Unstructured adaptive meshes: bad for your memory?

The most important performance bottleneck in modern high-end computers is access to memory. Many forms of hardware and software support for reducing memory latency exist, but certain important applications defy these. Examples of such applications are unstructured adaptive (UA) mesh problems, which feature irregular, dynamically changing memory access. We describe a new benchmark program, called UA, for measuring the performance of computer systems when solving such problems. It complements the existing NAS Parallel Benchmarks suite that deals mainly with static, regular-stride memory references. The UA benchmark involves the solution of a stylized heat transfer problem in a cubic domain, discretized on an adaptively refined, unstructured mesh. We describe the numerical and implementation issues, and also present some interesting performance results.     

Optimal convex hull algorithms on enhanced meshes

In this paper we propose time-optimal convex hull algorithms for two classes of enhanced meshes. Our first algorithm computes the convex hull of an arbitrary set ofn points in the plane inO (logn) time on a mesh with multiple broadcasting of sizen×n. The second algorithm shows that the same problem can be solved inO (1) time on a reconfigurable mesh of sizen×n. Both algorithms achieve time lower bounds for their respective model of computation.This work was supported by NASA under grant NCCI-99.Additional support by the National Science Foundation under grant CCR-8909996 is gratefully acknowledged.     

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L ₂ norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.     

A coarsening algorithm on adaptive red-green-blue refined meshes

Adaptive meshing is a fundamental component of adaptive finite element methods. This includes refining and coarsening meshes locally. In this work, we are concerned with the red-green-blue refinement strategy in two dimensions and its counterpart-coarsening. In general, coarsening algorithms are mostly based on an explicitly given refinement history. In this work, we present a coarsening algorithm on adaptive red-green-blue meshes in two dimensions without explicitly knowing the refinement history. To this end, we examine the local structure of these meshes, find an easy-to-verify criterion to adaptively coarsen red-green-blue meshes, and prove that this criterion generates meshes with the desired properties. We present a MATLAB implementation built on the red-green-blue refinement routine of the ameshref-package (Funken and Schmidt 2018, 2019).

     

Relaxing the CFL condition for the wave equation on adaptive meshes

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. A simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence results for continuous-time adaptive stochastic filtering algorithms

The adaptive stochastic filtering problem for Gaussian processes is considered. The self-tuning synthesis procedure is used to derive two algorithms for this problem. Almost sure convergence for the parameter estimate and the filtering error will be established. The convergence analysis is based on an almost-supermartingale convergence lemma that allows a stochastic Lyapunov-like approach.     

A new adaptive method for extrapolative forecasting algorithms

A quite serious problem when using time series forecasting methods is choosing the smoothing parameter (or parameters). Several methods have been developed, which employ variable, adaptively determined, smoothing factors. A new adaptive method for updating the value of smoothing parameters is introduced in this paper. The proposed model for exponential smoothing methods using one, two and three smoothing parameters is described and the accuracy of the method is measured.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

Mesh adaptive direct search algorithms for mixed variable optimization

This paper introduces a new derivative-free class of mesh adaptive direct search (MADS) algorithms for solving constrained mixed variable optimization problems, in which the variables may be continuous or categorical. This new class of algorithms, called mixed variable MADS (MV-MADS), generalizes both mixed variable pattern search (MVPS) algorithms for linearly constrained mixed variable problems and MADS algorithms for general constrained problems with only continuous variables. The convergence analysis, which makes use of the Clarke nonsmooth calculus, similarly generalizes the existing theory for both MVPS and MADS algorithms, and reasonable conditions are established for ensuring convergence of a subsequence of iterates to a suitably defined stationary point in the nonsmooth and mixed variable sense.     

Conservative semi‐Lagrangian advection on adaptive unstructured meshes

A conservative semi‐Lagrangian method is designed in order to solve linear advection equations in two space variables. The advection scheme works with finite volumes on an unstructured mesh, which is given by a Voronoi diagram. Moreover, the mesh is subject to adaptive modifications during the simulation, which serves to effectively combine good approximation quality with small computational costs. The required adaption rules for the refinement and the coarsening of the mesh rely on a customized error indicator. The implementation of boundary conditions is addressed. Numerical results finally confirm the good performance of the proposed conservative and adaptive advection scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 388–411, 2004     

Fejer algorithms with an adaptive step

For Fejer processes with attractants, a general adaptive scheme for step multiplier control is proposed and the convergence of this class of algorithms to stationary points is proved. Numerical results demonstrating that the convergence rate is generally linear are presented.     

Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics

Techniques that improve the accuracy of numerical solutions and reduce their computational costs are discussed as applied to continuum mechanics problems with complex time-varying geometry. The approach combines shock-capturing computations with the following methods: (1) overlapping meshes for specifying complex geometry; (2) elastic arbitrarily moving adaptive meshes for minimizing the approximation errors near shock waves, boundary layers, contact discontinuities, and moving boundaries; (3) matrix-free implementation of efficient iterative and explicit–implicit finite element schemes; (4) balancing viscosity (version of the stabilized Petrov–Galerkin method); (5) exponential adjustment of physical viscosity coefficients; and (6) stepwise correction of solutions for providing their monotonicity and conservativeness.     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast adaptive algorithms in the non-standard form for multidimensional problems

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class (−Δ+μ²I)^−α, where μ0 and 0<α<3/2, and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has the advantage of being easily extendable to higher dimensions.     

Optimization of adaptive direct algorithms for approximate solution of integral equation

The concern of this paper is to study the optimization of adaptive direct algorithms for approximate solution of the second kind Fredholm integral equations. We derive the exact order of the problem with integral kernels belonging to the Besov classes. Furthermore, we also present an almost optimal adaptive direct algorithm for above equation class.     

On using meshes of mixed element types in adaptive finite element analysis

Four different automatic mesh generators capable of generating either triangular meshes or hybrid meshes of mixed element types have been used in the mesh generation process. The performance of these mesh generators were tested by applying them to the adaptive finite element refinement procedure. It is found that by carefully controlling the quality and grading of the quadrilateral elements, an increase in efficiency over pure triangular meshes can be achieved. Furthermore, if linear elements are employed, an optimal hybrid mesh can be obtained most economically by a combined use of the mesh coring technique suggested by Lo and Lau and a selective removal of diagonals from the triangular element mesh. On the other hand, if quadratic elements are used, it is preferable to generate a pure triangular mesh first, and then obtain a hybrid mesh by merging of triangles.     

New stochastic approximation algorithms with adaptive step sizes

This paper considers the stochastic approximation problem in which the gradient of the function is disturbed by noise. In other words, at each point x _k , instead of the exact gradient g _k , only noisy measurement ${\tilde g_k=g_k+\xi_k}$ is available, where ?? _k denotes the noise. To accelerate the classical Robbins?CMonro algorithm, Kesten (Ann Math Stat 29:41?C59, 1958) and Delyon and Juditsky (SIAM J Optim 3:868?C881, 1993) considered the use of the quantity s _k that stands for the number of changes of the sign of ${\tilde{g}_k^{T}\tilde{g}_{k-1}}$ . Assuming the presence of state independent noise, we discuss in this paper the properties of the quantity ${\frac{s_k}{k}}$ that stands for the change frequency of the sign of ${\tilde{g}_{k}^{T}\tilde{g}_{k-1}}$ and design new stochastic approximation algorithms based on the quantity ${\frac{s_k}k}$ . The almost sure convergence of the new algorithms are also established. The numerical results show that the algorithms are promising.     

On the discrete adjoints of adaptive time stepping algorithms

We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incorrect derivatives. To regain consistency, one has to cancel out the contributions of the non-physical derivatives in the discrete adjoint model. We demonstrate that the discrete adjoint models of one-step, explicit adaptive algorithms, such as the Runge–Kutta schemes, can be made consistent with their continuous counterparts using simple code modifications. Furthermore, we extend the analysis to cover second order adjoint models derived through an extra forward mode differentiation of the discrete adjoint code. Several numerical examples support the mathematical derivations.     

Convergence qualification of adaptive partition algorithms in global optimization

Following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems, necessary and sufficient convergence conditions are formulated, in terms of respectively implied or postulated properties of the partition operator. The convergence results obtained are pertinent to a number of deterministic algorithms in global optimization, permitting their diverse modifications and generalizations.