scholarly journals Extreme Learning Machines as Encoders for Sparse Reconstruction

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 88 ◽  
Author(s):  
S Al Mamun ◽  
Chen Lu ◽  
Balaji Jayaraman
Keyword(s):  
Fluid Flows ◽  
Data Driven ◽  
Sparse Reconstruction ◽  
Learning Machines ◽  
Physical Mechanisms ◽  
The Face ◽  
Ill Posed ◽  
Learning Machine ◽  
Proper Orthogonal ◽  
Well Posed

Reconstruction of fine-scale information from sparse data is often needed in practical fluid dynamics where the sensors are typically sparse and yet, one may need to learn the underlying flow structures or inform predictions through assimilation into data-driven models. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches encode the physics into an underlying sparse basis space that spans the manifold to generate well-posedness. To achieve this, one commonly uses a generic orthogonal Fourier basis or a data specific proper orthogonal decomposition (POD) basis to reconstruct from sparse sensor information at chosen locations. Such a reconstruction problem is well-posed as long as the sensor locations are incoherent and can sample the key physical mechanisms. The resulting inverse problem is easily solved using l 2 minimization or if necessary, sparsity promoting l 1 minimization. Given the proliferation of machine learning and the need for robust reconstruction frameworks in the face of dynamically evolving flows, we explore in this study the suitability of non-orthogonal basis obtained from extreme learning machine (ELM) auto-encoders for sparse reconstruction. In particular, we assess the interplay between sensor quantity and sensor placement in a given system dimension for accurate reconstruction of canonical fluid flows in comparison to POD-based reconstruction.

scholarly journals Extreme Learning Machines as Encoders for Sparse Reconstruction

2018 ◽  
Author(s):  
Abdullah Al-Mamun ◽  
Chen Lu ◽  
Balaji Jayaraman
Keyword(s):  
Fluid Flows ◽  
Data Driven ◽  
Sparse Reconstruction ◽  
Learning Machines ◽  
Physical Mechanisms ◽  
The Face ◽  
Ill Posed ◽  
Learning Machine ◽  
Proper Orthogonal ◽  
Well Posed

Reconstruction of fine-scale information from sparse data is often needed in practical fluid dynamics where the sensors are typically sparse and yet, one may need to learn the underlying flow structures or inform predictions through assimilation into data-driven models. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches encode the physics into an underlying sparse basis space that spans the manifold to generate well-posedness. To achieve this, one commonly uses generic orthogonal Fourier basis or data specific proper orthogonal decomposition (POD) basis to reconstruct from sparse sensor information at chosen locations. Such a reconstruction problem is well-posed as long as the sensor locations are incoherent and can sample the key physical mechanisms. The resulting inverse problem is easily solved using $l_2$ minimization or if necessary, sparsity promoting $l_1$ minimization. Given the proliferation of machine learning and the need for robust reconstruction frameworks in the face of dynamically evolving flows, we explore in this study the suitability of non-orthogonal basis obtained from Extreme Learning Machine (ELM) auto-encoders for sparse reconstruction. In particular, we assess the interplay between sensor quantity and sensor placement for a given system dimension for accurate reconstruction of canonical fluid flows in comparison to POD-based reconstruction.

scholarly journals Interplay of Sensor Quantity, Placement and System Dimension in POD-based Sparse Reconstruction of Fluid Flows

2019 ◽  
Author(s):  
Balaji Jayaraman ◽  
S. M. Abdullah Al-Mamun ◽  
Chen Lu
Keyword(s):  
Fluid Flows ◽  
Sparse Recovery ◽  
Training Data ◽  
Sparse Reconstruction ◽  
Control Parameters ◽  
Underlying Basis ◽  
Ill Posed ◽  
Basis System ◽  
Using Data ◽  
Value Decomposition

Sparse recovery of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale PDE dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying basis space spanning the manifold in which the system resides. In this study, we employ an approach that learns basis from singular value decomposition (SVD) of training data to reconstruct sparsely sensed information. This results in a set of four control parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of control parameters implicitly determines the choice of algorithm as either $l_2$ minimization reconstruction or sparsity promoting $l_1$ norm minimization reconstruction. In this work, we systematically explore the implications of these control parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement provides the best balance of computational complexity and robust reconstruction for marginally oversampled cases which happens to be the most challenging regime in the explored parameter design space.

On a characteristic property of conditionally well-posed problems

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Inverse Problems ◽  
Input Data ◽  
Characteristic Property ◽  
Optimization Problems ◽  
Data Driven ◽  
Error Models ◽  
Regularizing Algorithms ◽  
Stochastic Error ◽  
Ill Posed ◽  
Well Posed

AbstractThe aim of this paper is to discuss and illustrate the fact that conditionally well-posed problems stand out among all ill-posed problems as being regularizable via an operator independent of the level of errors in input data. We give examples of corresponding purely data driven regularizing algorithms for various classes of conditionally well-posed inverse problems and optimization problems in the context of deterministic and stochastic error models.

scholarly journals Cholesky Factorization Based Online Sequential Extreme Learning Machines with Persistent Regularization and Forgetting Factor

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 801
Author(s):  
Xinran Zhou ◽  
Xiaoyan Kui
Keyword(s):  
Extreme Learning Machine ◽  
Cholesky Factorization ◽  
Extreme Learning Machines ◽  
Forgetting Factor ◽  
Learning Machines ◽  
Chunk Size ◽  
Ill Posed ◽  
Regression Problems ◽  
Simulation Results ◽  
Learning Machine

The online sequential extreme learning machine with persistent regularization and forgetting factor (OSELM-PRFF) can avoid potential singularities or ill-posed problems of online sequential regularized extreme learning machines with forgetting factors (FR-OSELM), and is particularly suitable for modelling in non-stationary environments. However, existing algorithms for OSELM-PRFF are time-consuming or unstable in certain paradigms or parameters setups. This paper presents a novel algorithm for OSELM-PRFF, named “Cholesky factorization based” OSELM-PRFF (CF-OSELM-PRFF), which recurrently constructs an equation for extreme learning machine and efficiently solves the equation via Cholesky factorization during every cycle. CF-OSELM-PRFF deals with timeliness of samples by forgetting factor, and the regularization term in its cost function works persistently. CF-OSELM-PRFF can learn data one-by-one or chunk-by-chunk with a fixed or varying chunk size. Detailed performance comparisons between CF-OSELM-PRFF and relevant approaches are carried out on several regression problems. The numerical simulation results show that CF-OSELM-PRFF demonstrates higher computational efficiency than its counterparts, and can yield stable predictions.

scholarly journals On Data-Driven Sparse Sensing and Linear Estimation of Fluid Flows

Sensors ◽  
2020 ◽  
Vol 20 (13) ◽  
pp. 3752
Author(s):  
Balaji Jayaraman ◽  
S M Abdullah Al Mamun
Keyword(s):  
Fluid Flows ◽  
Sparse Data ◽  
Data Driven ◽  
Estimation Methods ◽  
Natural Environments ◽  
Fine Scale ◽  
Linear Estimation ◽  
Performance Improvements ◽  
Learning Machine ◽  
Low Dimensional

The reconstruction of fine-scale information from sparse data measured at irregular locations is often needed in many diverse applications, including numerous instances of practical fluid dynamics observed in natural environments. This need is driven by tasks such as data assimilation or the recovery of fine-scale knowledge including models from limited data. Sparse reconstruction is inherently badly represented when formulated as a linear estimation problem. Therefore, the most successful linear estimation approaches are better represented by recovering the full state on an encoded low-dimensional basis that effectively spans the data. Commonly used low-dimensional spaces include those characterized by orthogonal Fourier and data-driven proper orthogonal decomposition (POD) modes. This article deals with the use of linear estimation methods when one encounters a non-orthogonal basis. As a representative thought example, we focus on linear estimation using a basis from shallow extreme learning machine (ELM) autoencoder networks that are easy to learn but non-orthogonal and which certainly do not parsimoniously represent the data, thus requiring numerous sensors for effective reconstruction. In this paper, we present an efficient and robust framework for sparse data-driven sensor placement and the consequent recovery of the higher-resolution field of basis vectors. The performance improvements are illustrated through examples of fluid flows with varying complexity and benchmarked against well-known POD-based sparse recovery methods.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

Damage Detection in Flexible Propeller Beam Structures by Exploiting Impact-Induced Coupled Acceleration Signals

2011 ◽  
Author(s):  
Ioannis T. Georgiou
Keyword(s):  
Data Driven ◽  
Distributed Information ◽  
Modal Expansion ◽  
Shape Difference ◽  
Spatio Temporal ◽  
Temporal Sample ◽  
High Level ◽  
Proper Orthogonal ◽  
Acceleration Signals

A local damage at the tip of a composite propeller is diagnosed by properly comparing its impact-induced free coupled dynamics to that of a pristine wooden propeller of the same size and shape. This is accomplished by creating indirectly via collocated measurements distributed information for the coupled acceleration field of the propellers. The powerful data-driven modal expansion analysis delivered by the Proper Orthogonal Decomposition (POD) Transform reveals that ensembles of impact-induced collocated coupled experimental acceleration signals are underlined by a high level of spatio-temporal coherence. Thus they furnish a valuable spatio-temporal sample of coupled response induced by a point impulse. In view of this fact, a tri-axial sensor was placed on the propeller hub to collect collocated coupled acceleration signals induced via modal hammer nondestructive impacts and thus obtained a reduced order characterization of the coupled free dynamics. This experimental data-driven analysis reveals that the in-plane unit components of the POD modes for both propellers have similar shapes-nearly identical. For the damaged propeller this POD shape-difference is quite pronounced. The shapes of the POD modes are used to compute indices of difference reflecting directly damage. At the first POD energy level, the shape-difference indices of the damaged composite propeller are quite larger than those of the pristine wooden propeller.

scholarly journals A Gaussian Process Regression approach within a data-driven POD framework for engineering problems in fluid dynamics

2021 ◽  
Vol 4 (3) ◽  
pp. 1-16
Author(s):  
Giulio Ortali ◽  
◽  
Nicola Demo ◽  
Gianluigi Rozza ◽  
Keyword(s):  
Gaussian Process ◽  
Stokes Problem ◽  
Gaussian Process Regression ◽  
Engineering Problem ◽  
Data Driven ◽  
Regression Approach ◽  
Data Driven Approach ◽  
Naval Engineering ◽  
Proper Orthogonal ◽  
Industrial Problem

<abstract><p>This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the Stokes problem, and in the following to a real-world industrial problem, within a shape optimization pipeline for a naval engineering problem.</p></abstract>

Sign in / Sign up

Export Citation Format

Share Document