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Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study ...

• Implicit neural representations help establish a mesh-free assimilation framework.
• Spherical harmonics bring theoretical guarantee for the neural network.
• Uncertainty of trained model is evaluated; compatible with existing ...

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of ...

• A fast approximation algorithm for variable-order spectral fractional Laplacian.

High-index saddle dynamics provide an effective means to compute the any-index saddle points and construct the solution landscape. In this paper, we prove error estimates for Euler discretization of high-index saddle dynamics with respect to the time step ...

In this paper, we consider a variable-separation (VS) method to solve the nonlinear partial differential equations (PDEs) with random inputs. The aim of the VS method is to get a sep- arated representation of the Galerkin solution for nonlinear PDEs with ...

The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit process of isotropic Lévy ...

Confined nematic liquid crystals can admit multiple stable equilibria. These equilibrium states usually exhibit peculiar defect patterns in order to satisfy the topological constraints imposed by the boundary conditions. In this work, ...

We present a high-index optimization-based shrinking dimer (HiOSD) method to compute index-$k$ saddle points as a generalization of the optimization-based shrinking dimer method for index-1 saddle points [L. Zhang, Q. Du, and Z. Zheng, SIAM J. Sci. ...

We propose a fast algorithm for evaluating the moments of Bingham distribution. The calculation is done by piecewise rational approximation, where interpolation and Gaussian integrals are utilized. Numerical tests show that the algorithm reaches the ...

We consider approximations on SO(3) by Wigner D-matrix. We establish basic approximation properties of Wigner D-matrix, develop efficient numerical schemes using Wigner D-matrix for elliptic and parabolic equations on SO(3), and establish corresponding ...

To characterize the Brownian motion in a bounded domain $\Omega$, it is well known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the ...

We construct a tensor model for nematic phases of bent-core molecules from molecular theory. The form of free energy is determined by molecular symmetry, which includes the couplings and derivatives of a vector and two second-order tensors, with the ...

In this paper, we study the new Q-tensor model previously derived from Onsager's molecular theory by Han et al. [Arch. Rational Mech. Anal., 215 (2015), pp. 741--809] for static liquid crystal modeling. Taking density and Q-tensor as order parameters, the ...

In this paper, we derive the sharp interface model of the nematic-isotropic phase transition from the Landau--de Gennes theory by using the matched asymptotic expansion method. The model includes the evolution equation of the velocity and director field of ...

Starting from the Beris--Edwards system for the liquid crystal, we present a rigorous derivation of the Ericksen--Leslie system with general Ericksen stress and Leslie stress by using the Hilbert expansion method.

The conventionally used Modified Self-consistent Field Theory (MSCFT) for confined system is known to have two major difficulties: numerical instability and dependence on artificial constraints. Here we use Compressible Model based Self-consistent Field ...

Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come from two parts: the numerical discretization, and the ...

In this study, we simulate double-gate MOSFET using a 2-D direct Boltzmann transport equation solver. Simulation results are interpreted by quasi-ballistic theory. It is found that the relation between average carrier velocity at virtual source and back-...

A numerical method in Fourier-space is developed to solve the polymeric self-consistent field equations. The method does not require a priori symmetric information. More significantly, periodic structure can be adjusted automatically during the ...

A numerical approach based on the string method is developed to study nucleation of ordered phases in first-order phase transitions. Among other things, this method allows an efficient computation of the minimum energy path (MEP) during the nucleation ...

Non-equilibrium Green's function (NEGF) is a general method for modeling non-equilibrium quantum transport in open mesoscopic systems with many body scattering effects. In this paper, we present a unified treatment of quantum device boundaries in the ...