De-noising by thresholding operator adapted wavelets

Yoo, Gene Ryan; Owhadi, Houman

doi:10.1007/s11222-019-09893-x

Cited by 8 publications

(5 citation statements)

References 27 publications

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“…G implies that K(q, q) is invertible if q i are pairwise distinct and α = 0, i.e., K is non-degenerate. (ii) in Proposition 1 follows from the identity (21) and…”

Section: Gphmentioning

confidence: 97%

“…Two main approaches are available for solving PDEs as learning problems: (i) artificial neural network (ANN)-based approaches, with physics-informed neural networks [14][15] as a prototypical example and (ii) GP-based approaches, with Gamblets [16][17][18] as a prototypical example. Although GP-based approaches are more theoretically well-founded [9] and have a long history of interplay with numerical approximation [8,[19][20][21] , they were essentially limited to linear/quasi-linear/time-dependent PDEs and have only recently been generalized to arbitrary nonlinear PDEs [22] (and computational graphs [23] ).…”

Section: Solving Partial Differential Equations (Pdes) As Learning Pr...mentioning

confidence: 99%

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Gaussian process hydrodynamics

Owhadi

2023

Appl. Math. Mech.-Engl. Ed.

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We present a Gaussian process (GP) approach, called Gaussian process hydrodynamics (GPH) for approximating the solution to the Euler and Navier-Stokes (NS) equations. Similar to smoothed particle hydrodynamics (SPH), GPH is a Lagrangian particle-based approach that involves the tracking of a finite number of particles transported by a flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior ξ on the velocity field and conditioning it on the vorticity at the particle locations. Known physics (e.g., the Richardson cascade and velocity increment power laws) is incorporated into the GP prior by using physics-informed additive kernels. This is equivalent to expressing ξ as a sum of independent GPs ξl, which we call modes, acting at different scales (each mode ξl self-activates to represent the formation of eddies at the corresponding scales). This approach enables a quantitative analysis of the Richardson cascade through the analysis of the activation of these modes, and enables us to analyze coarse-grain turbulence statistically rather than deterministically. Because GPH is formulated by using the vorticity equations, it does not require solving a pressure equation. By enforcing incompressibility and fluid-structure boundary conditions through the selection of a kernel, GPH requires significantly fewer particles than SPH. Because GPH has a natural probabilistic interpretation, the numerical results come with uncertainty estimates, enabling their incorporation into an uncertainty quantification (UQ) pipeline and adding/removing particles (quanta of information) in an adapted manner. The proposed approach is suitable for analysis because it inherits the complexity of state-of-the-art solvers for dense kernel matrices and results in a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on the accuracy and stability. Because the proposed approach uses a Bayesian interpretation, it naturally enables data assimilation and predictions and estimations by mixing simulation data and experimental data.

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“…G implies that K(q, q) is invertible if q i are pairwise distinct and α = 0, i.e., K is non-degenerate. (ii) in Proposition 1 follows from the identity (21) and…”

Section: Gphmentioning

confidence: 97%

Section: Solving Partial Differential Equations (Pdes) As Learning Pr...mentioning

confidence: 99%

Gaussian process hydrodynamics

Owhadi

2023

Appl. Math. Mech.-Engl. Ed.

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“…The coefficients of the three vectors decomposed by wavelet in Formula (4) correspond to the vectors w * j,k , w j,k , q, respectively, and the corresponding mathematical model is [29]:…”

Section: Hierarchical Adaptive Thresholdmentioning

confidence: 99%

Multi-Leakage Source Localization of Safety Valve Based on Uniform Circular AE Array and Improved MUSIC Algorithm

Hou

Liu

2023

Sensors

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The safety valve is the core component of the pressure-relief protection device for pressure-bearing special equipment. When the safety valve leaks, the medium of the pressure vessel will be lost and wasted, which may cause safety accidents. With the aim to solve the problem of accurately locating the multiple leakage sources of safety valves, a localization method combining a uniform circular array acoustic emission detection and an improved multiple signal classification (MUSIC) algorithm is proposed. First, an improved wavelet threshold function denoising method is introduced to extract acoustic emission signals with high SNR, thereby reducing the rank of the covariance matrix, weakening the noise dispersion caused by eigenvalue reconstruction, avoiding signal and noise cross-confusion, and improving positioning accuracy. By introducing a windowed fast Fourier transform (FFT) frequency division processing link to obtain narrowband signal, the premise of using MUSIC positioning algorithm is established. In addition, a forward/backward spatial smoothing algorithm is introduced in the decoherence link to reduce co-channel interference, reduce the rank loss of the signal covariance matrix, and improve the positioning accuracy of the algorithm. The results show that when the working pressure is 0.70 MPa, 0.75 MPa, and 0.80 MPa, the deviation between the azimuth angle and elevation angle positioning results of each leakage source obtained by the improved MUSIC algorithm and the actual angle does not exceed 2°, and the relative error does not exceed 3.5%. Therefore, the improved MUSIC algorithm can accurately locate multiple leakage sources of the safety valve, and as the working pressure of the safety valve increases, the positioning accuracy of the improved MUSIC algorithm also increases accordingly.

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“…For PDEs, resent research includes [Chkrebtii et al, 2016, Cockayne et al, 2016, 2017, Owhadi, 2015, with these contributions making substantial use of reproducing kernel Hilbert space (RKHS) structure and Gaussian processes. Unsurprisingly, given the deep connections between linear algebra and numerical methods for PDEs, the probabilistically-motivated theory of gamblets for PDEs [Owhadi, 2017, Owhadi and Scovel, 2017a, Owhadi and Zhang, 2017 has gone hand-in-hand with the development of fast solvers for structured matrix inversion and approximation problems [Schäfer et al, 2017]; see also [Yoo and Owhadi, 2019].…”

Section: Probabilistic Numericalmentioning

confidence: 99%

A modern retrospective on probabilistic numerics

Oates

Sullivan

2019

Stat Comput

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This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.

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De-noising by thresholding operator adapted wavelets

Cited by 8 publications

References 27 publications

Gaussian process hydrodynamics

Gaussian process hydrodynamics

Multi-Leakage Source Localization of Safety Valve Based on Uniform Circular AE Array and Improved MUSIC Algorithm

A modern retrospective on probabilistic numerics

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