Dirichlet Kernel Research Articles

Sampling discretization of the uniform norm and applications

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any N-dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in N number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on N. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

Mollification of Fourier spectral methods with polynomial kernels

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter is analogous to that in the Dirichlet mollifier, which is detailed by analyzing the numerical solution. Further, they are extended to linear scalar conservation law problems.

Asymptotics and criticality for a space-dependent branching process

We investigate a non-conservative semigroup determined by a branching process tracing the evolution of particles moving in a domain in . When a particle is killed at the boundary, a new generation of particles with mean number is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of , we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles , with depending explicitly on . We prove the Yaglom limit , where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point . The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by , the model provides an explicit example of self-organized criticality.

Compressive beamforming for direction-of-arrival estimation of cyclostationary propeller noise

A compressive beamformer is proposed for direction-of-arrival estimation of cyclostationary ship-radiated propeller noise sources. The sources are recovered by iterative hard thresholding algorithm which is adapted for seeking the sparsity spatially and in cyclic frequency domain. Exploiting this double sparsity, the proposed cyclic compressive beamformer enables high-resolution localization of sources with different cyclic frequencies. Spectral averaging is suggested to reduce the adverse effect of the spatial Dirichlet kernel when there are multiple sources. Numerical examples demonstrate that the proposed scheme outperforms the well-known methods especially for low signal-to-noise ratios, numerous closely-spaced sources and few sensors. Performance of the proposed estimators are verified by explicit analytical derivations for several array and signal parameters.

A mollification regularization method with Dirichlet kernel to solve potential-free field inverse Schrödinger Cauchy problem

We consider solving the Cauchy problem of the Schrödinger equation with potential-free field by a mollification regularization method in this work. By convolving the measured data with the Dirichlet kernel, the ill-posed case is turned into a well-posed one. Convergent estimates are gained via the priori and the posteriori parameter selection rules. Finally, three simulation experiment results are shown to prove the feasibility and stability of our presented procedure.

Sampling Discretization of the Uniform Norm

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N-dimensional subspace of the space of continuous functions it is sufficient to use $$e^{CN}$$ sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate the application of our technique on the example of trigonometric polynomials.

Minimax properties of Dirichlet kernel density estimators

This paper considers the asymptotic behavior in β-Hölder spaces, and under Lp losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison–Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever (p,β)∈[1,3)×(0,2] or (p,β)∈Ad, where Ad is a specific subset of [3,4)×(0,2] that depends on the dimension d of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either p∈[4,∞) or β∈(2,∞). These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.

Power of Dirichlet kernels and approximation by discrete linear operators $\textit{I}$: direct results

The second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered.

Moment estimates for the exponential sum with higher divisor functions

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $\tau_k$, and asymptotics for high moments of the Dirichlet kernel.

A multivariate normal approximation for the Dirichlet density and some applications

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison and Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem is briefly mentioned but left open for future research.

Asymptotic properties of Dirichlet kernel density estimators

We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate O(n−4/(d+4)) for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

A new image-quality evaluating and enhancing methodology for bridge inspection using an unmanned aerial vehicle

This paper proposes a new methodology to address the image quality problem encountered as the use of an unmanned aerial vehicle (UAV) in the field of bridge inspection increased. When inspecting a bridge, the image obtained from the UAV was degraded by various interference factors such as vibration, wind, and motion of UAV. Image quality degradation such as blur, noise, and low-resolution is a major obstacle in utilizing bridge inspection technology based on UAV. In particular, in the field of bridge inspection where damages must be accurately and quickly detected based on data obtained from UAV, these quality issues weaken the advantage of using UAVs by requiring re-take of images through re-flighting. Therefore, in this study, image quality assessment (IQA) based on local blur map (LBM) and image quality enhancement (IQE) using the variational Dirichlet (VD) kernel estimation were proposed as a solution to address the quality issues. First, image data was collected by setting different camera parameters for each bridge member. Second, a blur map was generated through discrete wavelet transform (DWT) and a new quality metric to measure the degree of blurriness was proposed. Third, for low-quality images with a large degree of blurriness, the blind kernel estimation and blind image deconvolution were performed to enhance the quality of images. In the validation tests, the proposed quality metric was applied to material image sets of bridge pier and deck taken from UAV, and its results were compared with those of other quality metrics based on singular value decomposition (SVD), sum of gray-intensity variance (SGV) and high-frequency multiscale fusion and sort transform (HiFST) methods. It was validated that the proposed IQA metric showed better classification performance on UAV images for bridge inspection through comparison with the classification results by human perception. In addition, by performing IQE, on average, 26% of blur was reduced, and the images with enhanced quality showed better damage detection performance through the deep learning model (i.e., mask and region-based convolutional neural network).

A special approach to derive new formulas for some special numbers and polynomials

By applying Laplace differential operator to harmonic conjugate components of the analytic functions and using Wirtinger derivatives, some identities and relations including Bernoulli and Euler polynomials and numbers are obtained. Next, using the Legendre identity, trigonometric functions and the Dirichlet kernel, some formulas and relations involving Bernoulli and Euler numbers, cosine-type Bernoulli and Euler polynomials, and sine-type Bernoulli and Euler polynomials are driven. Then, by using the generating functions method and the well-known Euler identity, many new identities, formulas, and combinatorial sums among the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, the Chebyshev polynomials, and Bernoulli and Euler type polynomials are given. Finally, some infinite series representations for these special numbers and polynomials and their numerical examples are presented.

Heuristic Global Optimization in Chemical Compound Space.

We report developments in combinatorial optimization under constraints in chemical space. We considered random functions, which serve as a baseline to measure performance, and constrained optimizations over two databases of electrochromic molecules (∼5000 and 1012, respectively). These problems were optimized using sequential heuristic next-neighbor search (HNNS) (introduced in Elward and Rinderspacher Phys. Chem. Chem. Phys. 2015, 17, 24322 and Elward and Rinderspacher Mol. Syst. Des. Eng. 2018, 3, 485) and kernel-based efficient global optimization (EGO) with two reordering strategies. In addition to the average ordering method introduced with sequential HNNS, a new reordering based on a locally separable linear model is formulated and applied. Presented is the analysis of periodic kernels for EGO: the modified Dirichlet kernels D̃n(x) = ∑in[cos(2πix) - (-1)i]/i, the minimalist kernel KN(x) = cos(πx/N)2/2 - δx/2, and an analogue to the popular Gaussian kernel, GNσ(x) = e-|xmodN|2/(2σ2), where n is the order of the Dirichlet kernel, N is the number of choices on a combinatorial site, and σ is a broadening factor. We find that reordering is pivotal for high optimization efficiency, in both the average and worst-case scenarios. The new linear-estimation ordering paradigm is superior in the chemistry context. Furthermore, with judicious use of hyperparameters and algorithmic choices, EGO outperforms HNNS. The global optimum for the small chemical problem is found with >98% likelihood for all global search methods employing a linear reordering heuristic for organizing the search space.

SH-waveform modeling of small local seismic events in Ladoga lake

A lot of seismic phenomena with small magnitude has been taking place in the Ladoga lake region (northwestern Russia). One of them is the earthquake which was recorded on the 31st of July, 2010 followed by the swarm of microearthquakes near Nikonovsky cape. There is only one station in the region that is why it is difficult to process the data. In this work, we propose a new approach to construct synthetic seismograms of the main event with minimum data. Also this approach applies us to calculate parameters of the source. The approximate magnitude of the main event is $$M_L=-0.8$$ . For swarm events, we have magnitude values from $$-\,1.0$$ to $$-\,2.6$$ . The data of swarm events (microearthquakes) is used to test the technique of constructing synthetic seismograms. The approach we propose is based on Green’s function method with using two temporal $$\delta$$ -sequences (the Lorentzian function and the continuous Dirichlet kernel). A comparison of synthetic and real seismograms shows that the Lorentzian function is better to describe small local events, and the continuous Dirichlet kernel is more appropriate for earthquakes. This approach can be generalized to various problems of different nature.

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

The problem of computing the Sparse Fast Fourier Transform(sFFT) of a $K$ -sparse signal of size $N$ has received significant attention for a long time. The first stage of sFFT is hashing the frequency coefficients into $B(\approx {K})$ buckets named frequency bucketization. The process of frequency bucketization is achieved through the use of filters: Dirichlet kernel filter, aliasing filter, flat filter, etc. The frequency bucketization through these filters can decrease runtime and sampling complexity in low dimensions. It is a hot topic about sFFT algorithms using the flat filter because of its convenience and efficiency since its emergence and wide application. The next stage of sFFT is the spectrum reconstruction by identifying frequencies that are isolated in their buckets. Up to now, there are more than thirty different sFFT algorithms using the sFFT idea as mentioned above by their unique methods. An important question now is how to analyze and evaluate the performance of these sFFT algorithms in theory and practice. In this paper, it is mainly discussed about sFFT algorithms using the flat filter. In the first part, the paper introduces the techniques in detail, including two types of frameworks, five different methods to reconstruct spectrum and corresponding algorithms. We get the conclusion of the performance of these five algorithms, including runtime complexity, sampling complexity and robustness in theory. In the second part, we make three categories of experiments for computing the signals of different SNR, different $N$ , and different $K$ by a standard testing platform and record the run time, percentage of the signal sampled, and $L_{0},L_{1},L_{2}$ error both in the exactly sparse case and general sparse case. The result of experiments is consistent with the inferences obtained in theory. It can help us to optimize these algorithms and use them correctly in the right areas.

A mollification regularization method with the Dirichlet kernel for two Cauchy problems of three-dimensional Helmholtz equation

In this paper, two Cauchy problems of Helmholtz equation in a three-dimensional case are considered. To address these problems, a mollification method with bivariate Dirichlet kernel is proposed. Stable errors estimates are obtained based on appropriate a priori choices of mollification parameters. Convergence estimates show that the regularization solution depends continuously on the data and wavenumber. Numerical examples of our interest show that Dirichlet kernel is more effective than the Gaussian kernel under the same parameter selection rule, and our procedure is stable with respect to perturbations noise in the data.

A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation

In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

Comparison of V- and S-Dini Tests. Counterexamples for Symmetric Dini Tests with Respect to Generalized Haar and Walsh Systems

In this paper, we prove that in symmetric Dini tests with respect to Price and generalized Haar systems both conditions are essential; no one of them alone guarantees the convergence of the Fourier series independently of the majorant of Dirichlet kernels. We give the corresponding counterexamples.