Jun Li

We formulate the problem of estimating hydraulic conductivity from measured hydraulic head data in groundwater flow within the unified framework of parameter estimation using the extended Kalman filter (EKF) and least-squares optimization. We first formulate the EKF and its variants as problems of least-squares optimization. We then show that the successive linear estimator can be interpreted as a variant of the EKF, with its initial estimate obtained through co-Kriging. Furthermore, by showing that the EKF is equivalent to a one-step Newton iteration for solving a least-squares problem, we reformulate the hydraulic conductivity estimation problem as an associated least-squares optimization problem. Finally, we present an algorithm for estimating hydraulic conductivity from unsaturated soil data.

We formulate the problem of estimating hydraulic conductivity from measured hydraulic-head data in groundwater flow within the unified framework of parameter estimation using the extended Kalman filter (EKF) and least-squares optimization. We first formulate the EKF and its variants as problems of least-squares optimization. We then show that the successive linear estimator can be interpreted as a variant of the EKF, with its initial estimate obtained through Co-Kriging. Furthermore, by showing that the EKF is equivalent to a one-step Newton iteration for solving a least-squares problem, we reformulate the hydraulic-conductivity estimation problem as an associated least-squares optimization problem. Finally, we present an algorithm for estimating hydraulic conductivity from unsaturated soil data.

In this study, we focus on the collective dynamics of polar active particles navigating across three distinct surfaces, each characterized by its own unique blend of topological and geometrical properties. The behavior of these active particles is influenced by a multitude of factors, including self-propulsion, inter-particle interactions, surface constraints, and under-damped stochastic forces simulated via Ornstein–Uhlenbeck processes. Our exploration unveils the prevailing collective patterns observed within these systems across three surface types: a sphere, a torus, and a landscape featuring hills and valleys, each distinguished by its specific topological and geometrical attributes. We underscore the profound impact of surface curvature and symmetry on the sustainable spatial-temporal dynamics witnessed. Our findings illuminate how the interplay between substantial surface curvature and particular symmetrical characteristics gives rise to a diverse spectrum of spatial-temporal patterns. Notably, we discern that high curvature tends to drive collective motion toward cyclic rotation on spheres and tori, or spatial-temporal periodic traveling ring patterns on landscapes with hills and valleys. Additionally, we observe that rough surfaces and the incorporation of excluded volume effects can disrupt the complexity of these collective spatial-temporal patterns. Through this investigation, we provide invaluable insight into the intricate interplay of curvature and symmetry, profoundly shaping collective behaviors among active particles across varied surfaces.

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn-Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.

We articulate and apply the generalized Onsager principle to derive transport equations for active liquid crystals in a fixed domain as well as in a free surface domain adjacent to a passive fluid matrix. The Onsager principle ensures fundamental variational structure of the models as well as dissipative properties of the passive component in the models, irrespective of the choice of scale (kinetic to continuum) and of the physical potentials. Many popular models for passive and active liquid crystals in a fixed domain subject to consistent boundary conditions at solid walls, as well as active liquid crystals in a free surface domain with consistent transport equations along the free boundaries, can be systematically derived from the generalized Onsager principle. The dynamical boundary conditions are shown to reduce to the static boundary conditions for passive liquid crystals used previously.