We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients. As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm. A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.
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The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved.
We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.
Parameter inference is essential when interpreting observational data using mathematical models. Standard inference methods for differential equation models typically rely on obtaining repeated numerical solutions of the differential equation(s). Recent results have explored how numerical truncation error can have major, detrimental, and sometimes hidden impacts on likelihood-based inference by introducing false local maxima into the log-likelihood function. We present a straightforward approach for inference that eliminates the need for solving the underlying differential equations, thereby completely avoiding the impact of truncation error. Open-access Jupyter notebooks, available on GitHub, allow others to implement this method for a broad class of widely-used models to interpret biological data.
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments, including the Prothero-Robinson problem.
The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.
Unlabeled sensing is a linear inverse problem with permuted measurements. We propose an alternating minimization (AltMin) algorithm with a suitable initialization for two widely considered permutation models: partially shuffled/$k$-sparse permutations and $r$-local/block diagonal permutations. Key to the performance of the AltMin algorithm is the initialization. For the exact unlabeled sensing problem, assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we bound the initialization error in terms of the number of blocks $s$ and the number of shuffles $k$. Experimental results show that our algorithm is fast, applicable to both permutation models, and robust to choice of measurement matrix. We also test our algorithm on several real datasets for the linked linear regression problem and show superior performance compared to baseline methods.
The computation of integrals is a fundamental task in the analysis of functional data, which are typically considered as random elements in a space of squared integrable functions. Borrowing ideas from recent advances in the Monte Carlo integration literature, we propose effective unbiased estimation and inference procedures for integrals of uni- and multivariate random functions. Several applications to key problems in functional data analysis (FDA) involving random design points are studied and illustrated. In the absence of noise, the proposed estimates converge faster than the sample mean and the usual algorithms for numerical integration. Moreover, the proposed estimator facilitates effective inference by generally providing better coverage with shorter confidence and prediction intervals, in both noisy and noiseless setups.
An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.
We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error analysis in [9] for Runge-Kutta methods for nonlinear parabolic equations; in analogy to [9], the proofs are based on maximal regularity properties of discontinuous Galerkin methods for non-autonomous linear parabolic equations.
Districting is a complex combinatorial problem that consists in partitioning a geographical area into small districts. In logistics, it is a major strategic decision determining operating costs for several years. Solving districting problems using traditional methods is intractable even for small geographical areas and existing heuristics often provide sub-optimal results. We present a structured learning approach to find high-quality solutions to real-world districting problems in a few minutes. It is based on integrating a combinatorial optimization layer, the capacitated minimum spanning tree problem, into a graph neural network architecture. To train this pipeline in a decision-aware fashion, we show how to construct target solutions embedded in a suitable space and learn from target solutions. Experiments show that our approach outperforms existing methods as it can significantly reduce costs on real-world cities.
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