Matrix-factorization (MF)-based approaches prove to be highly accurate and scalable in addressing collaborative filtering (CF) problems. During the MF process, the non-negativity, which ensures good ...representativeness of the learnt model, is critically important. However, current non-negative MF (NMF) models are mostly designed for problems in computer vision, while CF problems differ from them due to their extreme sparsity of the target rating-matrix. Currently available NMF-based CF models are based on matrix manipulation and lack practicability for industrial use. In this work, we focus on developing an NMF-based CF model with a single-element-based approach. The idea is to investigate the non-negative update process depending on each involved feature rather than on the whole feature matrices. With the non-negative single-element-based update rules, we subsequently integrate the Tikhonov regularizing terms, and propose the regularized single-element-based NMF (RSNMF) model. RSNMF is especially suitable for solving CF problems subject to the constraint of non-negativity. The experiments on large industrial datasets show high accuracy and low-computational complexity achieved by RSNMF.
The present paper is devoted to recovering the source term and the initial data simultaneously for a time-fractional diffusion equation from additional temperature data at two fixed times t=T1 and ...t=T2. Firstly, we prove the uniqueness result for the direct problem. And then we apply a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm. Three numerical examples in both one-dimensional and two-dimensional cases are provided to demonstrate the effectiveness and feasibility of the proposed method.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Regularization is possibly the most popular method for solving discrete ill-posed problems, whose solution is less sensitive to the error in the observed vector in the right hand than the original ...solution. This paper presents a new modified truncated randomized singular value decomposition (TR-MTRSVD) method for large Tikhonov regularization in standard form. The proposed TR-MTRSVD algorithm introduces the idea of randomized algorithm into the improved truncated singular value decomposition (MTSVD) method to solve large Tikhonov regularization problems. The approximation matrix Ãℓ produced by randomized SVD is replaced by the closest matrix Ãk̃ in a unitarily invariant matrix norm with the same spectral condition number. The regularization parameters are determined by the discrepancy principle. Numerical examples show the effectiveness and efficiency of the proposed TR-MTRSVD algorithm for large Tikhonov regularization problems.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most ...popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, a novel electromagnetic coil design method is proposed, which has a large uniform region for calibrating array optically pumped magnetometers (OPMs). The weighted Tikhonov ...regularization method, whose weight matrix is optimized by particle swarm optimization, is to balance the field error distribution in the uniform region. The cube coils system with a 25 cm side length can produce uniform fields in a cube uniform region with a 15 cm side length. The simulation result shows that the maximum field error ɛmax of the cube coils system is 1% in the uniform region, which is greatly reduced compared with Ruben’s coils system(4.8%) and bi-planar coils system (22.7%). Due to the optimized weight matrix, the errors of the coils are mainly distributed in the range of 0–0.5%. The testing results were consistent with simulation results. The result of calibration experiment proves the superiority of cube coil in this application.
•The ill-posed problem in the target field method is mainly due to the distribution of the coils on two finite planes. Therefore, the cube coils designed in this paper are innovatively designed on six planes of a cube.•In order to expand the volume of the uniform region, the weighted least squares method is used to balance the distribution of field errors in the uniform region, which is combined with particle swarm optimization algorithm.•Because the cube coil system designed in this paper has a large uniform region and high uniformity, it can obtain a higher-precision gain coefficient than the on-board coil of OPM, which is crucial for MCG and MEG experiments.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The classical function expansion method based on minimizing l2-norm of the response residual employs various basis functions to represent the unknown force. Its difficulty lies in determining the ...optimum number of basis functions. Considering the sparsity of force in the time domain or in other basis space, we develop a general sparse regularization method based on minimizing l1-norm of the coefficient vector of basis functions. The number of basis functions is adaptively determined by minimizing the number of nonzero components in the coefficient vector during the sparse regularization process. First, according to the profile of the unknown force, the dictionary composed of basis functions is determined. Second, a sparsity convex optimization model for force identification is constructed. Third, given the transfer function and the operational response, Sparse reconstruction by separable approximation (SpaRSA) is developed to solve the sparse regularization problem of force identification. Finally, experiments including identification of impact and harmonic forces are conducted on a cantilever thin plate structure to illustrate the effectiveness and applicability of SpaRSA. Besides the Dirac dictionary, other three sparse dictionaries including Db6 wavelets, Sym4 wavelets and cubic B-spline functions can also accurately identify both the single and double impact forces from highly noisy responses in a sparse representation frame. The discrete cosine functions can also successfully reconstruct the harmonic forces including the sinusoidal, square and triangular forces. Conversely, the traditional Tikhonov regularization method with the L-curve criterion fails to identify both the impact and harmonic forces in these cases.
•Spare representation is extended to the field of force identification.•SpaRSA is developed to solve l1 regularization problem in force identification.•The Dirac, Db6, Sym4 and B-spline dictionaries are used to represent impact force.•The discrete cosine dictionary is used to represent three types of harmonic force.•Compared with Tikhonov regularization, SpaRSA is highly accurate and efficient.
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In this paper, we are interested in solving the data completion problem for the Laplace equation. It consists to determine the missing data on the inaccessible part of the boundary from overspecified ...conditions in the accessible part. Knowing that this problem is severely ill-posed, we consider its formulation as an optimization problem using Tikhonov regularization. Then, we consider an optimization approach based on adapted Real Coded Genetic Algorithm (RCGA) to minimize the cost function and recover the missing data. The performed numerical simulations, with different domains, illustrate the accuracy and efficiency of the proposed method with an adequate regularization parameter, in addition to the good agreement between the numerical solutions and different noise level of the given data.
Discretization and regularization are required steps to provide a stable approximation when solving integral equations of the first kind. The integral operator involved may be approximated by a ...sequence of finite rank operators and then the regularization procedure is applied. On the other hand, a regularization procedure can be conceived prior to the discretization. Both approaches are developed, implemented and compared for certain projection based methods.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The Cauchy problem associated with the Helmholtz equation is an ill-posed inverse problem that is challenging to solve due to its instability and sensitivity to noise. In this paper, we propose a ...metaheuristic approach to solve this problem using Genetic Algorithms in conjunction with Tikhonov regularization. Our approach is able to produce stable, convergent, and accurate solutions for the Cauchy problem, even in the presence of noise. Numerical results on both regular and irregular domains show the effectiveness and accuracy of our approach.
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