An Adaptive Sampling Procedure for Training a Neural Network Based On a Gaussian Process

A novel approach is presented for efficiently training a neural network (NN)-based surrogate model when the training data set is to be generated using a computationally intensive high-fidelity computational model. The approach consists in using a Gaussian Process (GP), and more specifically, its acquisition function, to adaptively sample the parameter space of interest and generate the minimum amount of training data needed to achieve the desired level of approximation accuracy. The overall approach is explained and illustrated with numerical experiments associated with the prediction of the lift-over-drag ratio for a NACA airfoil in a large, two-dimensional parameter space of free-stream Mach number and free-stream angle of attack. The obtained numerical results demonstrate the superior accuracy delivered by the proposed training over standard trainings using uniform and random samplings

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

MPAS-Albany Land Ice (MALI): a variable-resolution ice sheet model for Earth system modeling using Voronoi grids

We introduce MPAS-Albany Land Ice (MALI) v6.0, a new variable-resolution land ice model that uses unstructured Voronoi grids on a plane or sphere. MALI is built using the Model for Prediction Across Scales (MPAS) framework for developing variable-resolution Earth system model components and the Albany multi-physics code base for the solution of coupled systems of partial differential equations, which itself makes use of Trilinos solver libraries. MALI includes a three-dimensional first-order momentum balance solver (Blatter–Pattyn) by linking to the Albany-LI ice sheet velocity solver and an explicit shallow ice velocity solver. The evolution of ice geometry and tracers is handled through an explicit first-order horizontal advection scheme with vertical remapping. The evolution of ice temperature is treated using operator splitting of vertical diffusion and horizontal advection and can be configured to use either a temperature or enthalpy formulation. MALI includes a mass-conserving subglacial hydrology model that supports distributed and/or channelized drainage and can optionally be coupled to ice dynamics. Options for calving include eigencalving, which assumes that the calving rate is proportional to extensional strain rates. MALI is evaluated against commonly used exact solutions and community benchmark experiments and shows the expected accuracy. Results for the MISMIP3d benchmark experiments with MALI's Blatter–Pattyn solver fall between published results from Stokes and L1L2 models as expected. We use the model to simulate a semi-realistic Antarctic ice sheet problem following the initMIP protocol and using 2&thinsp;km resolution in marine ice sheet regions. MALI is the glacier component of the Energy Exascale Earth System Model (E3SM) version 1, and we describe current and planned coupling to other E3SM components.</p

LIVVkit 2.1: automated and extensible ice sheet model validation

A collection of scientific analyses, metrics, and visualizations for robust validation of ice sheet models is presented using the Land Ice Verification and Validation toolkit (LIVVkit), version 2.1, and the LIVVkit Extensions repository (LEX), version 0.1. This software collection targets stand-alone ice sheet or coupled Earth system models, and handles datasets and analyses that require high-performance computing and storage. LIVVkit aims to enable efficient and fully reproducible workflows for postprocessing, analysis, and visualization of observational and model-derived datasets in a shareable format, whereby all data, methodologies, and output are distributed to users for evaluation. Extending from the initial LIVVkit software framework, we demonstrate Greenland ice sheet simulation validation metrics using the coupled Community Earth System Model (CESM) as well as an idealized stand-alone high-resolution Community Ice Sheet Model, version 2 (CISM2), coupled to the Albany/FELIX velocity solver (CISM-Albany or CISM-A). As one example of the capability, LIVVkit analyzes the degree to which models capture the surface mass balance (SMB) and identifies potential sources of bias, using recently available in situ and remotely sensed data as comparison. Related fields within atmosphere and land surface models, e.g., surface temperature, radiation, and cloud cover, are also diagnosed. Applied to the CESM1.0, LIVVkit identifies a positive SMB bias that is focused largely around Greenland's southwest region that is due to insufficient ablation.</p

Automatic Substructuring for Domain Decomposition Using Neural Networks.

learning. -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 -2 0 2 4 6 Fig. 4 (a) -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 -2 0 2 4 6 Fig. 4 (b) -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 -2 0 2 4 6 Fig. 4 (c) Fig. 4. Substructuring by neural networks. (a) Output of an existing greedy algorithm. (b) Output of Hopfield network. (c) Output of competitive learning. 5 Subdomain Epoch % Improvement 15 35.12 3 30 40.69 40 44.99 20 8.33 6 40 11.67 80 18.69 Table 1.(a). Substructure generation in the 2-D finite element mesh shown in Fig. 2 using Hopfield network. Subdomain Epoch % Improvement 5 37.52 3 10 42.48 15 44.99 20 6.33 6 40 11.15 60 19.60 Table 1.(b). Substructure generation in the 2-D finite element mesh shown in Fig. 2 using Competitive learning. Subdomain Epoch % Improvement 20 17.21 60 26.75 8 80 30.07 100 32.36 20 9.79 20 40 15.17 60 17.71 80 20.67 Table 2.(a). Substructure generation in the

A DOMAIN DECOMPOSITION METHOD FOR A CLASS OF DISCONTINUOUS GALERKIN DISCRETIZATIONS OF HELMHOLTZ PROBLEMS

Recently, a discontinuous Galerkin finite element method with plane wave basis functions was introduced for the efficient solution of Helmholtz problems. The method uses Lagrange multipliers to enforce a weak continuity of the solution at the element interfaces. Here, a preconditioned iterative solution procedure based on a domain decomposition is proposed for the resulting systems of linear equations. Numerical experiments study the iterative solution of a two-dimensional model problem

An Algebraic Theory for Primal and Dual Substructuring Methods by Constraints

FETI and BDD are two widely used substructuring methods for the solution of large sparse systems of linear algebraic equations arizing from discretization of elliptic boundary value problems. The two most advanced variants of these methods are the FETI-DP and the BDDC methods, whose formulation does not require any information beyond the algebraic system of equations in a substructure form. We formulate the FETI-DP and the BDDC methods in common framework as methods based on general constraints between the substructures, and provide a simplified algebraic convergence theory. The basic implementation blocks including transfer operators are common to both methods. It is shown that commonly used properties of the transfer operators in fact determine the operators uniquely. Identical algebraic condition number bounds for both methods are given in terms of a single inequality, and, under natural additional assumptions, it is proved that the eigenvalues of the preconditioned problems are the same. The algebraic bounds imply the usual polylogarithmic bounds for finite elements, independent of coefficient jumps between substructures. Computational experiments confirm the theory

Analysis and Numerical Realization of Coupled BEM and FEM for Nonlinear Exterior Problems

The paper presents main results of the investigation of the coupled BEM and FEM applied to a nonlinear generally nonmonotone exterior boundary value problem. The problem consists of a nonlinear differential equation considered in an annular bounded domain and the Laplace equation outside. These equations are equipped with boundary and transmission conditions. The problem is reformulated in a weak sense and combined with an integral equation. The discretization is carried out by the coupled finite element -- boundary element method. The attention is paid to the existence of the solution, the convergence of the method and the solution of the coupled discrete problem. The method is applied to compressible inviscid flow past an airfoil and the solution of the discrete problem is treated. 1 Formulation of the Problem Let\Omega \Gamma ae IR 2 be a bounded domain with a Lipschitz continuous boundary @\Omega \Gamma = \Gamma 0 [\Gamma. Here, \Gamma 0 ; \Gamma are simple closed curves, \..