Fourier Operator Research Articles

A visual digital twin framework based on residual-based fourier neural operator online simulation method

ABSTRACT To enjoy a wonderful cooking experience in a smart kitchen, users and designers need a visual interaction platform. This paper proposes a DT framework incorporating data processing, flow field online simulation, equipment monitoring, interaction and visualization. Specifically, the DT online simulation and visualization of the kitchen fume flow field serve as the foundation for appliance design and control. Additionally, users can gain deeper insight into the state and change trends of the kitchen. To address the online simulation of the flow field, a RFNO online simulation method is proposed. In addition, this paper proposes an Echarts-based 2D and UE5-based 3D flow field visualization method to enable dynamic visualization and interaction of the flow field. The proposed DT framework was successfully verified in the case of the smart kitchen, demonstrating its efficiency and effectiveness.

Optimal sensor placement for leak location in water distribution networks based on graph learning algorithm

Abstract The effective monitoring of urban water distribution networks (WDNs) relies heavily on pressure sensor placement. Nevertheless, a WDN may consist of hundreds of nodes, and it is not economically feasible to install sensors at each node. Therefore, how to identify an optimal location for sensor placement becomes a crucial issue. We use graph signal processing to analyze the pressure signals and introduce a framework for optimal sensor placement specifically designed for pressure signals. To address the limitation that pressure signals cannot be sampled directly, we propose a method to convert the signal into a band-limited signal that meets the requirements. Central to the method is learning a graph Fourier operator, and the effectiveness of the proposed method is proved theoretically. The graph Fourier operator enables the pressure data to become a smooth graph signal with variations in its topology. In addition, we design a graph filter based on the energy of the signal and obtain a band-limited signal that meets the requirements. To ensure the selection of representative nodes, we use a noise-robust graph sampling method to obtain the sensor node. Our method is further evaluated using the pressure data from Anytown versus Net3, showing strong performance in leak identification and signal reconstruction capabilities.

Machine learning-based reduced-order reconstruction method for flow fields

The real-time prediction of flow fields has scientific and engineering significance, although it is currently challenging. To address this issue, we propose a nonintrusive supervised reduced-order machine learning framework for flow-field reconstruction, referred to as ROR, to achieve real-time flow-field prediction. The model predicts a signed distance function of the domain and uses a typical flow field as feature extraction objects. Utilizing a cross-fit method, it efficiently combines these features, enabling rapid prediction of the full-order flow field. During the model validation phase, we assess the performance of our model by reconstructing steady-state two-dimensional indoor flows in different room layouts. The results indicate that our model accurately predicts the flow field in the target indoor layout within a short timeframe (approximately 5 s) and demonstrates robustness. To delve deeper into the model performance, we discuss the specific parameters of the model framework and test the effectiveness of the flow-field reconstruction under different air supply modes, with the results showing a mean squared error (MSE) of less than 1.5 %. Additionally, we compare our model with the fourier neural operator (FNO) model and find that it exhibited superior performance with the same number of training steps. The outcomes of this study not only bear significant theoretical implications for the field of flow-field prediction but also provide robust support for practical engineering applications.

Certain Fourier operators and their associated Poisson summation formulae on GL1

Certain Fourier operators and their associated Poisson summation formulae on GL1

Approximation of the Lebesgue constant of the Fourier operator by a logarithmic-fractional-rational function

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (non-monotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude.

Approximation of the Lebesgue Constant of the Fourier Operator by a Logarithmic-Fractional-Rational Function

Approximation of the Lebesgue Constant of the Fourier Operator by a Logarithmic-Fractional-Rational Function

Fourier analysis of membrane locking and unlocking

Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler–Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.

Deep learning‐assisted model‐based off‐resonance correction for non‐Cartesian SWI

PurposePatient‐induced inhomogeneities in the static magnetic field cause distortions and blurring (off‐resonance artifacts) during acquisitions with long readouts such as in SWI. Conventional versatile correction methods based on extended Fourier models are too slow for clinical practice in computationally demanding cases such as 3D high‐resolution non‐Cartesian multi‐coil acquisitions.TheoryMost reconstruction methods can be accelerated when performing off‐resonance correction by reducing the number of iterations, compressed coils, and correction components. Recent state‐of‐the‐art unrolled deep learning architectures could help but are generally not adapted to corrupted measurements as they rely on the standard Fourier operator in the data consistency term. The combination of correction models and neural networks is therefore necessary to reduce reconstruction times.MethodsHybrid pipelines using UNets were trained stack‐by‐stack over 99 SWI 3D SPARKLING 20‐fold accelerated acquisitions at 0.6 mm isotropic resolution using different off‐resonance correction methods. Target images were obtained using slow model‐based corrections based on self‐estimated field maps. The proposed strategies, tested over 11 volumes, are compared to model‐only and network‐only pipelines.ResultsThe proposed hybrid pipelines achieved scores competing with two to three times slower baseline methods, and neural networks were observed to contribute both as pre‐conditioner and through inter‐iteration memory by allowing more degrees of freedom over the model design.ConclusionA combination of model‐based and network‐based off‐resonance correction was proposed to significantly accelerate conventional methods. Different promising synergies were observed between acceleration factors (iterations, coils, correction) and model/network that could be expanded in the future.

Mapping properties of pseudodifferential and Fourier operators

The composition of the Fourier transform in \mathbb{R}^n with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on mapping properties of the Fourier transform as developed in a preceding paper and related assertions for pseudodifferential operators.

An approach to the Gaussian RBF kernels via Fock spaces

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.

An O(1) algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.

Upscaling acoustic wave equation using renormalization group theory

Seismic waves interact with a broad range of heterogeneities as they propagate through the earth. Simulating this full range of scales for wave propagation requires capturing heterogeneities of all scales, which can be computationally unaffordable. In such cases, we have relied on macroscopic representations of media obtained through an upscaling process that preserves the effects of small-scale heterogeneities (in comparison with the wavelengths of interest). Here, we discuss the application of the renormalization group (RG) theory-based upscaling to the 2D acoustic wave equation. RG-based upscaling requires constructing a special Fourier operator and is implemented using a domain decomposition in conjunction with an “expansion and truncation” method to mitigate edge effects. We test this upscaling method on several benchmark models and in the context of reverse time migration. The upscaled models obtained using this method indicate a good consistency for generated waveforms, whereas the runtime for simulations is reduced by at least an order of magnitude.

Approximation of the Lebesgue Constant of the Fourier Operator by a Logarithmic Function

Approximation of the Lebesgue Constant of the Fourier Operator by a Logarithmic Function

Extreme Problems Related to the Approximation of the Lebesgue Constant of a Fourier Operator by a Logarithmic Function

The Lebesgue constant corresponding to the classical Fourier operator is approximated by a logarithmic function depending on two parameters. The difference between the Lebesgue constant and this function is studied, various extreme problems are considered, algorithms of successive reduction of values of the obtained best uniform approximations are given.

ESTIMATES AND RIGIDITY FOR STABLE SOLUTIONS TO SOME NONLINEAR ELLIPTIC PROBLEMS

My thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabre, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincare sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.

Multiscale frequency-domain seismic inversion for fracture weakness

Amplitude variation with offset and azimuth (AVOAz) inversion is an effective tool to invert the azimuthal seismic reflection data for fracture weaknesses, which play an important role in seismic characterization of fractures. Conventional AVOAz inversion is implemented in the time domain, which simultaneously uses full-frequency components of the seismic data causing an unstable and inaccurate inversion result in the case of noisy seismic data. Here we propose a novel multiscale frequency-domain seismic inversion approach to improve resolution and accuracy of fracture weakness estimates. Based on new derived frequency-domain expression of azimuthal seismic amplitude difference for a horizontal transversely isotropic (HTI) medium, we build the kernel function incorporated with the Fourier operator, which avoids the Fourier inverse transform in the frequency-domain inversion. Low frequency information in seismic data is often missing or contaminated with noises, thus we establish an inversion objective function combining low frequency regularization term and Cauchy sparse regularization term in the Bayesian framework. We adopt the multiscale strategy and perform inversion successively from low to high frequency group utilizing a limited number of frequency components with a high signal-to-noise (S/N) ratio. In addition, we compare the multiscale frequency-domain inversion approach, the frequency-domain simultaneous inversion approach, and the time-domain inversion approach using synthetic data and field data. Synthetic example demonstrates that the cross-correlation coefficients (CCs) between the true fracture weaknesses and the inversion results obtained by multiscale frequency-domain inversion approach are above 0.83 even with a S/N of 2. Compared to the time-domain inversion approach and the frequency-domain simultaneous inversion approach, the inversion results with multiscale frequency-domain inversion approach have higher resolution and accuracy. Field data example further validates the feasibility and superiority of the proposed multiscale frequency-domain inversion approach.

On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

The Norm of the Fourier Transform on Compact or Discrete Abelian Groups

We calculate the norm of the Fourier operator from $L^p(X)$ to $L^q(\hat{X})$ when $X$ is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff-Young inequality on such groups. In particular, we identify the region in $(p,q)$-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case of finite abelian groups, where the norm was determined by Gilbert and Rzeszotnik. As an application, uncertainty principles on such groups expressed in terms of R\'enyi entropies are discussed.

Пространства Соболева и краевые задачи для операторов ротор и градиент дивергенции

В ограниченной области $G\subset \mathbb{R}^3$ с гладкой границей изучаются краевые и спектральные задачи для операторов $\operatorname{rot} +\lambda I$ и $\nabla \operatorname{div} +\lambda I$ в пространствах Соболева. При $\lambda\neq 0$ операторы расширяются (методом Б. Вайнберга и В. Грушина) до эллиптических матриц, а краевые задачи удовлетворяют условиям эллиптичности В. Солонникова. Из теории и оценок вытекают полезные свойства решений спектральных задач. Операторы $\nabla \operatorname{div}$ и $ \operatorname{rot}$ имеют самосопряженные расширения $\mathcal{N}_d$ и $\mathcal{S}$ в ортогональные подпространства $\mathcal{A}_{\gamma }$ и $V^0$ потенциальных и вихревых полей в $\mathbf{L}_{2}(G)$, а их собственные векторы задают ортогональные базисы в $\mathcal{A}_{\gamma }$ и $V^0$, элементы которых представляются рядами Фурье, а операторы - преобразованиями рядов. Определены аналоги пространств Соболева $\mathbf{A}^{2k}_{\gamma }$ и $\mathbf{W}^m$ порядков $2k$ и $m$ в классах потенциальных и вихревых полей и классы $ C(2k,m)$ их прямых сумм. Доказано, что при $\lambda\neq \operatorname{Sp}(\operatorname{rot})$ оператор $ \operatorname{rot}+\lambda I$ отображает класс $C(2k,m+1)$ на класс $C(2k,m)$ взаимно однозначно и непрерывно, а при $\lambda\neq \operatorname{Sp}(\nabla \operatorname{div})$ оператор $\nabla \operatorname{div}+\lambda I$ отображает $C(2(k+1), m)$ на $C(2k,m)$ соответственно.

Full Optical Fiber Link Characterization With the BSS-Lasso

Manipulation of the detected backscattered Rayleigh signal inside the bandwidth of a frequency-swept optical sub-carrier propagating into an optical fiber permits an efficient localization of faults through a Fourier operator. When the bandwidth is restricted, analysis in the frequency domain can overcome the spatial resolution limitation while also inducing a high-dimensional problem. Introducing the Lasso as a signal processing technique paired with the Baseband Subcarrier Sweep (BSS) framework allows for a methodology to consistently evaluate fiber defects. In this work, a novel technique for optical fiber monitoring within the BSS framework, hereinafter called the BSS-Lasso, is proposed and tested in simulated and real-world environments, taking into account both reflective and non-reflective events. The results show that, for fiber links ranging from 2 to 15 km with up to 3 faults, over 80% of faults are detected within a 50 m range, and indicate that the proposed methodology significantly outperforms current state-of-the-art BSS-based supervision techniques. Finally, the BSS-Lasso allows for precise, low-cost, transmitter-embedded full characterization of optical fiber links.