Data fitting method combining radial basis function optimization boundary and multivariate quadratic quasi-interpolation

Abstract

The invention discloses a radial basis function boundary optimization and multivariate quadratic quasi-interpolation combined data fitting method, which comprises the following steps of: firstly, selecting m data points at two ends from a given data set, fitting boundary data by using Gaussian radial basis function interpolation, and adjusting function characteristics through shape parameters; and fitting all data including the boundary data by using a multivariate quadratic quasi-interpolation method, calculating a weight function of the data and the corrected data, and combining a boundary fitting function to obtain a final result. The method provided by the invention solves the problems that the traditional interpolation calculation is complex, the numerical value is unstable and the boundary processing precision is low, the calculation complexity is low, and the numerical value is stable and reliable; the application effect in the fields of mechanical design, fluid dynamics simulation and the like is remarkable, compared with traditional polynomial fitting, the boundary fitting error is reduced by at least two orders of magnitude, and the calculation efficiency is greatly improved.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical analysis and function approximation, and particularly to a data fitting method combining radial basis function optimized boundary and multivariate quadratic quasi-interpolation. Background Art

[0002] When constructing an approximation function by traditional interpolation methods, it is usually necessary to solve large-scale equations, which inevitably involves large-scale matrix inversion operations. With the continuous increase in the amount of data, the computational complexity of such matrix inversion operations rises sharply, not only consuming a large amount of computing resources but also significantly prolonging the computing time, severely restricting the efficiency and feasibility of interpolation methods in practical applications.

[0003] To overcome this problem, the multivariate quadratic quasi-interpolation method has been proposed and widely applied in various scenarios. The multivariate quadratic quasi-interpolation method indirectly estimates the target function by constructing an approximation function, skillfully avoiding the large-scale matrix inversion problem that must be solved in traditional interpolation. This method not only has high computational efficiency but also stronger numerical stability, providing a more robust approximation means for numerical analysis. However, the multivariate quadratic quasi-interpolation method has certain limitations in dealing with boundary data.

[0004] In the prior art, boundary data is usually processed by polynomial fitting methods. Although this method is relatively simple to implement, it lacks sufficient degrees of freedom and is difficult to adapt to the complex changes of boundary data, resulting in low fitting accuracy. In addition, using spline function interpolation for boundary data processing and internal data processing also has deficiencies. A spline function is a function composed of piecewise polynomials, and it fits the data by selecting appropriate polynomials in different intervals and ensuring continuity and smoothness at the connection points. However, the calculation of spline functions usually involves solving a system of linear equations for spline coefficients, which may have a large amount of calculation for large-scale data and is not as flexible as radial basis function interpolation in dealing with highly irregular boundaries. The accuracy of interpolation depends to a large extent on the distribution of data points. For the case where the data points are unevenly distributed, there may be a problem of large interpolation errors.

[0005] To solve the problems existing in the prior art, the present invention proposes a data fitting method combining radial basis function optimized boundary and multivariate quadratic quasi-interpolation. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to overcome the defects of the prior art and provide a data fitting method combining radial basis function optimized boundary and multivariate quadratic quasi-interpolation, which not only solves the problems of high computational complexity, numerical instability and insufficient boundary processing accuracy existing in traditional interpolation methods, but also realizes efficient and accurate data interpolation by introducing the combination of radial basis function interpolation method and MQ quasi-interpolation method, providing new tools and technical support for the research and application in related fields.

[0007] The present invention provides the following technical solutions:

[0008] The present invention provides a data fitting method combining radial basis function optimized boundary and multivariate quadratic quasi-interpolation, comprising the following steps:

[0009] S1. Boundary data processing: Select m data points at each end from the given data set as boundary data points, a total of 2m data points, and use the radial basis function interpolation method to interpolate and fit these 2m boundary data points to obtain a boundary fitting function

[0010] S2. Internal data processing: Based on the boundary data points and the remaining internal data points processed in step S1, use the multivariate quadratic quasi-interpolation method to fit all the data to obtain a final fitting function f * (x);

[0011] Among them, the radial basis function interpolation method uses a Gaussian radial basis function , ∈ is the shape parameter, and the multivariate quadratic quasi-interpolation method multiplies and accumulates with the weight function and the corrected data, and combines the contribution of the boundary fitting function to obtain the overall fitting result.

[0012] As a preferred technical solution of the present invention, in step S1, the mathematical expression of the radial basis function interpolation is:

[0013]

[0014] In formula 1, ||x - x i || represents the Euclidean distance between x and the boundary data point x i , λ i is a parameter to be determined, which is determined by solving a linear equation system and the linear equation system is only solved for the boundary data points.

[0015] As a preferred technical solution of the present invention, the shape parameter ∈ is used to adjust the local influence range and smoothness of the Gaussian radial basis function, and is dynamically optimized according to the distribution characteristics of the boundary data.

[0016] As a preferred technical solution of the present invention, in the step S2, the calculation method of the weight function of the multivariate quadratic quasi-interpolation is as follows:

[0017]

[0018] In Formula 2, c is the shape parameter;

[0019] The expression of the final fitting function f * (x) is:

[0020]

[0021] As a preferred technical solution of the present invention, in the step S1, the value range of m is 3 ≤ m ≤ 10.

[0022] As a preferred technical solution of the present invention, the multivariate quadratic quasi-interpolation method indirectly estimates the objective function by constructing an approximation function, bypasses the large matrix inversion step, reduces the computational complexity to a linear level, and is applicable to real-time data processing scenarios.

[0023] As a preferred technical solution of the present invention, the combined method of radial basis function interpolation and multivariate quadratic quasi-interpolation accurately captures local irregular boundary features under the global smoothness constraint.

[0024] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0025] 1: The present invention has efficient calculation, breaks through the bottleneck of traditional interpolation methods, uses the multivariate quadratic quasi-interpolation method to avoid large-scale matrix inversion, and when the data volume increases, the calculation efficiency is greatly improved and the consumption of computing resources is significantly reduced.

[0026] 2: The present invention has stable numerical values. With the help of the multivariate quadratic quasi-interpolation method, the numerical results are ensured to be stable and reliable, effectively avoiding error accumulation in large-scale data processing and real-time applications, and providing a stable basis for data processing.

[0027] 3: The present invention has precise boundary processing. It uses radial basis function interpolation to process boundary data, and through flexible adjustment of the shape parameter, it accurately fits the complex changes of the boundary. Compared with traditional polynomial fitting, the accuracy is significantly improved.

[0028] 4. The present invention has wide applications. It can be applied to fields such as scientific computing, engineering design, and data analysis, can improve simulation accuracy, optimize product design, and help mine data value, providing strong support for the development of multiple fields. Description of the Drawings

[0029] The accompanying drawings are used to provide a further understanding of the present invention and form a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation to the present invention. In the accompanying drawings:

[0030] Figure 1 is the overall data fitting flow chart of the present invention;

[0031] Figure 2 is the boundary data processing flow chart of the present invention;

[0032] Figure 3 is the internal data processing flow chart of the present invention;

[0033] Figure 4 is the comparison chart of the multivariate quadratic quasi-interpolation error between the polynomial interpolation processing boundary and the radial basis function processing boundary; Detailed implementation manners

[0034] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present invention and are not used to limit the present invention.

[0035] Embodiment

[0036] As Figures 1-4 shown, the present invention provides a data fitting method combining a radial basis function optimized boundary and a multivariate quadratic quasi-interpolation, including the following steps:

[0037] S1. Boundary data processing: Select m data points at both ends from the given data set as boundary data points, a total of 2m data points. Use the radial basis function interpolation method to interpolate and fit these 2m boundary data points to obtain the boundary fitting function

[0038] S2. Internal data processing: Based on the boundary data points processed in step S1 and the remaining internal data points, use the multivariate quadratic quasi-interpolation method to fit all the data to obtain the final fitting function f * (x);

[0039] wherein, the radial basis function interpolation method uses the Gaussian radial basis function ∈ is the shape parameter, and the multivariate quadratic quasi-interpolation method multiplies and accumulates with the weight function and the modified data, and combines the contribution of the boundary fitting function to obtain the overall fitting result.

[0040] Furthermore, in step S1, the mathematical expression of the radial basis function interpolation is:

[0041]

[0042] In Equation 1, ||x - x i || represents the Euclidean distance between x and the boundary data point x i , and λ i is a parameter to be determined, which is determined by solving a system of linear equations , and the system of linear equations is only solved for boundary data points.

[0043] Furthermore, the shape parameter ∈ is used to adjust the local influence range and smoothness of the Gaussian radial basis function, and is dynamically optimized according to the distribution characteristics of the boundary data.

[0044] Furthermore, in step S2, the calculation method of the weight function for multiquadratic quasi-interpolation is as follows:

[0045]

[0046] In Equation 2, c is the shape parameter;

[0047] The expression of the final fitting function f * (x) is:

[0048]

[0049] Furthermore, in step S1, the value range of m is 3 ≤ m ≤ 10, and preferably m = 5.

[0050] Furthermore, the multiquadratic quasi-interpolation method indirectly estimates the objective function by constructing an approximation function, bypasses the step of inverting a large matrix, reduces the computational complexity to a linear level, and is applicable to real-time data processing scenarios.

[0051] Furthermore, the combined method of radial basis function interpolation and multiquadratic quasi-interpolation accurately captures local irregular boundary features under the global smoothness constraint, and the error range is reduced by at least two orders of magnitude compared with polynomial fitting.

[0052] Specifically,

[0053] I. Boundary data processing, refer to Figure 2 ;

[0054] Data point selection: First, for the given data set where x i represents the position of the data point, and f(x i ) is the corresponding data value. It is necessary to determine m data points at both ends from this data set as boundary data points. The determination of the m value needs to comprehensively consider the data scale and the boundary complexity.

[0055] For datasets with a relatively small data scale and simple boundary changes, m can take a relatively small value, such as m = 3; while for datasets with a large data scale and complex boundary changes, m needs to be taken as 8 - 10.

[0056] Radial basis function selection and parameter setting: The Gaussian radial basis function is selected to process the boundary data, where r = ||x - x i || represents the Euclidean distance between x and the boundary data point x i , and ∈ is the shape parameter. The shape parameter ∈ has important effects on the interpolation result in many aspects.

[0057] When ∈ is relatively small, the width of the Gaussian function is larger, the influence range is wider, the function is smoother, but it may cause the loss of local details, making the interpolation result not accurately reflect the local changes of the boundary; when ∈ is larger, the width of the Gaussian function is smaller, the influence range is limited, and it can better capture local details, but it may cause oscillations in the interpolation result and reduce the stability. Therefore, the initial value of ∈ needs to be determined according to the characteristics of the data. Generally speaking, for boundaries with relatively gentle data changes, ∈ can take a relatively small value, such as 0.1 - 0.5; for boundaries with drastic data changes and many local features, ∈ can take a larger value, such as 5 - 10. Usually, ∈ can be set to 1 first, and then adjusted according to the subsequent fitting effect.

[0058] Construction and solution of the linear equations: Construct the function where λ i is the parameter to be determined. Use the conditions of the boundary data points (only valid at the boundary) to construct a linear equation system to solve these parameters.

[0059] Specifically, for each boundary data point x j (j ranges from 1 to 2m), there is Thus, a linear equation system containing equations is obtained. Common linear equation system solving methods, such as Gaussian elimination method, matrix inversion method, etc., can be used for solution.

[0060] II. Internal data processing, refer to Figure 3 ;

[0061] Data merging: After completing the processing of the boundary data, merge the processed boundary data with the internal data to form a new dataset for subsequent multivariate quadratic quasi-interpolation processing. Ensure unified fitting within the entire data range, so that the fitting result can maintain good continuity and accuracy at both the boundary and the interior.

[0062] For data-intensive regions, a density-based sampling method can be used to remove redundant data points. The density-based sampling method selects appropriate sampling points according to the distribution density of data points, removes those overly dense data points to reduce the computational amount, while maintaining the main features of the data.

[0063] Multivariate quadratic quasi-interpolation calculation: Use multivariate quadratic basis functions as kernel functions to calculate the weight function

[0064]

[0065] c is the shape parameter.

[0066] The value of the shape parameter c affects the distribution of the weight function and the smoothness of the interpolation result. When c is small, the distribution of the weight function is more concentrated, and the interpolation result is more dependent on local data, which may lead to insufficient smoothness of the interpolation result; when c is large, the distribution of the weight function is more dispersed, and the interpolation result is smoother, but some local details may be lost. Generally, c is first set to a moderate value, such as 1, and then adjusted according to the actual fitting effect.

[0067] Final fitting result Through this formula, the fitting result of the entire data set can be obtained to achieve an accurate approximation of the data.

[0068] Application examples in different fields:

[0069] 1. Apply the present invention to the fitting of complex surfaces in mechanical design, specifically:

[0070] Step 1: Boundary data processing:

[0071] Data selection: From the surface boundary containing n = 1000 data points, select m = 5 points at each end as boundary data (a total of 10 points).

[0072] Radial basis function interpolation: Use Gaussian radial basis functions Initial setting ∈ = 0.5.

[0073] Dynamically adjust ∈ according to the distribution density of boundary data: If the data points are dense (spacing < 0.1), then increase ∈ to 1.0 to enhance local smoothness; if sparse (spacing > 0.5), then decrease ∈ to 0.3 to improve flexibility.

[0074] Solve the linear equations Obtain the coefficient λ i , and generate the boundary fitting function f(x i ).

[0075] Step 2: Internal data processing

[0076] Weight function calculation: Define multivariate quadratic basis functions Set c = 0.8.

[0077] According to the formula Calculate the weight function.

[0078] Overall fitting: Multiply and accumulate the corrected data with the weight function and add the boundary contributions to obtain the final fitting function f * (x).

[0079] Verification and effect:

[0080] Accuracy verification: Compare the fitting surface with the actual measurement data. The maximum error in the boundary region is 2.3×10 -7 , which is two orders of magnitude lower than that of the traditional polynomial fitting (5.1×10 -5 ).

[0081] Efficiency verification: The calculation time is reduced from 12.5 seconds in the traditional method to 7.8 seconds (a reduction of 37.6%).

[0082] 2. Apply the present invention to the real-time flow field generation in fluid dynamics simulation, specifically as follows:

[0083] Step 1: Fluid boundary processing

[0084] Dynamic parameter adjustment: For the flow field boundary data, automatically adjust ∈ according to the velocity gradient:

[0085] High velocity gradient region (such as the vortex center): ∈ = 0.2 to capture high-frequency changes;

[0086] Low velocity gradient region (uniform flow): ∈ = 0.6 to maintain smoothness.

[0087] Radial basis function interpolation: Generate a high-precision boundary flow field

[0088] Step 2: Internal flow field generation

[0089] Weight function optimization: Set c = 1.0 to balance local and global features, and calculate

[0090] Real-time update: Update the internal flow field data once per second, and quickly generate f * (x) using MQ quasi-interpolation.

[0091] Verification and effect:

[0092] Stability test: Continuously run for 24 hours, and the numerical error fluctuation range remains within 10-7 Magnitude, no divergence phenomenon.

[0093] Real-time verification: Processing 10 4 data points per second, with a latency of less than 0.1 second, meeting the requirements of real-time simulation.

[0094] In summary, it can be seen that the multiquadratic quasi-interpolation method adopted by the present invention indirectly estimates the objective function by constructing an approximation function, skillfully avoiding the problem of large-scale matrix inversion, greatly improving the calculation efficiency and reducing the demand for computing resources; the multiquadratic quasi-interpolation method not only simplifies the calculation process, but also ensures the stability and reliability of the numerical results, is applicable to large-scale data sets and real-time applications, is particularly important when dealing with complex and changing data, can provide a more robust approximation method, and avoids the problem of error accumulation caused by numerical instability.

[0095] The present invention introduces a radial basis function interpolation method to process boundary data. The radial basis function interpolation method has high degrees of freedom and flexibility due to its parametric characteristics, can capture the changes of boundary conditions more accurately, and provides a higher-precision fitting result; the powerful approximation ability of the radial basis function interpolation method enables it to accurately capture local features while maintaining global smoothness, especially having significant advantages for complex or irregular boundaries. To make this advantage more intuitive, an error comparison graph as Figure 4 is drawn, comparing and showing the error of the radial basis function in processing the boundary with the error of the polynomial in processing the boundary. It can be clearly seen from the comparison that: the error range generated when the polynomial interpolation method processes boundary data is approximately in the order of 10 -5 magnitude. Although the polynomial interpolation method can fit the data well, its accuracy is limited when dealing with complex boundary conditions. When using the radial basis function interpolation method to process boundary data, the error range is significantly reduced to the order of 10 -7 magnitude, indicating that the radial basis function interpolation method has higher accuracy and reliability when dealing with complex boundary conditions.

[0096] The hybrid interpolation technical solution of the present invention has important application values in many fields. In the field of scientific computing, such as physical simulation and astronomical calculation, it is often necessary to perform accurate function approximation on a large amount of data. The present invention can efficiently process boundary data and ensure the accuracy and stability of the overall calculation, thereby improving the accuracy of simulation and calculation and providing a more reliable basis for scientific research. In engineering design, such as mechanical design and electronic circuit design, which involve the modeling and analysis of various complex shapes and boundary conditions, the present invention can accurately process boundary data, helping to design products that better meet actual needs and have better performance. In the field of data analysis, in the face of massive and complex data, the present invention can quickly and accurately perform data fitting and approximation, helping analysts better explore the laws and information behind the data and providing strong support for decision-making.

[0097] The present invention applies the radial basis function interpolation method to boundary data processing and combines the multiquadratic quasi-interpolation method to perform numerical approximation on the entire data set, forming a hybrid interpolation technical solution. This combination not only overcomes the computational bottlenecks existing in traditional interpolation methods but also greatly improves the accuracy and flexibility of boundary processing, providing a more effective solution for complex data sets. This method has broad application prospects in the fields of scientific computing, engineering design, and data analysis, and can significantly improve the effect and quality of data processing.

[0098] Finally, it should be noted that the above are only the preferred embodiments of the present invention and are not used to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions recorded in the foregoing embodiments or perform equivalent replacements on some of the technical features. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A data fitting method combining radial basis function optimization boundary and multivariate quadratic quasi-interpolation, characterized in that: The steps include: S1. Boundary data processing: From a given data set Select m data points at both ends as boundary data points, totaling 2m data points, and use radial basis function interpolation method to interpolate and fit these 2m boundary data points to get the boundary fitting function S2. Internal data processing Based on the boundary data points and the remaining internal data points processed in step S1, the multivariate quadratic quasi-interpolation method is used to fit all the data to obtain the final fitting function f * (x); Among them, the radial basis function interpolation method uses Gaussian radial basis function ∈ is the shape parameter, and the multivariate quadratic quasi-interpolation method uses the weight function The corrected data are multiplied and accumulated, and the contribution of the boundary fitting function is combined to obtain the overall fitting result.

2. The data fitting method according to claim 1, characterized in that: In step S1, the mathematical expression of radial basis function interpolation is: In formula 1, ||xx i || represents x and boundary data point x i The Euclidean distance, λ i To determine the parameters, solve the linear equations is determined, and the system of linear equations is solved only for the boundary data points.

3. The data fitting method according to claim 1 or 2, characterized in that: The shape parameter ∈ is used to adjust the local influence range and smoothness of the Gaussian radial basis function, and is dynamically optimized according to the distribution characteristics of the boundary data.

4. The data fitting method according to claim 1, characterized in that: In step S2, the weight function of the multivariate quadratic pseudo-interpolation is calculated as follows: In formula 2, c is the shape parameter; The final fitting function f * The expression of (x) is:

5. The data fitting method according to claim 1, characterized in that: In step S1, the value range of m is 3≤m≤10.

6. The data fitting method according to claim 1, characterized in that: The multivariate quadratic quasi-interpolation method indirectly estimates the target function by constructing an approximation function, bypassing the large matrix inversion step, reducing the computational complexity to a linear level, and is suitable for real-time data processing scenarios.

7. The data fitting method according to claim 6, characterized in that: The combined method of radial basis function interpolation and multivariate quadratic quasi-interpolation accurately captures local irregular boundary features under global smoothness constraints.

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