Direct Reconstruction of the Experimental Data in the Case of Ill-Conditioned Problems and in the Presence of Data Disto
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ICATION OF COMPUTERS IN EXPERIMENTS
Direct Reconstruction of the Experimental Data in the Case of Ill-Conditioned Problems and in the Presence of Data Distortions A. V. Novikov-Borodin* Institute for Nuclear Research (INR), Russian Academy of Sciences, Moscow, 117312 Russia *e-mail: [email protected] Received February 17, 2020; revised April 26, 2020; accepted May 1, 2020
Abstract—The possibilities of reconstructing discrete experimental data using direct inversion, i.e., deconvolution, are investigated. Methods for optimization, smoothing, compensation, and decomposition are proposed, with which it is possible to reconstruct data with minimal information losses in many cases of ill-conditioned problems and in the presence of data distortions. These methods are based on the formation of well-conditioned systems with the compensation of accidental distortions from overdetermined systems of equations corresponding to the convolution equations using the linear transformations. A comparative analysis of the proposed methods is carried out. Their potentialities and accuracies of reconstruction are analyzed. Examples of the reconstruction are presented. DOI: 10.1134/S0020441220050322
1. INTRODUCTION Real measuring systems are known to have finite speed of operation. In digital systems, it is limited primarily by the final sampling frequency of the signal. A distorted experimental signal is presented as the convolution of an undistorted signal and a pulse response of the measuring system, which describes the nature of the signal distortion. Inverse transformation, i.e., deconvolution, is required for reconstruction of the undistorted signal from experimental data. This transformation belongs to the class of ill-posed problems [1–3], whose general solution without a loss of information does not exist. The efficiency of the reconstruction depends on many factors, such as the type of the pulse response, the amount of processed data, the noise level in them, etc. Therefore, the development of various reconstruction methods that would be effective in various cases is a relevant line of research. In the case of discrete measurements, convolution equations are reduced to systems of linear algebraic equations and the deconvolution problem can be solved in a variety of different ways, starting with the direct matrix inversion methods and finishing with the integral transformation methods and regularization techniques [4–7]. The main encountered problems consist in the poor conditionality of systems and the presence of noise in experimental data, which lead to instability of the solutions and large errors.
Systems that correspond to convolution equations are fundamentally overdetermined, i.e., the number of equations in them exceeds the number of unknowns. This allows one to form many square systems from them in which the number of equations is equal to the number of unknowns and to select the systems that are most suitable for solving specific problems. The proposed methods for optimization, smoothing, compensation, and
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