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At the continuous level, we consider two types of tumor growth models: the cell density model, based on the fluid mechanical construction, is more favorable for scientific interpretation and numeri...

This paper introduces a novel data clustering algorithm based on Langevin dynamics, where the associated potential is constructed directly from the data. To introduce a self-consistent potential, we adopt the potential model from the established Quantum Clustering method. The first step is to use a radial basis function to construct a density distribution from the data. A potential function is then constructed such that this density distribution is the ground state solution to the time-independent Schrödinger equation. The second step is to use this potential function with the Langevin dynamics at subcritical temperature to avoid ergodicity. The Langevin equations take a classical Gibbs distribution as the invariant measure, where the peaks of the distribution coincide with minima of the potential surface. The time dynamics of individual data points lead to different metastable states, which are interpreted as cluster centers. Clustering is therefore achieved when subsets of the data aggregate—as a result of the Langevin dynamics for a moderate period of time—in the neighborhood of a particular potential minimum. While the data points are pushed towards potential minima by the potential gradient, Brownian motion allows them to effectively tunnel through local potential barriers and escape saddle points into locations of the potential surface otherwise forbidden. The algorithm’s feasibility is first established based on several illustrating examples and theoretical analyses, followed by a stricter evaluation using a standard benchmark dataset.

An efficient sampling method, the pmmLang + RBM, is proposed to compute the quantum thermal average in the interacting quantum particle system. Benefiting from the random batch method (RBM), the pmmLang + RBM has the potential to reduce the complexity due to interaction forces per time step from O(NP2) to O(NP), where N is the number of beads and P is the number of particles. Although the RBM introduces a random perturbation of the interaction forces at each time step, the long time effects of the random perturbations along the sampling process only result in a small bias in the empirical measure of the pmmLang + RBM from the target distribution, which also implies a small error in the thermal average calculation. We numerically study the convergence of the pmmLang + RBM and quantitatively investigate the dependence of the error in computing the thermal average on the parameters such as batch size, time step, and so on. We also propose an extension of the pmmLang + RBM, which is based on the splitting Monte Carlo method and is applicable when the interacting potential contains a singular part.

We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.

From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an “s-d” potential term, including the Schroźdinger--Landau--Lifshitz--Maxwell system, the Pauli--Landau--Lifshitz system, and the Schroźdinger--Landau--Lifshitz system with successive simplifications. For the latter two systems, we propose using the time splitting spectral method for the quantum spin current and the Gauss--Seidel projection method for the magnetization. Accuracy of the time splitting spectral method applied to the Pauli equation is analyzed and verified by numerous examples. Moreover, behaviors of the Schroźdinger--Landau--Lifshitz system in different “s-d” coupling regimes are explored numerically.

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density \begin{document}$ n$\end{document} is governed by the Darcy law via the pressure \begin{document}$ p(n) = n^{γ}$\end{document} . For finite \begin{document}$ γ$\end{document} , we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As \begin{document}$ γ \to ∞$\end{document} , the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

In this work, a novel ring polymer representation for a multi-level quantum system is proposed for thermal average calculations. The proposed representation keeps the discreteness of the electronic states: besides position and momentum, each bead in the ring polymer is also characterized by a surface index indicating the electronic energy surface. A path integral molecular dynamics with surface hopping (PIMD-SH) dynamics is also developed to sample the equilibrium distribution of the ring polymer configurational space. The PIMD-SH sampling method is validated theoretically and by numerical examples.

We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, the constitutive law connecting pressure $p$ and density $\rho$ is $p(\rho)=\frac{m}{m-1} \rho^{m-1}$, and when $m \gg 1$, the cell density $\rho$ may evolve its support due to a pressure-driven geometric motion with sharp interface along the boundary of its support. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations, let alone the capturing of the singular free boundary limit. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as $m\gg 1$. In this paper, we develope a numerical scheme based on a novel prediction-correction reformulation that can accurately approximate the front propagation even when the nonlinearity is extremely strong. We show that the semi-discrete scheme naturally connects to the free boundary limit equation as $m \rightarrow \infty$, and with proper spacial discretization, the fully discrete scheme has improved stability, preserves positivity, and implements without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties and showcase good performance in various applications.

In this paper, we present a time splitting scheme for the Schrodinger equation in the presence of electromagnetic eld in the semi-classical regime, where the wave function propagates O(e) oscillations in space and time. With the operator splitting technique, the time evolution of the Schrodinger equation is divided into three parts: the kinetic step, the convection step and the potential step. The kinetic and the potential steps can be handled by the classical time-splitting spectral method. For the convection step, we propose a semi-Lagrangian method in order to allow large time steps. We prove the unconditional stability conditions with spatially variant external vector potentials, and the error estimate in the l approximation of the wave function. By comparing with the semi-classical limit, the classical Liouville equation in the Wigner framework, we show that this method is able to capture the correct physical observables with time step ∆t ≫ e. We implement this method numerically for both one dimensional and two dimensional cases to verify that e−independent time steps can indeed be taken in computing physical observables.

In the spirit of the fewest switches surface hopping, the frozen Gaussian approximation with surface hopping (FGA-SH) method samples a path integral representation of the non-adiabatic dynamics in the semiclassical regime. An improved sampling scheme is developed in this work for FGA-SH based on birth and death branching processes. The algorithm is validated for the standard test examples of non-adiabatic dynamics., 14 pages, 9 figures

We derive the semiclassical Bloch dynamics with second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external electric field, the bi-characteristics system after a positional shift introduced by Berry connections agrees with the recent result in previous works. Moreover, for the case with a linear external electric field, we show that the extra terms arising in the bi-characteristics system after the positional shift are also gauge independent.

© American Institute of Mathematical Sciences. We consider a coupled system of Schrödinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semiclassically scaled Schrödinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.