Choice Of Basis Functions Research Articles

A dynamic programming framework for optimal delivery time slot pricing

We study the dynamic programming approach to revenue management in the context of attended home delivery. We draw on results from dynamic programming theory for Markov decision problems to show that the underlying Bellman operator has a unique fixed point. We then provide a closed-form expression for the resulting fixed point and show that it admits a natural interpretation. Moreover, we also show that – under certain technical assumptions – the value function, which has a discrete domain and a continuous codomain, admits a continuous extension, which is a finite-valued, concave function of its state variables, at every time step. Furthermore, we derive results on the monotonicity of prices with respect to the number of orders placed in our setting. These results open the road for achieving scalable implementations of the proposed formulation, as it allows making informed choices of basis functions in an approximate dynamic programming context. We illustrate our findings on a low-dimensional and an industry-sized numerical example using real-world data, for which we derive an approximately optimal pricing policy based on our theoretical results.

A comparison of parameter estimation in function-on-function regression

Recent technological developments have enabled us to collect complex and high-dimensional data in many scientific fields, such as population health, meteorology, econometrics, geology, and psychology. It is common to encounter such datasets collected repeatedly over a continuum. Functional data, whose sample elements are functions in the graphical forms of curves, images, and shapes, characterize these data types. Functional data analysis techniques reduce the complex structure of these data and focus on the dependences within and (possibly) between the curves. A common research question is to investigate the relationships in regression models that involve at least one functional variable. However, the performance of functional regression models depends on several factors, such as the smoothing technique, the number of basis functions, and the estimation method. This article provides a selective comparison for function-on-function regression models where both the response and predictor(s) are functions, to determine the optimal choice of basis function from a set of model evaluation criteria. We also propose a bootstrap method to construct a confidence interval for the response function. The numerical comparisons are implemented through Monte Carlo simulations and two real data examples.

Study of LCAO-MO calculation by using completely numerical basis functions

Choice of basis functions is decisive in LCAO-MO method to calculate accurately MOs of a molecule under complicated electric/magnetic potential. For such situation, calculating basis functions analytically is generally difficult. To obtain them, the new numerical calculating method based on matrix operation and the new sample point system based on FCC lattice were proposed. Further LCAO-MO calculation using basis functions calculated by this method was applied for small molecules under inner electric potential and showed reasonable results. Finally, two challenges to apply this method for arbitrary magnetic potential are clarified.

Continuum electromagnetic gyrokinetic simulations of turbulence in the tokamak scrape-off layer and laboratory devices

We present algorithms and results from Gkeyll, a full-f continuum, electromagnetic gyrokinetic code, designed to study turbulence in the edge region of fusion devices. The edge is computationally very challenging, requiring robust algorithms that can handle large-amplitude fluctuations and stable interactions with plasma sheaths. We present an energy-conserving high-order discontinuous Galerkin scheme that solves gyrokinetic equations in Hamiltonian form. Efficiency is improved by a careful choice of basis functions and automatically generated computation kernels. Previous verification tests were performed in the straight-field-line large plasma device [Shi et al., J. Plasma Phys. 83, 905830304 (2017)] and the Texas Helimak, a simple magnetized torus [Bernard et al., Phys. Plasmas 26, 042301 (2019)], including the effect of end-plate biasing on turbulence. Results for the scrape-off layer for NSTX parameters with a model helical magnetic geometry with bad curvature have been obtained [Shi et al., Phys. Plasmas 26, 012307 (2019)]. In this paper, we present algorithms for the two formulations of electromagnetic gyrokinetics: the Hamiltonian and the symplectic. We describe each formulation and show results of benchmark tests. Although our scheme works for the Hamiltonian formulation, the presence of spurious numerical modes for high-β and large k⊥2ρs2 regimes shows that the symplectic formulation is more robust. We then review our recent algorithm for the symplectic formulation [Mandell et al., J. Plasma Phys. 86, 905860109 (2020)], along with example application of this new capability. Maintaining positivity of the distribution function can be challenging, and we describe a new and novel exponential recovery based algorithm to address this.

Possible evidence of induced repetitive magnetic reconnection in a superflare from a young solar-type star

We report the detection of multiple quasi-periodic pulsations (QPPs) observed during the flaring activity of KIC 8414845, a young, active solar-type star observed by the Kepler mission launched by NASA. We analyzed the QQP signal using a data-driven, nonparametric method called singular spectrum analysis (SSA), which has never been utilized previously for analyzing solar or stellar QPPs. Because it is not based on a prescribed choice of basis functions, SSA is particularly suitable for analyzing nonstationary, nonlinear signals such as those observed in QPPs during major flares. The analysis has revealed that the apparent anharmonic shape of the QPP in this superflare results from a superposition of two intrinsic modes of periods of 49 min and 86 min, which display quasi-harmonic behaviors and different modulation patterns. The two reconstructed signals are consistent with slow-mode transverse and/or longitudinal magnetohydrodynamic oscillations excited in a coronal loop inducing periodic releases of flaring energy in a nearby loop through a mechanism of repetitive reconnection. The peculiar amplitude modulation of the two modes evinced by SSA favors the interpretation of the observed QPP pattern as due to the excitation in a coronal loop of the second harmonic of a standing slow-mode magnetoacoustic oscillation and a global kink oscillation periodically triggering magnetic reconnection in a nearby loop. Concurrent interpretations cannot however be ruled out on the basis of the available data.

A category theoretical interpretation of discretization in Galerkin finite element method

The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.

Option values of quantity flexibility in nuclear fuel long-term contracts: an application of the least squares Monte Carlo

ABSTRACTOne of the important features in uranium contracts is that the quantity flexibility option is included in long-term contracts, so that buyer has the flexibility to increase or decrease the quantity purchase and adjust the delivery month depending on the market conditions. While it is important to address the value of the flexibility in the contracts, the literature on the contract valuation in uranium markets is scant. In this paper, we evaluate the quantity flexibility in the context of Korean long-term uranium contracts. Specifically, we adapt the least squares Monte Carlo method to simulate the contract value while incorporating the characteristics of option structure and the dynamics of uranium prices. Our simulation results show that the quantity flexibility can play an important role in uranium long-term contracts. The results are also robust to the choice of basis functions, which implies that the method we applied in this paper could provide correct valuation of long-term contracts. Moreover, our results provide some policy implications related to first, the KHNP’s strategy to mix of long-term and spot-market uranium contracts, and second, correct valuation of contract parameters when negotiations take place.

Statistical Inference on Panel Data Models: A Kernel Ridge Regression Method

We propose statistical inferential procedures for nonparametric panel data models with interactive fixed effects in a kernel ridge regression framework. Compared with the traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters. The model complexity is controlled by a continuous regularization parameter which can be automatically selected by the generalized cross-validation. Based on the empirical process theory and functional analysis tools, we derive the joint asymptotic distributions for the estimators in the heterogeneous setting. These joint asymptotic results are then used to construct the confidence intervals for the regression means and the prediction intervals for future observations, both being the first provably valid intervals in literature. The marginal asymptotic normality of the functional estimators in a homogeneous setting is also obtained. Our estimators can also be readily modified and applied to other widely used semiparametric models, such as partially linear models. Simulation and real data analyses demonstrate the advantages of our method. Supplementary materials for this article are available online.

Variable Besov Spaces: Continuous Version

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calderon reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We characterize these function spaces by so-called Peetre maximal functions and we obtain the Sobolev embeddings for these function spaces. We use these results to prove the atomic decomposition for these spaces.

Variable Triebel–Lizorkin-Type Spaces

In this paper, we study Triebel–Lizorkin-type spaces with variable smoothness and integrability. We show that our space is well-defined, i.e., independent of the choice of basis functions and we obtain their atomic characterization. Moreover, the Sobolev embeddings for these function spaces are obtained.

Space–time Trefftz-DG approximation for elasto-acoustics

ABSTRACTDiscontinuous finite element methods have proven their numerical accuracy and flexibility, but they are criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin approximations, but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore is Trefftz method, which distinguishes itself by the choice of basis functions: they are local solutions of initial equation. Thus, in case of time-dependent problems, space-time meshes are required. Classical Trefftz-type approximation uses the exact solutions of acoustic and elastodynamic systems taken with different frequencies, in order to obtain better numerical errors. We have computed a polynomial basis using the Taylor expansions of generating exponential functions that are the exact solutions of the initial acoustic and elastodynamic systems. This basis, compared to the classical one based on trigonometric functions, requires less degrees of freedom for the same level of accuracy. In the present work, we develop the theory for coupled elasto-acoustic system. We confirm well-posedness of the problem, based on error estimates in mesh-dependent norms. We consider a space-time polynomial basis to implement numerically the method and to perform some numerical experiments showing promising results.

Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth

In this paper, we consider the Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth. We prove that such function spaces are well defined in the sense that their definitions are independent of the choice of basis functions under some specific assumptions. Then we show some embeddings related to them.

Sparse Bayesian Learning-Based Seismic Denoise by Using Physical Wavelet as Basis Functions

Attenuating random noise is a fundamental yet necessary step for subsequent seismic image processing and interpretation. We introduce a sparse Bayesian learning (SBL)-based seismic denoise method by using the physical wavelet as the basis function. The physical wavelet estimated from seismic and well logging data can appropriately describe the characteristics of the seismic data. Thus, it is an appropriate choice of basis function. Moreover, the tradeoff regularization parameter for determining denoise quality can be adaptively estimated according to the updated data misfit and sparseness degree during the iterative process of the SBL algorithm. The motivation behind the denoise method using sparse representations is that seismic signals can be sparsely represented by using several physical wavelets, whereas noise cannot. Both synthetic and real seismic data examples are adopted to demonstrate the effectiveness of the method.

Separating metric perturbations in near-horizon extremal Kerr spacetimes

Linear perturbation theory is a powerful toolkit for studying black hole spacetimes. However, the perturbation equations are hard to solve unless we can use separation of variables. In the Kerr spacetime, metric perturbations do not separate, but curvature perturbations do. The cost of curvature perturbations is a very complicated metric-reconstruction procedure. This procedure can be avoided using a symmetry-adapted choice of basis functions in highly symmetric spacetimes, such as near-horizon extremal Kerr. In this paper, we focus on this spacetime, and (i) construct the symmetry-adapted basis functions; (ii) show their orthogonality; and (iii) show that they lead to separation of variables of the scalar, Maxwell, and metric perturbation equations. This separation turns the system of partial differential equations into one of ordinary differential equations over a compact domain, the polar angle.

ON THE CALCULATION OF RISK MEASURES USING LEAST-SQUARES MONTE CARLO

The paper establishes a convergence result for the Least-Squares Monte Carlo method, applied to the estimation of risk measures (such as VaR) depending on the distribution of the future value of some derivative contract (or portfolio). We extend previous results in the literature by focusing on the specific case where the distribution of the underlying factors is known. In particular, we are able to remove the requirement of bounded conditional variance which is usually not satisfied by common financial models and payoffs at the cost of using a weaker notion of convergence. Our main result shows the convergence of the empirical distribution function when the number of simulations [Formula: see text] and of basis variables [Formula: see text] are jointly sent to infinity under certain conditions on the payoff and the basis. With a suitable choice of basis functions, we finally prove that convergence can be ensured when [Formula: see text].

Modelling Seabed Ploughing Using the Material Point Method

Ploughing of the seabed is needed for the installation of cables and pipelines and is an area of construction likely to increase given plans in the UK and elsewhere for offshore marine energy (wind and tidal). Seabed ploughing is an expensive and sometimes risky operation for which there are limited guidelines as to what the tow force and speed is for an expected trenching profile within a specified ground condition. Most ploughing schemes are designed using semi-empirical approaches, with very few computational tools able to take into account the geometric and material nonlinearity inherited by the ploughing problem. In this paper we describe how the Material Point Method (MPM) is being used as a numerical tool to model seabed ploughing with the aim of providing an improved predictive tool for the future, via an EPSRC-funded research project at Durham and Dundee Universities. Various issues are discussed in the paper including the means of modelling moving essential boundaries and the choice of basis functions, and this new method in MPM is demonstrated on a simple ploughing problem.

Optimal Weighted Pointwise Ensemble of Radial Basis Functions with Different Basis Functions

The radial basis functions (RBF) interpolation model has been extensively used in various engineering fields. All these applications call for accurate RBF models. The RBF predictions are affected by the choice of basis functions, whereas the proper basis function is problem dependent. To avoid the choice of basis functions and improve the predictions, this paper presents an optimal weighted pointwise ensemble (OWPE) to combine the locally accurate predictions of RBF models built with different basis functions together. The key to the success of OWPE is to construct proper pointwise weight functions for the component RBF models. At the observed points, the weights of one or zero were used to sufficiently highlight the locally accurate predictions of component RBF models. At the unobserved points, the optimal pointwise weight functions were constructed by using an optimized coefficient that can adapt to the characteristics of component RBF models. Numerical experiments on 14 analytical functions and an axial compressor blade design example show that OWPE provides more accurate and robust predictions. Additionally, OWPE performs better when having more observed points and component RBF models. It is notable that OWPE can also be used with other types of interpolation metamodels.

Presentation of special series with computed recurrently coefficients of solutions of nonlinear evolution equations

One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basis functions. The coefficients of such series are found successively as solutions of linear differential equations. To find recurrence, the coefficient is achieved by the choice of basis functions, which may also contain arbitrary functions. By using such functional arbitrariness, it allows in some cases to prove the global convergence of the corresponding constructed series, as well as the solvability of the boundary value problem.

A short note on the error estimates of Yuan–Shu discontinuous Galerkin method based on non-polynomial approximation spaces

In the article [Yuan and Shu (2006) [17]], Yuan and Shu have developed discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for solving time dependent problems. The authors established L2 error estimates of the proposed methods and also presented a criterion for the choice of basis functions of the non-polynomial spaces to have the same approximation rates as those of polynomial finite element spaces of the same dimension. However, the verification that some of approximation spaces do satisfy the criterion has been performed only to limited types of approximation spaces. In this regards, the aim of this short note is to fill the gap. We prove that the criterion of Yuan and Shu can be satisfied by a wide class of basis functions, including the well-known basis functions such as trigonometric and exponential polynomials.

Robust Monte Carlo Method for R&D Real Options Valuation

This paper is devoted to developing a robust numerical analysis of least squares Monte Carlo (LSM) in valuing R&D investment opportunities. As it is well known, R&D projects are characterized by sequential investments and therefore they can be considered as compound options involving a set of interacting American-type options. The basic Monte Carlo simulation takes a long time and it is computationally intensive and inefficient. In this context, LSM method is a powerful and flexible tool for capital budgeting decisions and for valuing R&D investments. In particular way, numerical tests are performed to examine the optimal choice of basis function and polynomial degree in terms of reduction of the execution time, accuracy and improvement in the simulation.