bounded by $k^{12kd}\cdot poly(n, k, d, 1/σ)$ when the points in the instance are perturbed by independent $d$-dimensional Gaussian random variables of mean $0$ and standard deviation $σ$.

cs.DS↗

arXiv2023

Improved Smoothed Analysis of 2-Opt for the Euclidean TSP

The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case running time is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey \& Veenstra, who obtained smoothed complexity bounds polynomial in $n$, the dimension $d$, and the perturbation strength $σ^{-1}$. However, their analysis only works for $d \geq 4$. The only previous analysis for $d \leq 3$ was performed by Englert, Röglin \& Vöcking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in $n$ and $σ^{-d}$, and super-exponential in $d$. As no direct analysis existed for Gaussian perturbations that yields polynomial bounds for all $d$, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt.

cs.DS↗

arXiv2023

Approximation Ineffectiveness of a Tour-Untangling Heuristic

We analyze a tour-uncrossing heuristic for the Travelling Salesperson Problem, showing that its worst-case approximation ratio is $Ω(n)$ and its average-case approximation ratio is $Ω(\sqrt{n})$ in expectation. We furthermore evaluate the approximation performance of this heuristic numerically on average-case instances, and find that it performs far better than the average-case lower bound suggests. This indicates a shortcoming in the approach we use for our analysis, which is a rather common approach in the analysis of local search heuristics.

cs.DS↗

arXiv2022

Towards a Lower Bound for the Average Case Runtime of Simulated Annealing on TSP

We analyze simulated annealing (SA) for simple randomized instances of the Traveling Salesperson Problem. Our analysis shows that the theoretically optimal cooling schedule of Hajek explores members of the solution set which are in expectation far from the global optimum. We obtain a lower bound on the expected length of the final tour obtained by SA on these random instances. In addition, we also obtain an upper bound on the expected value of its variance. These bounds assume that the Markov chain that describes SA is stationary, a situation that does not truly hold in practice. Hence, we also formulate conditions under which the bounds extend to the nonstationary case. These bounds are obtained by comparing the tour length distribution to a related distribution. We furthermore provide numerical evidence for a stochastic dominance relation that appears to exist between these two distributions, and formulate a conjecture in this direction. If proved, this conjecture implies that SA stays far from the global optimum with high probability when executed for any sub-exponential number of iterations. This would show that SA requires at least exponentially many iterations to reach a global optimum with nonvanishing probability.

cs.DS↗

arXiv2022

A midpoint projection algorithm for stochastic differential equations on manifolds

Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold are often computationally impractical, and numerical projections are useful in many cases. We show that the Stratonovich interpretation of the stochastic calculus is obtained using adiabatic elimination with a constraint potential. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes, and review some earlier projection algorithms. In this paper, a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. Numerical examples are given for a range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, as well as higher-order polynomial constraints and a ten-dimensional hypersphere. We show that in all cases the combined midpoint method has greatly reduced errors compared to methods using a combined Euler projection approach or purely tangential projection. Our technique can handle multiple constraints. This allows manifolds that embody several conserved quantities. The algorithm is accurate, simple and efficient. An order of magnitude error reduction in diffusion distance is typically found compared to the other methods, with reductions of several orders of magnitude in constraint errors.

math.NA↗

arXiv2021

Energy derivatives in real-space diffusion Monte Carlo

We present unbiased, finite--variance estimators of energy derivatives for real--space diffusion Monte Carlo calculations within the fixed--node approximation. The derivative $d_λE$ is fully consistent with the dependence $E(λ)$ of the energy computed with the same time step. We address the issue of the divergent variance of derivatives related to variations of the nodes of the wave function, both by using a regularization for wave function parameter gradients recently proposed in variational Monte Carlo, and by introducing a regularization based on a coordinate transformation. The essence of the divergent variance problem is distilled into a particle-in-a-box toy model, where we demonstrate the algorithm.

cond-mat.mtrl-sci↗