Toward Grid-Based Models for Molecular Association

This paper presents a grid-based approach to model molecular association processes as an alternative to sampling-based Markov models. Our method discretizes the six-dimensional space of relative translation and orientation into grid cells. By discretizing the Fokker–Planck operator governing the system dynamics via the square-root approximation, we derive analytical expressions for the transition rate constants between grid cells. These expressions depend on geometric properties of the grid, such as the cell surface area and volume, which we provide. In addition, one needs only the molecular energy at the grid cell center, circumventing the need for extensive MD simulations and reducing the number of energy evaluations to the number of grid cells. The resulting rate matrix is closely related to the Markov state model transition matrix, offering insights into metastable states and association kinetics. We validate the accuracy of the model in identifying metastable states and binding mechanisms, though improvements are necessary to address limitations like ignoring bulk transitions and anisotropic rotational diffusion. The flexibility of this grid-based method makes it applicable to a variety of molecular systems and energy functions, including those derived from quantum mechanical calculations. The software package MolGri, which implements this approach, offers a systematic and computationally efficient tool for studying molecular association processes

Integrating Agent-Based and Compartmental Models for Infectious Disease Modeling: A Novel Hybrid Approach

This study investigates the spatial integration of agent-based models (ABMs) and compartmental models for infectious disease modeling, presenting a novel hybrid approach and examining its implications. ABMs offer detailed insights by simulating interactions and decisions among individuals but are computationally expensive for large populations. Compartmental models capture population-level dynamics more efficiently but lack granular detail. We developed a hybrid model that aims to balance the granularity of ABMs with the computational efficiency of compartmental models, offering a more nuanced understanding of disease spread in diverse scenarios, including large populations. This model spatially couples discrete and continuous populations by integrating an ordinary differential equation model with a spatially explicit ABM. Our key objectives were to systematically assess the consistency of disease dynamics and the computational efficiency across various configurations. For this, we evaluated two experimental scenarios and varied the influence of each sub-model via spatial distribution. In the first, the ABM component modeled a homogeneous population; in the second, it simulated a heterogeneous population with landscape-driven movement. Results show that the hybrid model can significantly reduce computational costs but is sensitive to between-model differences, highlighting the importance of model equivalence in hybrid approaches. The code is available at: git.zib.de/ibostanc/hybrid_abm_ode

Filtered data based estimators for stochastic processes driven by colored noise

We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario

Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body

This paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier–Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields

Lattice Rules Meet Kernel Cubature

Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed partial derivatives. In this work, we explore the impact of replacing the equal weights in lattice rules with optimized cubature weights derived using the reproducing kernel. We establish a theoretical result demonstrating a doubled convergence rate in the one-dimensional case and provide numerical investigations of convergence rates in higher dimensions. We also present numerical results for an uncertainty quantification problem involving an elliptic partial differential equation with a random coefficient

Explicit solutions of Genz test integrals

Abstract A collection of test integrals introduced by Genz (1984) has remained popular to this day for assessing the robustness of high-dimensional numerical integration algorithms. However, the explicit solutions to these integrals do not appear to be readily available in the existing literature: typically the true values of the test integrals are simply approximated using “overkill” numerical solutions. In this paper, analytic solutions are presented for the Genz test integral

Doubling the rate: improved error bounds for orthogonal projection with application to interpolation. To appear in BIT Numerical Mathematics, 2025.

Convergence rates for approximation in a Hilbert space H are a central theme in numerical analysis. The present work is inspired by Schaback (Math Comp, 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the best rate for functions in the “native space” H. Motivated by this, we obtain a general result for H-orthogonal projection onto a finite dimensional subspace of H: namely, that any known convergence rate for all functions in H translates into a doubled convergence rate for functions in a smoother normed space B, along with a similarly improved error bound in the H-norm, provided that , H and B are suitably related. As a special case we improve the known and H-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space B

Assessing Lagrangian coherence in atmospheric blocking

Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of the origin of air parcels that define the blocking region, especially with respect to non-adiabatic processes such as latent heating. So far, an objective method of clustering the individual Lagrangian trajectories passing through a blocking into larger and, more importantly, spatially coherent air streams has not been established. This is the focus of our study. To this end, we determine coherent sets of trajectories, which are regions in the phase space of dynamical systems that keep their geometric integrity in time and can be characterized by robustness under small random perturbations. We approximate a dynamic diffusion operator on the available Lagrangian data and use it to cluster the trajectories into coherent sets. Our implementation adapts the existing methodology to the non-Euclidean geometry of Earth's atmosphere and its challenging scaling properties. The framework also allows for statements about the spatial behavior of the trajectories as a whole. We discuss two case studies differing with respect to season and geographic location. The results confirm the existence of spatially coherent feeder air streams differing with respect to their dynamical properties and, more specifically, their latent heating contribution. Air streams experiencing a considerable amount of latent heating (warm conveyor belts) occur mainly during the maturing phase of the blocking and contribute to its stability. In our example cases, trajectories also exhibit an altered evolution of general coherence when passing through the blocking region, which is in line with the common understanding of blocking as a region of low dispersion

Dynamics of systems with varying number of particles: From Liouville equations to general master equations for open systems

A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics

Chemical potential and variable number of particles control the quantum state: Quantum oscillators as a showcase

Despite their simplicity, quantum harmonic oscillators are ubiquitous in the modeling of physical systems. They are able to capture universal properties that serve as references for the more complex systems found in nature. In this spirit, we apply a model of a Hamiltonian for open quantum systems in equilibrium with a particle reservoir to ensembles of quantum oscillators. By treating (i) a dilute gas of vibrating particles and (ii) a chain of coupled oscillators as showcases, we demonstrate that the property of varying numbers of particles leads to a mandatory condition on the energy of the system. In particular, the chemical potential plays the role of a parameter of control that can externally manipulate the spectrum of a system and the corresponding accessible quantum states