Mean-field molecular dynamics based on path integrals is used to approximate canonical quantum observables for particle systems consisting of nuclei and electrons. A computational bottleneck is the sampling from the Gibbs density of the electron operator, which due to the fermion sign problem has a computational complexity that scales exponentially with the number of electrons. In this work we construct an algorithm that approximates the mean-field Hamiltonian by path integrals for fermions. The algorithm is based on the determinant of a matrix with components based on Brownian bridges connecting permuted electron coordinates. The computational work for $n$ electrons is $\mathcal O(n^3)$, which reduces the computational complexity associated with the fermion sign problem. We analyze a bias resulting from this approximation and provide a computational error indicator. It remains to rigorously explain the surprisingly high accuracy.
相關內容
Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(\Delta t\right)^{2}+N^{-m})$, where $\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.
The method is based on the preliminary transformation of the traditionally used matrices or adjacency lists in the graph theory into refined projections free from redundant information, and their subsequent use in constructing shortest paths. Unlike adjacency matrices and lists based on enumerating binary adjacency relations, the refined projection is based on enumerating more complex relations: simple paths from a given graph vertex that are shortest. The preliminary acquisition of such projections reduces the algorithmic complexity of applications using them and improves their volumetric and real-time characteristics to linear ones for a pair of vertices. The class of graphs considered is extended to mixed graphs.
The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pad\'e approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.
We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They arise e.g.~in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples.
We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.
We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.
The structured $\varepsilon$-stability radius is introduced as a quantity to assess the robustness of transient bounds of solutions to linear differential equations under structured perturbations of the matrix. This applies to general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz matrices. The notion conceptually combines unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. We propose and study an algorithm for computing the structured $\varepsilon$-stability radius. This algorithm solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations that originate from a gradient system. The proposed algorithm has essentially the same computational cost as the known rank-1 algorithms for computing unstructured and structured stability radii. Numerical experiments illustrate the behavior of the algorithm.
Deep neural network models for image segmentation can be a powerful tool for the automation of motor claims handling processes in the insurance industry. A crucial aspect is the reliability of the model outputs when facing adverse conditions, such as low quality photos taken by claimants to document damages. We explore the use of a meta-classification model to assess the precision of segments predicted by a model trained for the semantic segmentation of car body parts. Different sets of features correlated with the quality of a segment are compared, and an AUROC score of 0.915 is achieved for distinguishing between high- and low-quality segments. By removing low-quality segments, the average mIoU of the segmentation output is improved by 16 percentage points and the number of wrongly predicted segments is reduced by 77%.
We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.
Dirac delta distributionally sourced differential equations emerge in many dynamical physical systems from neuroscience to black hole perturbation theory. Most of these lack exact analytical solutions and are thus best tackled numerically. This work describes a generic numerical algorithm which constructs discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae recovering higher order accuracy. It is shown by solving the distributionally sourced wave equation, which has analytical solutions, that numerical weak-form solutions can be recovered to high order accuracy by solving a first-order reduced system of ordinary differential equations. The method-of-lines framework is applied to the DiscoTEX algorithm i.e through discontinuous collocation with implicit-turned-explicit (IMTEX) integration methods which are symmetric and conserve symplectic structure. Furthermore, the main application of the algorithm is proved, for the first-time, by calculating the amplitude at any desired location within the numerical grid, including at the position (and at its right and left limit) where the wave- (or wave-like) equation is discontinuous via interpolation using DiscoTEX. This is shown, firstly by solving the wave- (or wave-like) equation and comparing the numerical weak-form solution to the exact solution. Finally, one shows how to reconstruct the scalar and gravitational metric perturbations from weak-form numerical solutions of a non-rotating black hole, which do not have known exact analytical solutions, and compare against state-of-the-art frequency domain results. One concludes by motivating how DiscoTEX, and related algorithms, open a promising new alternative Extreme-Mass-Ratio-Inspiral (EMRI)s waveform generation route via a self-consistent evolution for the gravitational self-force programme in the time-domain.
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