Related papers: Using Non-Linear Difference Equations to Study Qui…
In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formul\ae{} of this number of a…
We provide a smoothed analysis of Hoare's find algorithm and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect - is an easy-to-implement algorithm for finding the k-th smallest element of a…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
With the goal to provide absolute lower bounds for the best possible running times that can be achieved by $(1+\lambda)$-type search heuristics on common benchmark problems, we recently suggested a dynamic programming approach that computes…
This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The…
There is excitement within the algorithms community about a new partitioning method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly faster than the case when it runs under classic partitioning methods. We show that…
Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast-slow stochastic…
The prompt has become an effective linguistic tool for utilizing pre-trained language models. However, in few-shot scenarios, subtle changes in the prompt design always make the result widely different, and the prompt learning methods also…
We consider the problem of using experimental time-series data for parameter estimation in nonlinear ordinary differential equations, focusing on the case where the data is noisy, sparse, irregularly sampled, includes multiple experiments,…
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the…
Incorporating nonlinearity into quantum machine learning is essential for learning a complicated input-output mapping. We here propose quantum algorithms for nonlinear regression, where nonlinearity is introduced with feature maps when…
In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the…
We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters.…
Distributed optimization algorithms have been studied extensively in the literature; however, underlying most algorithms is a linear consensus scheme, i.e. averaging variables from neighbors via doubly stochastic matrices. We consider…
We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-$t$ probabilities arising in spatial statistics. The approach combines a tile-low-rank representation of covariance matrices with…
Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The…
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending…
In this paper we show some explicit results regarding non-linear diffusive equations on Poincar\'e half plane. We obtain exact solutions by using the generalized separation of variables and we also show the meaning of these results in the…
We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential…