Computational Speedup Research Articles

A Purely Real-Valued Fast Estimator of Dynamic Harmonics for Application in Embedded Monitoring Devices in Power-Electronic Grids

Dynamic harmonic estimation is important for the monitoring and control of power-electronic grids. But the high-precision dynamic harmonic estimation algorithms usually have a heavy computational burden and occupy a large memory space, making them difficult to implement in the embedded platform. Thus, the motivation of this paper lies in providing an estimator with low computational complexity and less storage space consumption. A purely real-valued fast dynamic harmonics estimator is proposed. Firstly, a purely real-valued estimation model is established based on the Taylor series expansion on the time-varying amplitude and phase angle. Secondly, the estimation filter bank is computed in the least-squares sense, and the corresponding estimation error is theoretically analyzed. Finally, the purely real-valued fast dynamic harmonics estimator is designed. The advantage includes significantly reducing the computational complexity and memory space consumption while maintaining high-precision estimation. The testing results show that the proposed estimator can achieve the highest harmonics estimation precision under dynamic conditions. The frequency error, magnitude error, and phase angle error are less than 5 × 10−2 Hz, 7 × 10−1%, and 8 × 10−2 degrees, respectively, which verifies the advantage of high-precision estimation. The proposed estimator achieves a computational speed-up of approximately 430, 396, and 330 times compared to the Prony method, ESPRIT method, and iterative Taylor Fourier transform method, respectively. The computational load rate for executing the proposed estimator on the embedded prototype using C6748 DSP for estimating 50 harmonics is approximately only 2.05%, which verifies the advantage of a low computational load rate.

Deep Learning Method for Airfoil Flow Field Simulation Based on Unet++

ABSTRACTThis paper investigates the accuracy of U‐Net++ networks in predicting Reynolds‐Averaged Navier‐Stokes (RANS) solutions. The study employs the symbolic distance function (SDF) to represent geometry and flow conditions, utilizing parameterized airfoil data from the UIUC (University of Illinois at Urbana‐Champaign) airfoil datasets. The research assesses the performance of multiple trained neural networks in predicting pressure and velocity distributions. Specifically, the study examines the influence of varying network weights on solution accuracy. Through the optimization of the model, the research demonstrates that the mean relative error is below 1.72% for a range of previously unseen wing shapes, with a computational speedup factor of up to 1,000× in certain scenarios. The accuracy achieved by this model underscores the significant potential of deep learning‐based approaches as reliable tools for aerodynamic design and optimization.

Lattice Multislice Algorithm for Fast Simulation of Scanning Transmission Electron Microscopy Images.

We introduce a new approach to the numerical simulation of Scanning Transmission Electron Microscopy images. The Lattice Multislice Algorithm takes advantage of the fact that the electron waves passing through the specimen have limited bandwidth and therefore can be approximated very well by a low-dimensional linear space spanned by translations of a well-localized function. Just like in the PRISM algorithm recently published by C. Ophus, we utilize the linearity of the Schrödinger equation but perform the approximations with functions that are well localized in real space instead of Fourier space. This way, we achieve a similar computational speedup as PRISM, but at a much lower memory consumption and reduced numerical error due to avoiding virtual copies of the probe waves interfering with the result. Our approach also facilitates faster recomputations if local changes are made to the specimen such as changing a single atomic column.

Computationally efficient collimator-detector response compensation in high energy SPECT using 1D convolutions and rotations

Objective. Modeling of the collimator-detector response (CDR) in single photon emission computed tomography (SPECT) reconstruction enables improved resolution and accuracy, and is thus important for quantitative imaging applications such as dosimetry. The implementation of CDR modeling, however, can become a computational bottleneck when there are substantial components of septal penetration and scatter in the acquired data, since a direct convolution-based approach requires large 2D kernels. This work proposes a 1D convolution and rotation-based CDR model that reduces reconstruction times but maintains consistency with models that employ 2D convolutions. To enable open-source development and use of these models in image reconstruction, we release a SPECTPSFToolbox repository for the PyTomography project on GitHub.Approach. A 1D/rotation-based CDR model was formulated and subsequently fit to Monte Carlo (MC) point source data representative of177Lu,131I, and225Ac imaging. Computation times of (i) the proposed 1D/rotation-based model and (ii) a traditional model that uses 2D convolutions were compared for typical SPECT matrix sizes. Both CDR models were then used in the reconstruction of MC, physical phantom, and patient data; the models were compared by quantifying total counts in hot regions of interest (ROIs) and activity contrast between hot ROIs and background regions.Results. For typical matrix sizes in SPECT reconstruction, application of the 1D/rotation-based model provides a two-fold computational speed-up over the 2D model when running on GPU. Only small differences between the 1D/rotation-based and 2D models (order of 1%) were obtained for count and contrast quantification in select ROIs.Significance. A technique for CDR modeling in SPECT was proposed that (i) significantly speeds up reconstruction times, and (ii) yields nearly identical reconstructions to traditional 2D convolution based CDR techniques. The released toolbox will permit open-source development of similar models for different isotopes and collimators.

Multifidelity uncertainty quantification for ice sheet simulations

Ice sheet simulations suffer from vast parametric uncertainties, such as the basal sliding boundary condition or geothermal heat flux. Quantifying the resulting uncertainties in predictions is of utmost importance to support judicious decision-making, but high-fidelity simulations are too expensive to embed within uncertainty quantification (UQ) computations. UQ methods typically employ Monte Carlo simulation to estimate statistics of interest, which requires hundreds (or more) of ice sheet simulations. Cheaper low-fidelity models are readily available (e.g., approximated physics, coarser meshes), but replacing the high-fidelity model with a lower fidelity surrogate introduces bias, which means that UQ results generated with a low-fidelity model cannot be rigorously trusted. Multifidelity UQ retains the high-fidelity model but expands the estimator to shift computations to low-fidelity models, while still guaranteeing an unbiased estimate. Through this exploitation of multiple models, multifidelity estimators guarantee a target accuracy at reduced computational cost. This paper presents a comprehensive multifidelity UQ framework for ice sheet simulations. We present three multifidelity UQ approaches—Multifidelity Monte Carlo, Multilevel Monte Carlo, and the Best Linear Unbiased Estimator—that enable tractable UQ for continental-scale ice sheet simulations. We demonstrate the techniques on a model of the Greenland ice sheet to estimate the 2015-2050 ice mass loss, verify their estimates through comparison with Monte Carlo simulations, and give a comparative performance analysis. For a target accuracy equivalent to 1 mm sea level rise contribution at 95 % confidence, the multifidelity estimators achieve computational speedups of two orders of magnitude.

Mitigating stop-and-go traffic congestion with operator learning

This paper presents a novel neural operator learning framework for designing boundary control to mitigate stop-and-go congestion on freeways. The freeway traffic dynamics are described by second-order coupled hyperbolic partial differential equations (PDEs), i.e. the Aw–Rascle–Zhang (ARZ) macroscopic traffic flow model. The proposed framework learns feedback boundary control strategies from the closed-loop PDE solution using backstepping controllers, which are widely employed for boundary stabilization of PDE systems. The PDE backstepping control design is time-consuming and requires intensive depth of expertise, since it involves constructing and solving backstepping control kernels. Existing machine learning methods for solving PDE control problems, such as physics-informed neural networks (PINNs) and reinforcement learning (RL), face the challenge of retraining when PDE system parameters and initial conditions change. To address these challenges, we present neural operator (NO) learning schemes for the ARZ traffic system that not only ensure closed-loop stability robust to parameter and initial condition variations but also accelerate boundary controller computation. The first scheme embeds NO-approximated control gain kernels within a analytical state feedback backstepping controller, while the second one directly learns a boundary control law from functional mapping between model parameters to closed-loop PDE solution. The stability guarantee of the NO-approximated control laws is obtained using Lyapunov analysis. We further propose the physics-informed neural operator (PINO) to reduce the reliance on extensive training data. The performance of the NO schemes is evaluated by simulated and real traffic data, compared with the benchmark backstepping controller, a Proportional Integral (PI) controller, and a PINN-based controller. The NO-approximated methods achieve a computational speedup of approximately 300 times with only a 1% error trade-off compared to the backstepping controller, while outperforming the other two controllers in both accuracy and computational efficiency. The robustness of the NO schemes is validated using real traffic data, and tested across various initial traffic conditions and demand scenarios. The results show that neural operators can significantly expedite and simplify the process of obtaining controllers for traffic PDE systems with great potential application for traffic management.

Generalised antenna functions for higher-order calculations

In this paper we discuss the definition, the construction and the implementation of generalised antenna functions for final-state radiation up to Next-to-Next-to-Leading Order (NNLO) in QCD. Generalised antenna functions encapsulate the singular behaviour of unresolved emissions when these occur within multiple hard radiators and not just two of them, as for traditional antenna functions. The construction of such objects is possible thanks to the recently proposed algorithm for building idealised antenna functions from a target set of infrared limits. Generalised antenna functions bring major simplifications in the assemblage of subtraction terms in the context of the antenna scheme at NNLO and beyond, as well as a substantial computational speedup of higher-order calculations. We discuss in detail the improvements on the formal and practical side for the computation of the NNLO correction to three-jet production at electron-positron colliders, providing a thorough numerical validation of the newly proposed scheme. For this calculation one can expect almost an order of magnitude speedup with respect to the original implementation.

Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing

Abstract Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis.
For instance in the domain of energy economics, accurate and timely risk management demands for
efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a
novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed
through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end
speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum
speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating
polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also
enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing
an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the
dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth
of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM
Quantum systems.

Predicting solvation free energies with an implicit solvent machine learning potential.

Machine learning (ML) potentials are a powerful tool in molecular modeling, enabling abinitio accuracy for comparably small computational costs. Nevertheless, all-atom simulations employing best-performing graph neural network architectures are still too expensive for applications requiring extensive sampling, such as free energy computations. Implicit solvent models could provide the necessary speed-up due to reduced degrees of freedom and faster dynamics. Here, we introduce a Solvation Free Energy Path Reweighting (ReSolv) framework to parameterize an implicit solvent ML potential for small organic molecules that accurately predicts the hydration free energy, an essential parameter in drug design and pollutant modeling. Learning on a combination of experimental hydration free energy data and abinitio data of molecules in vacuum, ReSolv bypasses the need for intractable abinitio data of molecules in an explicit bulk solvent and does not have to resort to less accurate data-generating models. On the FreeSolv dataset, ReSolv achieves a mean absolute error close to average experimental uncertainty, significantly outperforming standard explicit solvent force fields. Compared to the explicit solvent ML potential, ReSolv offers a computational speedup of four orders of magnitude and attains closer agreement with experiments. The presented framework paves the way for deep molecular models that are more accurate yet computationally more cost-effective than classical atomistic models.

Unconditional quantum magic advantage in shallow circuit computation

Quantum theory promises computational speed-ups over classical approaches. The celebrated Gottesman-Knill Theorem implies that the full power of quantum computation resides in the specific resource of “magic” states—the secret sauce to establish universal quantum computation. However, it is still questionable whether magic indeed brings the believed quantum advantage, ridding unproven complexity assumptions or black-box oracles. In this work, we demonstrate the first unconditional magic advantage: a separation between the power of generic constant-depth or shallow quantum circuits and magic-free counterparts. For this purpose, we link the shallow circuit computation with the strongest form of quantum nonlocality—quantum pseudo-telepathy, where distant non-communicating observers generate perfectly synchronous statistics. We prove quantum magic is indispensable for such correlated statistics in a specific nonlocal game inspired by the linear binary constraint system. Then, we translate generating quantum pseudo-telepathy into computational tasks, where magic is necessary for a shallow circuit to meet the target. As a by-product, we provide an efficient algorithm to solve a general linear binary constraint system over the Pauli group, in contrast to the broad undecidability in constraint systems. We anticipate our results will enlighten the final establishment of the unconditional advantage of universal quantum computation.

Benchmarking a dual-scale hybrid simulation framework for small globular proteins combining the CHARMM36 and Martini2 models.

Multi-scale models in which varying resolutions are considered in a single molecular dynamics simulation setup are gaining importance in integrative modeling. However, combining atomistic and coarse-grain resolutions, especially for coarse-grain force fields derived from top-down approaches, have not been well explored. In this study, we have implemented and tested a dual-resolution simulation approach to model globular proteins in atomistic detail (represented by the CHARMM36 model) with the surrounding solvent in Martini2 coarse-grain detail. The hybrid scheme considered is an extension of a model implemented earlier for mainly lipid and water molecules. We have considered a set of small globular proteins and have extensively compared to atomistic benchmark simulations as well as a host of experimental observables. We show that the protein structural dynamics sampled in the hybrid scheme is robust, and the intra-protein contact maps are reproduced, despite increased fluctuations of the loop regions. A good match is observed with experimental small angle X-ray scattering (SAXS) and NMR observables, such as chemical shifts and [Formula: see text] -coupling, with the best match obtained for the chemical shifts. However, deviations are observed in the water dynamics and protein-water interactions which we attribute to the limitation of solvent screening in the coarse-grain force field. The computational speed-up achieved is about 2-3 times compared to an all-atom system. Overall, the hybrid model is able to retain the main features of the underlying atomistic conformational landscape with a two-fold speed-up in computational cost.

Reduced-Order Modeling for Subsurface Flow Simulation in Fractured Reservoirs

Summary Reservoir simulation for fractured reservoirs is often challenging and time-consuming due to the strong heterogeneity and complex flow dynamics introduced by fracture-matrix interactions. In this study, we introduce a novel reduced-order modeling procedure to speed up the flow simulation of fractured reservoirs. The reduced-order model (ROM) is developed based on proper orthogonal decomposition (POD) in conjunction with the embedded discrete fracture model (EDFM) that provides full-order simulation results. With the full-order training simulation, snapshots of reservoir pressure and saturation state at different timesteps are captured and assembled into separate data matrices. Singular value decomposition (SVD) is then applied to these data matrices to obtain a reduced set of orthogonal base vectors for pressure and saturation solutions, respectively. These base vectors enable the projection of high-dimensional linear equations into much lower-dimensional spaces, which significantly accelerates the process of solving nonlinear governing equations under the EDFM approach. The developed reduced-order modeling procedure is implemented in the MATLAB reservoir simulation toolbox (MRST) and tested via multiple cases for both 2D and 3D fractured reservoirs under different boundary and well control scenarios. In certain challenging cases, the use of multiple training simulations is explored and is shown to provide improved predictions. Overall, the proposed ROM approach is able to provide simulation results that are very consistent with those obtained from the full-order simulations while achieving computational speedups of about an order of magnitude for large-scale cases. These observations indicate that the proposed ROM exhibits satisfactory generalization performance, making it suitable for problems that require many flow simulations under different settings, such as production optimization.

Operator learning with Gaussian processes

Operator learning focuses on approximating mappings G†:U→V between infinite-dimensional spaces of functions, such as u:Ωu→R and v:Ωv→R. This makes it particularly suitable for solving parametric nonlinear partial differential equations (PDEs). Recent advancements in machine learning (ML) have brought operator learning to the forefront of research. While most progress in this area has been driven by variants of deep neural networks (NNs), recent studies have demonstrated that Gaussian process (GP)/kernel-based methods can also be competitive. These methods offer advantages in terms of interpretability and provide theoretical and computational guarantees. In this article, we introduce a hybrid GP/NN-based framework for operator learning, leveraging the strengths of both deep neural networks and kernel methods. Instead of directly approximating the function-valued operator G†, we use a GP to approximate its associated real-valued bilinear form G˜†:U×V∗→R. This bilinear form is defined by the dual pairing G˜†(u,φ)≔[φ,G†(u)], which allows us to recover the operator G† through G†(u)(y)=G˜†(u,δy). The mean function of the GP can be set to zero or parameterized by a neural operator and for each setting we develop a robust and scalable training mechanism based on maximum likelihood estimation (MLE) that can optionally leverage the physics involved. Numerical benchmarks demonstrate our method’s scope, scalability, efficiency, and robustness; showing that (1) it enhances the performance of a base neural operator by using it as the mean function of a GP, and (2) it enables the construction of zero-shot data-driven models that can make accurate predictions without any prior training. Additionally, our framework (a) naturally extends to cases where G†:U→∏s=1SVs maps into a vector of functions, and (b) benefits from computational speed-ups achieved through product kernel structures and Kronecker product matrix representations of the underlying kernel matrices.11GitHub repository: https://github.com/Bostanabad-Research-Group/GP-for-Operator-Learning.

Research on Quantum Computing Acceleration of Support Vector Machines in Multi-dimensional Nonlinear Feature Spaces

The current development of Support Vector Machines (SVM) has reached a bottleneck, with issues such as long training times and weak interpretability when dealing with large-scale, multi-dimensional data. This paper introduces the concept of Quantum Support Vector Machines (QSVM) and achieves efficient solutions through quantum algorithms such as the HHL algorithm. The research designs a computational architecture that combines classical and quantum computing, utilizing the Pauli decomposition of Hermitian matrices to simulate the quantum simulation of Hamiltonian quantities and implementing the quantum simulation of unitary operators. Based on the conclusions, a complete quantum linear solver circuit is designed, achieving exponential growth in computational complexity. Experiments using the Iris dataset demonstrate the excellent classification performance of QSVM, which outperform classical SVM in classification accuracy, computational time complexity, and memory requirements. More efficient quantum mapping algorithms and quantum circuit optimization methods are implemented, providing new ideas and methods for the application of SVM to large-scale datasets.

A Data-Loader Tunable Knob to Shorten GPU Idleness for Distributed Deep Learning

Deep Neural Networks (DNNs) have been applied as an effective machine learning algorithm to tackle problems in different domains. However, the endeavor to train sophisticated DNN models can stretch from days into weeks, presenting substantial obstacles in the realm of research focused on large-scale DNN architectures. Distributed Deep Learning (DDL) contributes to accelerating DNN training by distributing training workloads across multiple computation accelerators, for example, graphics processing units (GPUs). Despite the considerable amount of research directed toward enhancing DDL training, the influence of data loading on GPU utilization and overall training efficacy remains relatively overlooked. It is non-trivial to optimize data-loading in DDL applications that need intensive central processing unit (CPU) and input/output (I/O) resources to process enormous training data. When multiple DDL applications are deployed on a system (e.g., Cloud and High-Performance Computing (HPC) system), the lack of a practical and efficient technique for data-loader allocation incurs GPU idleness and degrades the training throughput. Therefore, our work first focuses on investigating the impact of data-loading on the global training throughput. We then propose a throughput prediction model to predict the maximum throughput for an individual DDL training application. By leveraging the predicted results, A-Dloader is designed to dynamically allocate CPU and I/O resources to concurrently running DDL applications and use the data-loader allocation as a knob to reduce GPU idle intervals and thus improve the overall training throughput. We implement and evaluate A-Dloader in a DDL framework for a series of DDL applications arriving and completing across the runtime. Our experimental results show that A-Dloader can achieve a 28.9% throughput improvement and a 10% makespan improvement compared with allocating resources evenly across applications.

Multi-Resolution Real-Time Deep Pose-Space Deformation

We present a hard-real-time multi-resolution mesh shape deformation technique for skeleton-driven soft-body characters. Producing mesh deformations at multiple levels of detail is very important in many applications in computer graphics. Our work targets applications where the multi-resolution shapes must be generated at fast speeds ("hard-real-time", e.g., a few milliseconds at most and preferably under 1 millisecond), as commonly needed in computer games, virtual reality and Metaverse applications. We assume that the character mesh is driven by a skeleton, and that high-quality character shapes are available in a set of training poses originating from a high-quality (slow) rig such as volumetric FEM simulation. Our method combines multi-resolution analysis, mesh partition of unity, and neural networks, to learn the pre-skinning shape deformations in an arbitrary character pose. Combined with linear blend skinning, this makes it possible to reconstruct the training shapes, as well as interpolate and extrapolate them. Crucially, we simultaneously achieve this at hard real-time rates and at multiple mesh resolution levels. Our technique makes it possible to trade deformation quality for memory and computation speed, to accommodate the strict requirements of modern real-time systems. Furthermore, we propose memory layout and code improvements to boost computation speeds. Previous methods for realtime approximations of quality shape deformations did not focus on hard real-time, or did not investigate the multi-resolution aspect of the problem. Compared to a "naive" approach of separately processing each hierarchical level of detail, our method offers a substantial memory reduction as well as computational speedups. It also makes it possible to construct the shape progressively level by level and interrupt the computation at any time, enabling graceful degradation of the deformation detail. We demonstrate our technique on several examples, including a stylized human character, human hands, and an inverse-kinematics-driven quadruped animal.

Photonic-Electronic Integrated Circuits for High-Performance Computing and AI Accelerators

Photonic-Electronic Integrated Circuits for High-Performance Computing and AI Accelerators

Fast and Efficient Lunar Finite Element Gravity Model

In this paper, the finite element method (FEM) is integrated with orthogonal polynomial approximation in high-dimensional spaces to innovatively model the Moon’s surface gravity anomaly. The aim is to approximate solutions to Laplace’s classical differential equations of gravity, employing classical Chebyshev polynomials as basis functions. Using classical Chebyshev polynomials as basis functions, the least-squares approximation was used to approximate discrete samples of the approximation function. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. The first application of this project is to substitute the globally valid classical spherical harmonic series of approximations with locally valid series of orthogonal polynomial approximations (i.e., using the FEM approach). With an error tolerance set at 10−9ms−2, this method is used to adapt the gravity model radially upwards from the lunar surface. The results showcase a need for a higher degree of approximation on and near the lunar surface, with the necessity decreasing as the radius increases. Notably, this method achieves a computational speedup of five orders of magnitude when applying the method to radial adaptation. More intrinsically, the second application involves using the methodology as an effective tool in solving boundary value problems. Specifically, this approach is implemented to solve classical differential equations involved with high-precision, long-term orbit propagation. This application provides a four-order-of-magnitude speedup in computational time while maintaining an error within the 10−10ms−2 error range for various orbit propagation tests. Alongside the advancements in orthogonal approximation theory, the FEM enables revolutionary speedups in orbit propagation without compromising accuracy.

On a minimization problem of the maximum generalized eigenvalue: properties and algorithms

AbstractWe study properties and algorithms of a minimization problem of the maximum generalized eigenvalue of symmetric-matrix-valued affine functions, which is nonsmooth and quasiconvex, and has application to eigenfrequency optimization of truss structures. We derive an explicit formula of the Clarke subdifferential of the maximum generalized eigenvalue and prove the maximum generalized eigenvalue is a pseudoconvex function, which is a subclass of a quasiconvex function, under suitable assumptions. Then, we consider smoothing methods to solve the problem. We introduce a smooth approximation of the maximum generalized eigenvalue and prove the convergence rate of the smoothing projected gradient method to a global optimal solution in the considered problem. Also, some heuristic techniques to reduce the computational costs, acceleration and inexact smoothing, are proposed and evaluated by numerical experiments.

Normalizing Basis Functions: Approximate Stationary Models for Large Spatial Data

ABSTRACTIn geostatistics, traditional spatial models often rely on the Gaussian process (GP) to fit stationary covariances to data. It is well known that this approach becomes computationally infeasible when dealing with large data volumes, necessitating the use of approximate methods. A powerful class of methods approximate the GP as a sum of basis functions with random coefficients. Although this technique offers computational efficiency, it does not inherently guarantee a stationary covariance. To mitigate this issue, the basis functions can be ‘normalized’ to maintain a constant marginal variance, avoiding unwanted artefacts and edge effects. This allows for the fitting of nearly stationary models to large, potentially non‐stationary datasets, providing a rigorous base to extend to more complex problems. Unfortunately, the process of normalizing these basis functions is computationally demanding. To address this, we introduce two fast and accurate algorithms to the normalization step, allowing for efficient prediction on fine grids. The practical value of these algorithms is showcased in the context of a spatial analysis on a large dataset, where significant computational speedups are achieved. While implementation and testing are done specifically within the LatticeKrig framework, these algorithms can be adapted to other basis function methods operating on regular grids.