Empirical Eigenfunctions Research Articles

Three-Dimensional Analysis of Vocal Fold Oscillations: Correlating Superior and Medial Surface Dynamics Using Ex Vivo Human Hemilarynges.

The primary acoustic signal of the voice is generated by the complex oscillation of the vocal folds (VFs), whereby physicians can barely examine the medial VF surface due to its anatomical inaccessibility. In this study, we investigated possibilities to infer medial surface dynamics by analyzing correlations in the oscillatory behavior of the superior and medial VF surfaces of four human hemilarynges, each in 24 different combinations of flow rate, VF adduction, and elongation. The two surfaces were recorded synchronously during sustained phonation using two high-speed camera setups and were subsequently 3D-reconstructed. The 3D surface parameters of mean and maximum velocities and displacements and general phonation parameters were calculated. The VF oscillations were also analyzed using empirical eigenfunctions (EEFs) and mucosal wave propagation, calculated from medial surface trajectories. Strong linear correlations were found between the 3D parameters of the superior and medial VF surfaces, ranging from 0.8 to 0.95. The linear regressions showed similar values for the maximum velocities at all hemilarynges (0.69-0.9), indicating the most promising parameter for predicting the medial surface. Since excessive VF velocities are suspected to cause phono-trauma and VF polyps, this parameter could provide added value to laryngeal diagnostics in the future.

Reconstruction of Vocal Fold Medial Surface 3D Trajectories: Effects of Neuromuscular Stimulation and Airflow.

Analysis of medial surface dynamics of the vocal folds (VF) is critical to understanding voice production and treatment of voice disorders. We analyzed VF medial surface vibratory dynamics, evaluating the effects of airflow and nerve stimulation using 3D reconstruction and empirical eigenfunctions (EEF). In vivo canine hemilarynx phonation. An in vivo canine hemilarynx was phonated while graded stimulation of the recurrent and superior laryngeal nerves (RLN and SLN) was performed. For each phonatory condition, vibratory cycles were 3D reconstructed from tattooed landmarks on the VF medial surface at low, medium, and high airflows. Parameters describing medial surface trajectory shape were calculated, and underlying patterns were emphasized using EEFs. Fundamental frequency and smoothed cepstral peak prominence (CPPS) were calculated from acoustic data. Convex-hull area of landmark trajectories increased with increasing flow and decreasing nerve activation level. Trajectory shapes observed included circular, ellipsoid, bent, and figure-eight. They were more circular on the superior and anterior VF, and more elliptical and line-like on the inferior and posterior VF. The EEFs capturing synchronal opening and closing (EEF1) and alternating convergent/divergent (EEF2) glottis shapes were mostly unaffected by flow and nerve stimulation levels. CPPS increased with higher airflow except for low RLN activation and very dominant SLN stimulation. We analyzed VF vibration as a function of neuromuscular stimulation and airflow levels. Oscillation patterns such as figure-eight and bent trajectories were linked to high nerve activation and flow. Further studies investigating longer sections of 3D reconstructed oscillations are needed. N/A, Basic Science Laryngoscope, 134:1249-1257, 2024.

Histologic Examination of Vocal Fold Mucosal Wave and Vibration.

Despite gross anatomic and histologic differences between human and canine vocal folds, similar wave patterns have been described yet not fully characterized. We reconstructed vocal fold (VF) vibration in a canine hemilarynx and performed histologic examination of the same vocal fold. We demonstrate comparable wave patterns while exploring the importance of certain anatomic architectures. An in vivo canine hemilarynx was phonated against a glass prism at low and high muscle activation conditions. Vibration was captured using high-speed video, and trajectories of VF medial surface tattooed landmarks were 3D-reconstructed. The method of empirical eigenfunctions was used to capture the essential dynamics of vibratory movement. Histologic examination of the hemilarynx was performed. Oscillation patterns were highly similar between the in vivo canine and previous reports of ex vivo human models. The two most dominant eigenfunctions comprised over 90% of total variance of movement, representing opening/closing and convergent/divergent movement patterns, respectively. We demonstrate a vertical phase difference during the glottal cycle. The time delay between the inferior and superior VF was greater during opening than closing for both activation conditions. Histological examination of canine VF showed not only a thicker lamina propria layer superiorly but also a distinct pattern of thyroarytenoid muscle fibers and fascicles as described in human studies. Histologic and vibratory examination of the canine vocal fold demonstrated human vocal fold vibratory patterns despite certain microstructural differences. This study suggests that the multilayered lamina propria may not be fundamental to vibratory patterns necessary for human-like voice production. NA (Basic science study) Laryngoscope, 134:264-271, 2024.

An updated Gappy-POD to capture non-parameterized geometrical variation in fluid dynamics problems

In this work, we propose a new method to fill the gap within an incomplete turbulent and incompressible data field in such a way to satisfy the topological and intensity changes of the fluid flow after a non-parameterized geometrical variation in the fluid domain. This work extends the one that has been published as a conference proceeding to the 2018 AIAA Scitech Forum and Exposition (Akkari et al. in Geometrical reduced order modeling (ROM) by proper orthogonal decomposition (POD) for the incompressible navier stokes equations. In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1827), 2018). A single baseline large eddy simulation (LES) is assumed to be performed prior geometrical variations. The proposed method is an enhancement of the Gappy-POD method proposed by Everson and Sirovich in 1995, in the case where the given set of empirical eigenfunctions is not sufficient and is not interpolant for the recovering of the modal coefficients for each Gappy snapshot by a least squares procedure. This happens when the available data cannot be written as an interpolation of the baseline POD modes. This is typically the case when we introduce non-parameterized geometrical modifications in the fluid domain. Here, after the baseline simulation, additional solutions of the incompressible Navier–Stokes equations are solely performed over a restricted fluid domain, that contains the geometrical modifications. These local LESs that we will call hybrid simulations are performed by using the immersed boundary technique, which uses of a fluid boundary condition and the baseline velocity field. Then, we propose to update the POD modes using a local modification of the baseline POD modes in the restricted fluid domain. Furthermore, we will propose a physical correction of the latter enhanced Gappy-POD modal coefficients thanks to a Galerkin projection of the Navier–Stokes equations upon the new modes of the available data. This enhancement procedure on the global velocity reconstruction by the physical constraint was tested on a 3D semi-industrial test case of a typical aeronautical injection system and, a 2D laminar and unsteady incompressible test case. The speed-up relative to this new technique is equal to 100, which allows us to perform an exploration of two new designs of the aeronautical injection system.

Recurrent Neural-Network-Based Model Predictive Control of a Plasma Etch Process

In this article, we propose the development of a recurrent neural network (RNN)-based model predictive controller (MPC) for a plasma etch process on a three-dimensional substrate using inductive coupled plasma (ICP) analysis. Specifically, the plasma etch process is simulated by a multiscale model: (1) A macroscopic fluid model is applied to simulate the gas flows and chemical reactions of plasma. (2) A kinetic Monte Carlo (kMC) model is applied to simulate the etching process on the substrate. Subsequently, proper orthogonal decomposition (POD) is used to derive the empirical eigenfunctions of the plasma model. Then the empirical eigenfunctions are utilized as basis functions within a Galerkin’s model reduction framework to compute a low-order system capturing dominant dynamics of the plasma model. Additionally, RNN is introduced to approximate dynamics of both the reduced-order plasma system and the microscopic etch process. The training data for the RNN models are generated from discrete sampling of open-loop simulations. A probability distribution function is also involved to present the stochastic characteristic of the kMC model. The trained RNN models are then implemented as the prediction model in the development of MPC to achieve desired control objectives. Closed-loop simulation results are presented to compare the performance of the model predictive controller and a proportional-integral (PI) controller, which show that the proposed MPC framework is effective and exhibits better performance than does a PI controller.

On the significance of three-dimensional imaging for interpreting vocal fold dynamics

Physical mechanisms of regular and irregular vocal fold vibration were first studied in a finite element model of vocal fold vibration using the method of empirical eigenfunctions. Later the same method was used to successfully study physical mechanisms of vocal fold vibration in laboratory hemilarynx experiments (in which the medial surface of vocal fold was imaged). In subsequent clinical studies, however, the method exhibited significantly less interpretive power, presumably because of the limitations imposed by imaging from a 2D superior view. Our hypothesis is that the interpretive power of the method of empirical eigenfunctions will be significantly enhanced in clinical settings if 3D imaging is used instead of 2D imaging. To test this hypothesis, the method of empirical eigenfunctions was employed on the same finite element model implemented previously, as dynamical information was systematically removed from the analysis (interior tissue dynamics, medial surface dynamics, vertical dynamics of the superior surface, etc.). The natural modes of vibration were re-computed at each stage, and the deformations from the original natural modes were observed. The results suggest that in clinical settings, the method of empirical eigenfunctions retains significant interpretive power regarding physical mechanisms of irregular vibration when 3D imaging is used.

Machine learning-based model predictive control of diffusion-reaction processes

In this work, we develop a machine-learning-based predictive control design for nonlinear parabolic partial differential equation (PDE) systems using process state measurement time-series data. First, the Karhunen-Loève expansion is used to derive dominant spatial empirical eigenfunctions of the nonlinear parabolic PDE system from the data. Then, these empirical eigenfunctions are used as basis functions within a Galerkin's model reduction framework to derive the temporal evolution of a small number of temporal modes capturing the dominant dynamics of the PDE system. Subsequently, feedforward neural networks (FNN) are used to approximate the reduced-order dominant dynamics of the parabolic PDE system from the data within a desired operating region. Lyapunov-based model predictive control (MPC) scheme using FNN models is developed to stabilize the nonlinear parabolic PDE system. Finally, a diffusion-reaction process example is used to demonstrate the effectiveness of the proposed machine-learning-based predictive control method.

A Data-Driven Based Spatiotemporal Model Reduction for Microwave Heating Process with the Mixed Boundary Conditions

In this paper, a data-driven based spatiotemporal model reduction approach is proposed for predicting the temperature distribution and developing the computation speeds in the microwave heating process. Due to the mixed boundary conditions, it is difficult for the traditional spectral method to directly obtain the analytical eigenfunctions. Motivated by the time/space separation theory, we first propose a general framework of spatiotemporal model reduction, which can effectively develop the computation speeds in the numerical analysis of multi-physical fields. Subsequently, the empirical eigenfunctions are generated by applying the Karhunen–Loève theory to decompose the snapshots. Then, the partial differential Equation (PDE) model is discretized into a class of recursive equations and transformed as the reduced-order ordinary differential Equation (ODE) model. Finally, the effectiveness and superiority of the proposed approach is demonstrated by a comparison study with a traditional method on the microwave heating Debye medium.

Spatio-temporal proper orthogonal decomposition of turbulent channel flow

An extension of proper orthogonal decomposition is applied to the wall layer of a turbulent channel flow ($Re_{\unicode[STIX]{x1D70F}}=590$), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the eigenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200–300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12$u_{\unicode[STIX]{x1D70F}}$in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects.

Modified Empirical Eigenfunctions and Its Applications for Model Reduction of Nonlinear Spatiotemporal Systems

Model reduction can greatly reduce complexity and difficulty of control design for spatiotemporal systems (STS) in engineering applications. Empirical eigenfunctions (EEFs) are widely used for the model reduction of spatiotemporal systems, however, truncation of higher modes may describe the behaviours of nonlinear spatiotemporal systems inaccurately. In this paper, modified EEFs are proposed and applied to model reduction of nonlinear spatiotemporal systems. Modified EEFs are obtained via modifying the weights matrix in the method of snapshots, which can be rewritten as linear combinations of initial EEFs. The coefficient matrix for combinations is computed according to the nonlinear temporal dynamics of STSs. Thus, the effects of higher modes are considered into modified EEFs with less computational requirements. The reduced model can give a more accurate description for behaviours of the system. The performance of the proposed method is further proved theoretically, and a numerical example demonstrates the effectiveness of the proposed method.

Empirical eigenfunctions as a function of glottal adduction in excised hemilarynx experiments

For three excised human male hemilarynxes, vocal fleshpoints and empirical eigenfunctions were computed along the vocal fold surface. For two larynges, an increase in adduction resulted in an increase in lateral and vertical oscillation amplitudes and an improved energy transfer from the airflow to the vocal fold tissues. In contrast, the third larynx exhibited a decrease in oscillation amplitudes. By evaluating the empirical eigenfunctions, this decrease in oscillation amplitudes was associated with an unbalanced oscillation pattern with predominantly lateral amplitudes. These results suggest that adduction facilitates the phonatory process by increasing vibrational amplitudes. However, this relationship holds only when a balanced ratio between the vertical and lateral displacements is maintained. Indeed, it appears that a balanced vertical-lateral oscillation pattern may be more beneficial to phonation than strong periodicity with predominantly lateral vibrations.

Temporal clustering for order reduction of nonlinear parabolic PDE systems with time‐dependent spatial domains: Application to a hydraulic fracturing process

A temporally‐local model order‐reduction technique for nonlinear parabolic partial differential equation (PDE) systems with time‐dependent spatial domains is presented. In lieu of approximating the solution of interest using global (with respect to the time domain) empirical eigenfunctions, low‐dimensional models are derived by constructing appropriate temporally‐local eigenfunctions. Within this context, first of all, the time domain is partitioned into multiple clusters (i.e., subdomains) by using the framework known as global optimum search. This approach, a variant of Generalized Benders Decomposition, formulates clustering as a Mixed‐Integer Nonlinear Programming problem and involves the iterative solution of a Linear Programming problem (primal problem) and a Mixed‐Integer Linear Programming problem (master problem). Following the cluster generation, local (with respect to time) eigenfunctions are constructed by applying the proper orthogonal decomposition method to the snapshots contained within each cluster. Then, the Galerkin's projection method is employed to derive low‐dimensional ordinary differential equation (ODE) systems for each cluster. The local ODE systems are subsequently used to compute approximate solutions to the original PDE system. The proposed local model order‐reduction technique is applied to a hydraulic fracturing process described by a nonlinear parabolic PDE system with the time‐dependent spatial domain. It is shown to be more accurate and computationally efficient in approximating the original nonlinear system with fewer eigenfunctions, compared to the model order‐reduction technique with temporally‐global eigenfunctions. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3818–3831, 2017

On the necessity of using 3D imaging of vocal fold vibration

Physical mechanisms of regular and irregular vocal fold vibration were first studied using a computational model of vocal fold vibration. Later physical mechanisms of vocal fold vibration were studied in laboratory hemilarynx studies, where the medial surface of the vocal folds were imaged. While the method was also extended to clinical studies of vocal vibration, the method exhibited significantly less interpretive power in the clinical applications in which the superior surface of the vocal folds was imaged in 2D. Our hypothesis is that the interpretive power of the method of empirical eigenfunctions in studying the superior surface dynamics of the vocal folds will be significantly increased if one performs 3D imaging of the dynamics versus the standard 2D imaging. To test this hypothesis, the method of empirical eigenfunctions was employed on the same finite element model used in previous computational experiments. Our results confirm the hypothesis that the interpretive power of the method of empirical eigenfunctions was significantly increased in studying the superior surface dynamics of the vocal folds when 3D imaging was employed.

Dynamic vocal fold parameters with changing adduction in ex-vivo hemilarynx experiments.

Ex-vivo hemilarynx experiments allow the visualization and quantification of three-dimensional dynamics of the medial vocal fold surface. For three excised human male larynges, the vibrational output, the glottal flow resistance, and the sound pressure during sustained phonation were analyzed as a function of vocal fold adduction for varying subglottal pressure. Empirical eigenfunctions, displacements, and velocities were investigated along the vocal fold surface. For two larynges, an increase of adduction level resulted in an increase of the glottal flow resistance at equal subglottal pressures. This caused an increase of lateral and vertical oscillation amplitudes and velocity indicating an improved energy transfer from the airflow to the vocal folds. In contrast, the third larynx exhibited an amplitude decrease for rising adduction accompanying reduction of the flow resistance. By evaluating the empirical eigenfunctions, this reduced flow resistance was assigned to an unbalanced oscillation pattern with predominantly lateral amplitudes. The results suggest that adduction facilitates the phonatory process by increasing the glottal flow resistance and enhancing the vibrational amplitudes. However, this interrelation only holds for a maintained balanced ratio between vertical and lateral displacements. Indeed, a balanced vertical-lateral oscillation pattern may be more beneficial to phonation than strong periodicity with predominantly lateral vibrations.

Computer simulation of mucosal waves on vibrating human vocal folds

A three-dimensional (3D) finite element (FE) fully parametric model of the human larynx based on computer tomografy measurements was developed and specially adapted for numerical simulation of vocal folds vibrations with collisions.The complex model consists of the vocal folds, arytenoids, thyroid and cricoid cartilages. The vocal fold tissue is modeled as a three layered transversal isotropic material composed of the cover, ligament and muscle and compared with a four layered material where part of the cover was substituted by a liquid layer modelling the superficial layer of lamina propria.First, the basic frequency-modal properties of the model are presented for a given pretension of the vocal folds. The results of numerical simulation of the vocal folds oscillations excited by a prescribed intraglottal aerodynamic pressure are then presented. The results computed in time domain show the 3D motion of the vocal folds in all three directions (horizontal, vertical and anterior-posterior) and the mucosal waves are clearly modeled in the medial cross-section of the vocal folds. The proper orthogonal decomposition (POD) analysis of the excited modes of vibration shows that when taking account of the superficial sub-layer inside the lamina propria with liquid like properties the POD modes are in better agreement with the empirical eigenfunctions (EEF) obtained from measurements performed on excised human larynges.Finally, the usability of the POD analysis for simulation of pathological situations is demonstrated considering a vocal fold nodule located on the upper cranial margin of the right vocal fold.

Optimal eigen expansions and uniform bounds

Let \(\{X_k\}_{k \in \mathbb {Z}} \in {\mathbb {L}}^2({\mathcal {T}})\) be a stationary process with associated lag operators \({\varvec{\mathcal {C}}}_h\). Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal in a certain sense, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, convergence to an extreme value distribution is shown. We also discuss how the asymptotic expansions transfer to the long-run covariance operator \({\varvec{\mathcal {G}}}\) in a general framework.

Data-Driven H∞ Control for Nonlinear Distributed Parameter Systems.

The data-driven H∞ control problem of nonlinear distributed parameter systems is considered in this paper. An off-policy learning method is developed to learn the H∞ control policy from real system data rather than the mathematical model. First, Karhunen-Loève decomposition is used to compute the empirical eigenfunctions, which are then employed to derive a reduced-order model (ROM) of slow subsystem based on the singular perturbation theory. The H∞ control problem is reformulated based on the ROM, which can be transformed to solve the Hamilton-Jacobi-Isaacs (HJI) equation, theoretically. To learn the solution of the HJI equation from real system data, a data-driven off-policy learning approach is proposed based on the simultaneous policy update algorithm and its convergence is proved. For implementation purpose, a neural network (NN)- based action-critic structure is developed, where a critic NN and two action NNs are employed to approximate the value function, control, and disturbance policies, respectively. Subsequently, a least-square NN weight-tuning rule is derived with the method of weighted residuals. Finally, the developed data-driven off-policy learning approach is applied to a nonlinear diffusion-reaction process, and the obtained results demonstrate its effectiveness.

Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming.

Highly dissipative nonlinear partial differential equations (PDEs) are widely employed to describe the system dynamics of industrial spatially distributed processes (SDPs). In this paper, we consider the optimal control problem of the general highly dissipative SDPs, and propose an adaptive optimal control approach based on neuro-dynamic programming (NDP). Initially, Karhunen-Loève decomposition is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. These EEFs together with singular perturbation technique are then used to obtain a finite-dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the PDE system. Subsequently, the optimal control problem is reformulated on the basis of the slow subsystem, which is further converted to solve a Hamilton-Jacobi-Bellman (HJB) equation. HJB equation is a nonlinear PDE that has proven to be impossible to solve analytically. Thus, an adaptive optimal control method is developed via NDP that solves the HJB equation online using neural network (NN) for approximating the value function; and an online NN weight tuning law is proposed without requiring an initial stabilizing control policy. Moreover, by involving the NN estimation error, we prove that the original closed-loop PDE system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded. Finally, the developed method is tested on a nonlinear diffusion-convection-reaction process and applied to a temperature cooling fin of high-speed aerospace vehicle, and the achieved results show its effectiveness.

On empirical eigenfunction-based ranking with [formula omitted] norm regularization

The problem of ranking, in which the goal is to learn a real-valued ranking function that induces a ranking over an instance space, has recently gained increasing attention in machine learning. We study a learning algorithm for ranking generated by a regularized scheme with an ℓ1 regularizer. The algorithm is formulated in a data dependent hypothesis space. Such a space is spanned by empirical eigenfunctions which are constructed by a Mercer kernel and the learning data. We establish the computations of empirical eigenfunctions and the representer theorem for the algorithm. Particularly, we provide an analysis of the sparsity and convergence rates for the algorithm. The results show that our algorithm produces both satisfactory convergence rates and sparse representations under a mild condition, especially without assuming sparsity in terms of any basis.

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015