Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions

2013 ◽  
Vol 138 (20) ◽  
pp. 204113 ◽  
Author(s):  
Matthew S. Kelley ◽  
Toru Shiozaki
Keyword(s):  
Large Scale ◽  
Basis Functions ◽  
Density Fitting ◽  
Spinor Basis

Solving PDEs with radial basis functions

Acta Numerica ◽  
2015 ◽  
Vol 24 ◽  
pp. 215-258 ◽  
Author(s):  
Bengt Fornberg ◽  
Natasha Flyer
Keyword(s):  
Radial Basis Functions ◽  
Finite Differences ◽  
Large Scale ◽  
Numerical Approach ◽  
Basis Functions ◽  
Integration Time ◽  
Small Scale ◽  
Major Step ◽  
Mesh Free ◽  
Radial Basis

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

scholarly journals Predicting Longitudinal Traits Derived from High-Throughput Phenomics in Contrasting Environments Using Genomic Legendre Polynomials and B-Splines

2019 ◽  
Vol 9 (10) ◽  
pp. 3369-3380 ◽  
Author(s):  
Mehdi Momen ◽  
Malachy T. Campbell ◽  
Harkamal Walia ◽  
Gota Morota
Keyword(s):  
Abiotic Stress ◽  
High Throughput ◽  
Legendre Polynomials ◽  
Prediction Accuracy ◽  
Large Scale ◽  
Basis Functions ◽  
Random Regression ◽  
Growth Stages ◽  
Growth Environment ◽  
B Splines

Recent advancements in phenomics coupled with increased output from sequencing technologies can create the platform needed to rapidly increase abiotic stress tolerance of crops, which increasingly face productivity challenges due to climate change. In particular, high-throughput phenotyping (HTP) enables researchers to generate large-scale data with temporal resolution. Recently, a random regression model (RRM) was used to model a longitudinal rice projected shoot area (PSA) dataset in an optimal growth environment. However, the utility of RRM is still unknown for phenotypic trajectories obtained from stress environments. Here, we sought to apply RRM to forecast the rice PSA in control and water-limited conditions under various longitudinal cross-validation scenarios. To this end, genomic Legendre polynomials and B-spline basis functions were used to capture PSA trajectories. Prediction accuracy declined slightly for the water-limited plants compared to control plants. Overall, RRM delivered reasonable prediction performance and yielded better prediction than the baseline multi-trait model. The difference between the results obtained using Legendre polynomials and that using B-splines was small; however, the former yielded a higher prediction accuracy. Prediction accuracy for forecasting the last five time points was highest when the entire trajectory from earlier growth stages was used to train the basis functions. Our results suggested that it was possible to decrease phenotyping frequency by only phenotyping every other day in order to reduce costs while minimizing the loss of prediction accuracy. This is the first study showing that RRM could be used to model changes in growth over time under abiotic stress conditions.

Reduced Atmospheric Models Using Dynamically Motivated Basis Functions

2007 ◽  
Vol 64 (10) ◽  
pp. 3452-3474 ◽  
Author(s):  
Frank Kwasniok
Keyword(s):  
Total Energy ◽  
Large Scale ◽  
Empirical Orthogonal Functions ◽  
Basis Functions ◽  
Interaction Patterns ◽  
Order Model ◽  
Dynamical Equations ◽  
Reduced Models ◽  
Mean State ◽  
Low Order

Abstract Nonlinear deterministic reduced models of large-scale atmospheric dynamics are constructed. The dynamical framework is a quasigeostrophic three-level spectral model with realistic mean state and variability as well as Pacific–North America (PNA) and North Atlantic Oscillation (NAO) patterns. The study addresses the problem of finding appropriate basis functions for efficiently capturing the dynamics and a comparison between different choices of basis functions; it focuses on highly truncated models, keeping only 10–15 modes. The reduced model is obtained by a projection of the equations of motion onto a truncated basis spanned by empirically determined modes. The total energy metric is used in the projection; the nonlinear terms of the low-order model then conserve total energy. Apart from retuning the coefficient of horizontal diffusion, no empirical terms are fitted in the dynamical equations of the low-order model in order to properly preserve the physics of the system. Using the methodology of principal interaction patterns (PIPs), a basis is derived that carefully compromises minimizing tendency error with maximizing explained variance in the resolved modes. A new PIP algorithm is introduced that is more compact and robust than earlier PIP algorithms; a top-down approach is adopted, removing modes from the system one by one. The mean state and standard deviation of the streamfunction as well as transient momentum fluxes are well reproduced by a PIP model with only 10 modes. Probability density functions are accurately modeled and autocorrelation functions are captured fairly well using 15 modes. Reduced models based on PIPs are substantially superior to reduced models based on empirical orthogonal functions (EOFs). The leading PIPs have a higher projection onto the PNA and NAO teleconnection patterns than the corresponding EOFs. Both with EOFs and PIPs, the interactions between the resolved modes are predominantly linear and the improvement of PIP models on EOF models stems entirely from better modeling these linear interactions although the full nonlinear tendencies are optimized. There is considerable influence of smaller-scale modes on the large-scale modes due to nonlinear coupling that is not well captured by either EOFs or PIPs. This nonlinear backscattering possibly plays a role in generating the low-frequency variability of the model. The results call for a nonlinear and/or stochastic closure scheme in which PIPs may be suitable basis functions.

scholarly journals Analytical finite element matrix elements and global matrix assembly for hierarchical 3-D vector basis functions within the hybrid finite element boundary integral method

2014 ◽  
Vol 12 ◽  
pp. 1-11
Author(s):  
L. Li ◽  
K. Wang ◽  
H. Li ◽  
T. F. Eibert
Keyword(s):  
Finite Element ◽  
Large Scale ◽  
Higher Order ◽  
Boundary Integral ◽  
Basis Functions ◽  
Boundary Integral Method ◽  
Matrix Elements ◽  
Matrix Assembly ◽  
Time Accuracy ◽  
Element Boundary

Abstract. A hybrid higher-order finite element boundary integral (FE-BI) technique is discussed where the higher-order FE matrix elements are computed by a fully analytical procedure and where the gobal matrix assembly is organized by a self-identifying procedure of the local to global transformation. This assembly procedure applys to both, the FE part as well as the BI part of the algorithm. The geometry is meshed into three-dimensional tetrahedra as finite elements and nearly orthogonal hierarchical basis functions are employed. The boundary conditions are implemented in a strong sense such that the boundary values of the volume basis functions are directly utilized within the BI, either for the tangential electric and magnetic fields or for the asssociated equivalent surface current densities by applying a cross product with the unit surface normals. The self-identified method for the global matrix assembly automatically discerns the global order of the basis functions for generating the matrix elements. Higher order basis functions do need more unknowns for each single FE, however, fewer FEs are needed to achieve the same satisfiable accuracy. This improvement provides a lot more flexibility for meshing and allows the mesh size to raise up to λ/3. The performance of the implemented system is evaluated in terms of computation time, accuracy and memory occupation, where excellent results with respect to precision and computation times of large scale simulations are found.

scholarly journals An Approach Towards Generating Surrogate Models by Using RBFN With a Priori Bias

2014 ◽  
Author(s):  
Kaveh Amouzgar ◽  
Niclas Stromberg
Keyword(s):  
Radial Basis Functions ◽  
Large Scale ◽  
A Priori ◽  
Surrogate Models ◽  
Basis Functions ◽  
Normal Equation ◽  
A Posteriori ◽  
Mathematical Functions ◽  
Radial Basis ◽  
Price Functions

In this paper, an approach to generate surrogate models constructed by radial basis function networks (RBFN) with a priori bias is presented. RBFN as a weighted combination of radial basis functions only, might become singular and no interpolation is found. The standard approach to avoid this is to add a polynomial bias, where the bias is defined by imposing orthogonality conditions between the weights of the radial basis functions and the polynomial basis functions. Here, in the proposed a priori approach, the regression coefficients of the polynomial bias are simply calculated by using the normal equation without any need of the extra orthogonality prerequisite. In addition to the simplicity of this approach, the method has also proven to predict the actual functions more accurately compared to the RBFN with a posteriori bias. Several test functions, including Rosenbrock, Branin-Hoo, Goldstein-Price functions and two mathematical functions (one large scale), are used to evaluate the performance of the proposed method by conducting a comparison study and error analysis between the RBFN with a priori and a posteriori known biases. Furthermore, the aforementioned approaches are applied to an engineering design problem, that is modeling of the material properties of a three phase spherical graphite iron (SGI). The corresponding surrogate models are presented and compared.

scholarly journals Formulation and Implementation of Density Functional Embedding Theory using Products of Basis Functions

2020 ◽  
Author(s):  
Vladimir Rybkin
Keyword(s):  
Condensed Phase ◽  
Density Functional ◽  
Large Scale ◽  
Plane Waves ◽  
Basis Functions ◽  
Basis Set ◽  
Transfer Reactions ◽  
Molecular Systems ◽  
Embedding Theory ◽  
Proton Transfer Reactions

The representation of embedding potential in using products of AO basis functions has been developed in the context of density functional embedding theory (DFET). The formalism allows to treat pseudopotential and all-electron calculations on the same footing and enables simple transfer of the embedding potential in the compact matrix form. In addition, a simple cost-reduction procedure for basis set and potential reduction has been proposed. The theory has been implemented for the condensed-phase and molecular systems using Gaussian and Plane Waves (GPW) and Gaussian and Augmented Plane Waves (GAPW) formalisms and tested for proton transfer reactions in the cluster and the condensed phase. The computational scaling of the embedding potential optimization is similar to this of hybrid DFT with a significantly reduced prefactor and allows for large-scale applications.<div><br></div>

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