We generalize the transformations and duality found in incommensurate 2D systems between real space, configuration space, momentum space, and reciprocal space to study electronic observables of incommensurate bilayers in the tight-binding framework using a wide class of applicable Hamiltonians. We then apply this generalization to obtain the effects of mechanical relaxation on nearly aligned materials in momentum space, which produce in-plane incommensurate scattering. The relaxation scattering is long-ranged in this case, which likewise changes the momentum space numerical scheme convergence rate. We study this convergence theoretically, and perform a numerical study on twisted bilayer graphene at small angles with mechanical relaxation.
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The solution of time fractional partial differential equations in general exhibit a weak singularity near the initial time. In this article we propose a method for solving time fractional diffusion equation with nonlocal diffusion term. The proposed method comprises L1 scheme on graded mesh, finite element method and Newton's method. We discuss the well-posedness of the weak formulation at discrete level and derive \emph{a priori} error estimates for fully-discrete formulation in $L^2(\Omega)$ and $H^1(\Omega)$ norms. Finally, some numerical experiments are conducted to validate the theoretical findings.
We propose a staggered mesh method for correlation energy calculations of periodic systems under the random phase approximation (RPA), which generalizes the recently developed staggered mesh method for periodic second order M{\o}ller-Plesset perturbation theory (MP2) calculations [Xing, Li, Lin, JCTC 2021]. Compared to standard RPA calculations, the staggered mesh method introduces negligible additional computational cost. It avoids a significant portion of the finite-size error, and can be asymptotically advantageous for quasi-1D systems and certain quasi-2D and 3D systems with high symmetries. We demonstrate the applicability of the method using two different formalisms: the direct ring coupled cluster doubles (drCCD) theory, and the adiabatic-connection (AC) fluctuation-dissipation theory. In the drCCD formalism, the second order screened exchange (SOSEX) correction can also be readily obtained using the staggered mesh method. In the AC formalism, the staggered mesh method naturally avoids the need of performing "head/wing" corrections to the dielectric operator. The effectiveness of the staggered mesh method for insulating systems is theoretically justified by investigating the finite-size error of each individual perturbative term in the RPA correlation energy, expanded as an infinite series of terms associated with ring diagrams. As a side contribution, our analysis provides a proof that the finite-size error of each perturbative term of standard RPA and SOSEX calculations scales as $\mathcal{O}(N_{\mathbf{k}}^{-1})$, where $N_{\mathbf{k}}$ is the number of grid points in a Monkhorst-Pack mesh.
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled problem at a lower computational cost. We analysize a multiscale method for the nonlinear system where the fast system has a periodic applied force and the slow equation contains fractional derivatives as a simplication of the atherosclerosis with a plaque growth. A local periodic equation is derived to approximate the original system and the error estimates are given. Then a finite difference method is designed to approximate the original and the approximate problems. We construct four examples, including three with exact solutions and one following the original problem setting, to test the accuracy and computational efficiency of the proposed method. It is observed that, the computational time is very much reduced and the multiscale method performs very well in comparison to fully resolved simulation for the case of small time scale separation. The larger the time scale separation is, the more effective the multiscale method is.
A support structure is required to successfully create structural parts in the powder bed fusion process for additive manufacturing. In this study, we present the topology optimization of a support structure that improves the heat dissipation in the building process. First, we construct a numerical method that obtains the temperature field in the building process, represented by the transient heat conduction phenomenon with the volume heat flux. Next, we formulate an optimization problem for maximizing heat dissipation and develop an optimization algorithm that incorporates a level-set-based topology optimization. A sensitivity of the objective function is derived using the adjoint variable method. Finally, several numerical examples are provided to demonstrate the effectiveness and validity of the proposed method.
The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.
Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss--Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.
In this paper, we bring consumer theory to bear in the analysis of Fisher markets whose buyers have arbitrary continuous, concave, homogeneous (CCH) utility functions representing locally non-satiated preferences. The main tools we use are the dual concepts of expenditure minimization and indirect utility maximization. First, we use expenditure functions to construct a new convex program whose dual, like the dual of the Eisenberg-Gale program, characterizes the equilibrium prices of CCH Fisher markets. We then prove that the subdifferential of the dual of our convex program is equal to the negative excess demand in the associated market, which makes generalized gradient descent equivalent to computing equilibrium prices via t\^atonnement. Finally, we run a series of experiments which suggest that t\^atonnement may converge at a rate of $O\left(\frac{(1+E)}{t^2}\right)$ in CCH Fisher markets that comprise buyers with elasticity of demand bounded by $E$. Our novel characterization of equilibrium prices may provide a path to proving the convergence of t\^atonnement in Fisher markets beyond those in which buyers utilities exhibit constant elasticity of substitution.
In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.
Complex systems (stars, supernovae, galaxies, and clusters) often exhibit low scatter relations between observable properties (e.g., luminosity, velocity dispersion, oscillation period, temperature). These scaling relations can illuminate the underlying physics and can provide observational tools for estimating masses and distances. Machine learning can provide a systematic way to search for new scaling relations (or for simple extensions to existing relations) in abstract high-dimensional parameter spaces. We use a machine learning tool called symbolic regression (SR), which models the patterns in a given dataset in the form of analytic equations. We focus on the Sunyaev-Zeldovich flux$-$cluster mass relation ($Y_\mathrm{SZ}-M$), the scatter in which affects inference of cosmological parameters from cluster abundance data. Using SR on the data from the IllustrisTNG hydrodynamical simulation, we find a new proxy for cluster mass which combines $Y_\mathrm{SZ}$ and concentration of ionized gas ($c_\mathrm{gas}$): $M \propto Y_\mathrm{conc}^{3/5} \equiv Y_\mathrm{SZ}^{3/5} (1-A\, c_\mathrm{gas})$. $Y_\mathrm{conc}$ reduces the scatter in the predicted $M$ by $\sim 20-30$% for large clusters ($M\gtrsim 10^{14}\, h^{-1} \, M_\odot$) at both high and low redshifts, as compared to using just $Y_\mathrm{SZ}$. We show that the dependence on $c_\mathrm{gas}$ is linked to cores of clusters exhibiting larger scatter than their outskirts. Finally, we test $Y_\mathrm{conc}$ on clusters from simulations of the CAMELS project and show that $Y_\mathrm{conc}$ is robust against variations in cosmology, astrophysics, subgrid physics, and cosmic variance. Our results and methodology can be useful for accurate multiwavelength cluster mass estimation from current and upcoming CMB and X-ray surveys like ACT, SO, SPT, eROSITA and CMB-S4.
The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, maximizing variance and preservation of pairwise relative distances. The derivation of their asymptotic correlation and numerical experiments tell that a projection usually cannot satisfy both objectives. In a standard classification problem we determine projections on the input data that balance them and compare subsequent results. Next, we extend our application of orthogonal projections to deep learning frameworks. We introduce new variational loss functions that enable integration of additional information via transformations and projections of the target data. In two supervised learning problems, clinical image segmentation and music information classification, the application of the proposed loss functions increase the accuracy.
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