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Acquiring the field information on temperature, pressure, concentration, or velocity is crucial for the monitoring of chemical reactors, multiphase flow systems, heat transfer units, atmospheric pollutants diffusion, and underground pollutant migration. In this paper, a dimensionality reduction matrix completion (DRMC) method is proposed for the field information sensing (FIS) of objects of interest from the scattered point measurement data. An objective functional that casts the FIS task as an optimization problem is proposed. An iteration scheme is developed for solving the proposed objective functional. Numerical simulations are implemented to validate the feasibility and effectiveness of the proposed algorithm. It is found that differing from common inverse problems, numerical simulation approaches, and tomography based field measurement methods, in the proposed method the field information can be reconstructed without the knowledge on governing equations of the measurement objects, initial conditions, boundary conditions, and physical properties of materials, except the limited number of the measurement data. As a result, an alternative insight is introduced for the FIS problems.

Acquiring the field information on temperature, pressure, concentration, or velocity is crucial for the monitoring of chemical reactors, multiphase flow systems, heat transfer units, and the underground pollutant migration and thus facilitates the improvement of the system efficiency, the energy saving, and the reduction of the pollutant emission. With the increasing concerns of the energy and environment problems, especially, the key issue on the monitoring of the large-scale objects has attracted wide attention. In order to satisfy the above requirements, two kinds of approaches, including numerical simulation methods and measurement approaches, are available. Owing to the challenges, such as (1) the understanding of the complicated mechanisms of the underlying process and the acquisition of the reasonable mathematical model are challenging, (2) it is hard to provide reliable governing equations, boundary conditions, physical property parameters, and initial conditions due to the restrictions of real conditions and (3) numerical simulations are time-consuming; applying numerical simulation methods to achieve the monitoring and control of the large-scale dynamic objects may be impractical. Currently, different measurement methods, which can be approximately divided into two categories, such as point measurement methods and field measurement techniques, have been developed for acquiring such information. The point measurement methods can acquire the local measurement information. However, it is hard for point measurement methods to describe the comprehensive situation of the measurement objects. The field measurement methods can acquire the field information of the state variables of interest. However, the costs of the measurement methods are often higher owing to the additional devices. In particular, common tomography-based field measurement methods require the closure of the sensor array, which may be inappropriate for the large-scale measurement objects such as the atmospheric pollutants diffusion, the wind field reconstruction, the large-scale temperature distribution measurement, and the monitoring of the underground pollutant migration. In common tomography-based measurement method, additionally, the sensitivity of the sensing in the measurement domain may gradually decrease with the increase of the scale of the measurement domain, which restricts the improvement of the measurement accuracy. With the development of the wireless sensor networks technologies and the point measurement methods, it is possible to acquire a large number of the scattered point measurement information. In order to address the above challenges, naturally, one of main motivations in this paper is to seek a reliable method to reconstruct the field information from finite observation data.

Mathematically speaking, the FIS problem can be described as follows: for a given field, which can be formulated as a matrix

Differing from common point measurement methods and tomography-based field measurement approaches, in order to resolve the above challenges, this paper presents a RDMC method that integrates the finite measurement information and mathematical methods for the FIS problem, which does not require information on governing equations, initial conditions, boundary conditions, physical properties of materials, and additional and costly measurement devices, but finite point measurement data. In the proposed method, especially, the number of the unknown variables depends only on that of the measurement data, which will facilitate real applications since the number of the measurement data is always small. The main contributions of the paper can be summarized as follows.

(1) Differing from common measurement techniques or numerical simulation methods, a DRMC method is proposed for the FIS problem, which does not require the information on governing equations of the measurement objects, initial conditions, boundary conditions, and physical property parameters of materials, except the limited number of the measurement data.

(2) An objective functional that casts the FIS task as an optimization problem is proposed. An iteration scheme is developed for solving the proposed objective functional.

(3) This paper presents a general framework for the FIS problem from the partial observation data, which may be useful for other related problems.

Compared with common measurement techniques and numerical simulation methods, the significant properties of the proposed DRMC based FIS method can be outlined as follows.

(1) As compared to common inverse problems [

(2) As compared to common numerical simulation methods, such as the finite element method (FEM) and the finite difference method (FDM), the DRMC method acquires the field information without the knowledge on governing equations, initial conditions, boundary conditions, and physical property parameters of materials, except the limited number of the measurement data. It should be pointed out that acquiring the above-mentioned conditions that the implementations of the numerical computations require is not a trivial task owing to the restrictions of the real conditions and the lack of the adequate understandings of the underlying physical or chemical mechanisms of the dynamic behaviors of a dynamic object of interest.

(3) As compared to common tomography-based field measurement methods with the requirements of the additional and costly devices, the DRMC method only requires finite observation data that can be easily acquired by means of common point measurement methods. Common tomography-based measurement methods often require the closure of the sensor array, which may be inappropriate for the large-scale measurement objects such as the atmospheric pollutants diffusion, the wind field reconstruction, the large-scale temperature distribution measurement, and the monitoring of the underground pollutant migration. In common tomography-based measurement method, additionally, the sensitivity of the sensing in the measurement domain may decrease with the increase of the scale of the measurement domain, which restricts the improvement of the measurement accuracy. However, the DRMC method does not require such conditions, which is highly desirable for real applications.

(4) As compared to common point measurement techniques with a motivation of acquiring local measurement information, the DRMC method acquires the comprehensive field information from finite point measurement information.

(5) As compared to common CS method and the MC technique where the unknown variables are directly reconstructed, in the DRMC method the number of the unknown variables depends only on that of the measurement data.

The rest of this paper is organized as follows. In Section

The FIS problem aims at reconstructing the filed information from finite measurement data. Mathematically speaking, the FIS problem can be described as follows: for a given two-dimensional field that can be formulated as a matrix

Studies indicate that under some suitable conditions, one can recover an unknown low rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem [

We define a projection operator

Submitting (

Directly solving (

According to the optimization theory and the Tikhonov regularization method, (

Equation (

Currently, the MC method has been intensively studied and found wide applications in various fields [

In the case of the FIS problem, the number of the known measurement data is always smaller than that of the missing data, and directly reconstructing the missing data may be inappropriate. Naturally, seeking an efficient approach to reduce the number of the unknown variables is crucial. According to the radial basis function (RBF) method, an unknown function can be approximated as [

It can be found from (

In (

Equation (

In (

Following the above discussions, when the linearization dynamic evolution equation is used, the DRMC method can be formulated as

Equation (

The appealing properties of (

(1) Equation (

(2) In (

(3) It is found that the field information is often in low rank owing to the spatial redundancies and temporal correlations of the underlying measurement objects. In (

(4) The inaccurate properties on the measurement equation and the dynamic evolution equation of the measurement objects are emphasized in (

(5) In practice, the measurement objects are often in a dynamic evolution process and the measurement results at different time instants present the temporal correlations. In (

Equation (

It can be found that (

In order to solve (

Similarly, for the sake of easy computation, (

Following the above discussions, finally, an iteration scheme can be designed for solving the DRMC model, which can be summarized as follows.

With the development of the point measurement methods and the wireless sensor networks technology, acquiring a large number of the scattered measurement information is possible. Therefore, applying the scattered measurement information to reconstruct the field information is appropriate. With such considerations in mind, in this section a DRMC based method is proposed for the FIS tasks, which can be outlined in Figure

The DRMC method based FIS flowchart.

It can be found from Figure

It is worth emphasizing that compared with common measurement techniques and numerical simulation methods, the significant properties of the DRMC based FIS method can be summarized as follows.

(1) As compared to common inverse problems, in the DRMC method the measurement information is used to estimate the comprehensive field information without any other additional conditions, including governing equations, boundary conditions, physical property parameters, and initial conditions. It is worth emphasizing that the DRMC method does not solve complicated governing equations.

(2) As compared to common numerical simulation methods, such as the FEM and the FDM, the DRMC method acquires the field information without knowing governing equations, initial conditions, boundary conditions, and physical properties of materials, except the limited number of the observations.

(3) As compared to common tomography-based field measurement methods with the requirements of the additional and costly devices, the DRMC method does not require additional devices, but finite measurement data that can be easily obtained via common measurement methods.

(4) As compared to common point measurement techniques with a motivation of acquiring local measurement information, the DRMC method acquires the comprehensive field information from partial measurement information, which is highly attractive for real applications.

Similarly, compared with the MC method, the DRMC method has the following appealing properties.

(1) The DRMC method and the MC technique belong to the data driven methods, in which the estimation of the missing data does not require the knowledge on governing equations of underlying measurement objects, initial conditions, boundary conditions, and physical properties of materials.

(2) It is worth emphasizing that as compared to common MC method in which the unknown variables are directly reconstructed, the DRMC method reconstructs a low-dimensional coefficient vector rather than original unknown variables by introducing the dimensionality reduction method. It is found that the number of the unknown variables in the MC method is equivalent to that of the missing data. When the number of the unknown variables is large, the computational cost of the MC method is high. On the contrary, in the DRMC method the number of the unknown variables depends merely on that of the measurement data. Since the number of the measurement data is small in the FIS problems, the computational burden of the DRMC algorithm is smaller than the MC method.

(3) The DRMC method integrates the dynamic evolution information of the measurement objects and the measurement information. In the DRMC method, the temporal constraint is introduced to utilize the temporal correlations of the measurement objects, which is different from common MC method.

In previous sections, the DRMC model is proposed and an iteration scheme is developed for solving the model, and then the DRMC method based FIS flowchart is outlined. In this section, numerical simulations are implemented to evaluate the feasibility of the DRMC method. The simulation data is served as the measurement data. In order to simulate a real measurement environment, the simulation data is contaminated by the noises, which is defined as

In all cases, the stopping criterion of iterations for the DRMC method and the MC algorithm is defined as

To evaluate the feasibility of the DRMC algorithm, the results reconstructed by the DRMC method are compared with the MC method. The mean relative error (MRE) is used to evaluate the quality of a solution, which can be defined as

The sampling ratio is also used to evaluate the numerical performances of the DRMC method and the MC method, which is defined as

In this section, the DRMC method is used to reconstruct the temperature distribution described by the following equations from the partial temperature measurement data:

The original temperature distribution is solved by the FEM, which is shown in Figure ^{−7}, and

Original temperature distribution.

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 9 and 19.80%.

Temperature distribution reconstructed by the MC method when the noise variance and the sampling ratio are 9 and 19.80%.

Mean relative errors under different sampling ratios when the noise variance is 9.

Figure

The temperature distribution reconstructed by the MC method when the sampling ratio and the noise variance are 19.80% and 9 is shown in Figure

When the noise variance is 9, the MREs for the DRMC method and the MC algorithm under different sampling ratios are shown in Figure

It can be summarized from the above numerical results that as compared to common numerical simulation methods, such as the FEM and the FDM, the DRMC method acquires the field information without the knowledge on governing equations, initial conditions, boundary conditions, and physical properties of materials, except the limited number of the measurement data. As compared to common tomography-based field measurement methods with the requirements of the additional and costly devices, the DRMC method does not require additional devices but several measurement data that can be easily obtained via common measurement methods. As compared to common point measurement techniques, the DRMC method can reconstruct the field information from partial observation data, which will facilitate the understanding of the underlying physical or chemical mechanisms of the dynamic behaviors of the measurement objects. Meanwhile, the low rank property of the measurement objects is imposed in the DRMC method, which will facilitate the improvement of the reconstruction quality. Additionally, it is worth mentioning that in the MC method and the CS based methods, the number of the unknown variables is equivalent to that of the missing data. When the number of the unknown variables is large, the lack of the known information is serious, and thus the improvement of the reconstruction quality is restricted. In the DRMC method the number of the unknown variables depends on that of the measurement data. Since the number of the measurement data is small in the FIS problems, the computational cost of the DRMC algorithm is smaller than the MC method and the CS based methods.

In real applications, measurement noises are ubiquitous and complicated, a successful algorithm should be able to treat with the inaccurate properties of the measurement data. In this section, the noise-contaminated data with different variances is used to evaluate the robustness of the DRMC method. Figures

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 25 and 0.90%.

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 100 and 0.90%.

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 225 and 0.90%.

Mean relative errors under different noise variances.

The temperature distributions reconstructed by the DRMC method, when the noise variances are 25, 100, and 225 and the sampling ratio is 0.90%, are shown in Figures

In order to further evaluate the feasibility of the DRMC method, another FIS problem is simulated. The original temperature distribution is described by

In this case, the original temperature distribution, which is shown in Figure ^{−7}, and

Original temperature distribution.

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 9 and 19.80%.

Temperature distribution reconstructed by the MC method when the noise variance and the sampling ratio are 9 and 19.80%.

Mean relative errors under different sampling ratios.

Figure

When the noise variance is 9, the MREs of the DRMC method and the MC method under different sampling ratios are shown in Figure

In order to further evaluate the numerical performances of the DRMC method, another FIS problem is simulated. In this section, the original temperature distribution is described by [

Figure ^{−7}, and

Original temperature distribution.

Temperature distribution reconstructed by the DRMC method when the noise variance and the sampling ratio are 9 and 33.06%.

Temperature distribution reconstructed by the MC method when the noise variance and the sampling ratio are 9 and 33.06%.

Mean relative errors under different sampling ratios.

Figures

When the noise variance is 9, the MREs of the DRMC method and the MC method under different sampling ratios are shown in Figure

In real applications, the measurement objects are in a dynamic evolution process. A reliable reconstruction algorithm should be able to deal with such measurement objects. In this section, the DRMC method is used to reconstruct the dynamic temperature distributions described by the following equations from the partial temperature measurement data:

Figure ^{−7}, and

Original temperature distributions. (a) Original temperature distribution at

Temperature distributions reconstructed by the DRMC method when the noise variance and the sampling ratio are 9 and 19.80%. (a) Reconstructed temperature distribution at

Temperature distributions reconstructed by the MC method when the noise variance and the sampling ratio are 9 and 19.80%. (a) Reconstructed temperature distribution at

Mean relative errors under different sampling ratios at

Figures

Mean relative errors under different sampling ratios at

Following the results and discussions presented in previous sections, it can be concluded that the DRMC algorithm is competent in implementing the FIS tasks. Additionally, in the illustrated cases, in order to evaluate the feasibility of the DRMC method, the measurement data are randomly sampled, and numerical results indicate that the final solutions are not sensitive to the locations of the measurement data. This feature is highly desired for real applications. In this paper, merely the temperature distribution cases are shown. It is worth emphasizing that the DRMC algorithm is a general framework for the FIS tasks, which may be useful for other related problems.

Acquiring the field information on temperature, pressure, concentration, or velocity is crucial for the monitoring of chemical reactors, multiphase flow systems, heat transfer units, atmospheric pollutant diffusion, and underground pollutant migration, and thus facilitates the improvement of the system efficiency, the energy saving, and the reduction of the pollutant emission. In this paper, a DRMC method, which incorporates the point measurement method and numerical optimization approach, is proposed for solving the FIS problem. Numerical simulations are implemented to validate the feasibility and effectiveness of the proposed algorithm. It is found that differing from common inverse problems, numerical simulation approaches, and tomography based field measurement methods, in the proposed method the field information is reconstructed without the knowledge on governing equations of the measurement objects, initial conditions, boundary conditions, and physical property parameters of materials, except the limited number of the measurement data. For the cases simulated in this paper, the reconstruction results from the proposed method are in a good agreement with the original distributions. Furthermore, the reconstruction results from the noise-contaminated data show that the proposed algorithm is robust to the inaccurate property on the measurement data. As a result, a promising method is introduced for the FIS problems.

It is worth mentioning that the proposed method provides a general framework for the FIS tasks, which may be useful for other related problems. However, studies indicate that each method may show different numerical performances to different FIS problems. In practice, the selection of an appropriate reconstruction method depends mainly on a specific reconstruction object and the modeling of the related prior information. This work provides an alternative approach for the FIS problem, which needs to be further validated by more cases in the future.

The author declares that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank the National Natural Science Foundation of China (no. 51206048), the Fundamental Research Funds for the Central Universities (no. 13MS11), and the Open Foundation from Key Laboratory of Efficient Utilization of Low and Medium Grade Energy (Tianjin University), Ministry of Education of China (no. 201301-301) for supporting this research.

_{1}-norm-based low-rank matrix approximations for large-scale problems using alternating rectified gradient method