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.

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

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 109 ◽  
Author(s):  
Balaji Jayaraman ◽  
S M Abdullah Al Mamun ◽  
Chen Lu
Keyword(s):  
Interpolation Method ◽  
Dimensional Space ◽  
Fluid Flows ◽  
Training Data ◽  
Design Parameters ◽  
Sparse Reconstruction ◽  
Discrete Empirical Interpolation Method ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Accurate Reconstruction

Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (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 low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either l 2 minimization reconstruction or sparsity promoting l 1 minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate 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 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 Development of dynamic system response curve method for estimating initial conditions of conceptual hydrological models

2018 ◽  
Vol 20 (6) ◽  
pp. 1387-1400
Author(s):  
Yiqun Sun ◽  
Weimin Bao ◽  
Peng Jiang ◽  
Xuying Wang ◽  
Chengmin He ◽  
...  
Keyword(s):  
Dynamic System ◽  
Response Curve ◽  
Initial Conditions ◽  
Model Performance ◽  
System Response ◽  
Hydrological Models ◽  
Flood Prediction ◽  
Ill Posed ◽  
Value Decomposition

Abstract The dynamic system response curve (DSRC) has its origin in correcting model variables of hydrologic models to improve the accuracy of flood prediction. The DSRC method can lead to unstable performance since the least squares (LS) method, employed by DSRC to estimate the errors, often breaks down for ill-posed problems. A previous study has shown that under certain assumptions the DSRC method can be regarded as a specific form of the numerical solution of the Fredholm equation of the first kind, which is a typical ill-posed problem. This paper introduces the truncated singular value decomposition (TSVD) to propose an improved version of the DSRC method (TSVD-DSRC). The proposed method is extended to correct the initial conditions of a conceptual hydrological model. The usefulness of the proposed method is first demonstrated via a synthetic case study where both the perturbed initial conditions, the true initial conditions, and the corrected initial conditions are precisely known. Then the proposed method is used in two real basins. The results measured by two different criteria clearly demonstrate that correcting the initial conditions of hydrological models has significantly improved the model performance. Similar good results are obtained for the real case study.

scholarly journals Two-Step Root-MUSIC for Direction of Arrival Estimation without EVD/SVD Computation

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Feng-Gang Yan ◽  
Shuai Liu ◽  
Jun Wang ◽  
Ming Jin
Keyword(s):  
Super Resolution ◽  
Direction Of Arrival ◽  
Doa Estimation ◽  
Low Complexity ◽  
Music Algorithm ◽  
Correlation Matrices ◽  
Noise Subspace ◽  
Using Data ◽  
Eigen Decomposition ◽  
Value Decomposition

Most popular techniques for super-resolution direction of arrival (DOA) estimation rely on an eigen-decomposition (EVD) or a singular value decomposition (SVD) computation to determine the signal/noise subspace, which is computationally expensive for real-time applications. A two-step root multiple signal classification (TS-root-MUSIC) algorithm is proposed to avoid the complex EVD/SVD computation using a uniform linear array (ULA) based on a mild assumption that the number of signals is less than half that of sensors. The ULA is divided into two subarrays, and three noise-free cross-correlation matrices are constructed using data collected by the two subarrays. A low-complexity linear operation is derived to obtain a rough noise subspace for a first-step DOA estimate. The performance is further enhanced in the second step by using the first-step result to renew the previous estimated noise subspace with a slightly increased complexity. The new technique can provide close root mean square error (RMSE) performance to root-MUSIC with reduced computational burden, which are verified by numerical simulations.

scholarly journals A modified truncation singular value decomposition method for solving ill-posed problems

2018 ◽  
Vol 13 ◽  
pp. 174830181881360 ◽  
Author(s):  
Zhenyu Zhao ◽  
Riguang Lin ◽  
Zehong Meng ◽  
Guoqiang He ◽  
Lei You ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Decomposition Method ◽  
Singular Value ◽  
Numerical Results ◽  
New Method ◽  
Truncated Singular Value Decomposition ◽  
Singular Value Decomposition Method ◽  
Ill Posed ◽  
Value Decomposition

A modified truncated singular value decomposition method for solving ill-posed problems is presented in this paper, in which the solution has a slightly different form. Both theoretical and numerical results show that the limitations of the classical TSVD method have been overcome by the new method and very few additive computations are needed.

scholarly journals Using machine learning to predict extreme events in complex systems

2019 ◽  
Vol 117 (1) ◽  
pp. 52-59 ◽  
Author(s):  
Di Qi ◽  
Andrew J. Majda
Keyword(s):  
Dynamical Systems ◽  
Learning Strategies ◽  
Extreme Events ◽  
Water Waves ◽  
Extreme Values ◽  
Training Data ◽  
Partition Functions ◽  
Grand Challenge ◽  
Natural Systems ◽  
Using Data

Extreme events and the related anomalous statistics are ubiquitously observed in many natural systems, and the development of efficient methods to understand and accurately predict such representative features remains a grand challenge. Here, we investigate the skill of deep learning strategies in the prediction of extreme events in complex turbulent dynamical systems. Deep neural networks have been successfully applied to many imaging processing problems involving big data, and have recently shown potential for the study of dynamical systems. We propose to use a densely connected mixed-scale network model to capture the extreme events appearing in a truncated Korteweg–de Vries (tKdV) statistical framework, which creates anomalous skewed distributions consistent with recent laboratory experiments for shallow water waves across an abrupt depth change, where a remarkable statistical phase transition is generated by varying the inverse temperature parameter in the corresponding Gibbs invariant measures. The neural network is trained using data without knowing the explicit model dynamics, and the training data are only drawn from the near-Gaussian regime of the tKdV model solutions without the occurrence of large extreme values. A relative entropy loss function, together with empirical partition functions, is proposed for measuring the accuracy of the network output where the dominant structures in the turbulent field are emphasized. The optimized network is shown to gain uniformly high skill in accurately predicting the solutions in a wide variety of statistical regimes, including highly skewed extreme events. The technique is promising to be further applied to other complicated high-dimensional systems.

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