A method for inverting the size spectrum of ultrafine particles under non-negativity constraints

Abstract

The present invention relates to a method for inverting the particle size spectrum of ultrafine particles under non-negativity constraints, comprising: obtaining a transfer function matrix and a corresponding response matrix; establishing a relationship between the transfer function matrix and the response matrix; using a logarithmic regularization term matrix to provide a fixed positive size distribution; fitting the logarithmic regularization term matrix, substituting an initial number concentration matrix and an initial coefficient matrix into a regularization solution formula, iteratively obtaining N regularization solutions, residual terms, and regularization terms; obtaining an optimal regularization parameter λ, and then obtaining an optimal coefficient matrix, and plotting a particle size spectrum. During the particle size spectrum inversion process, an L-curve and regularization method are used to address the morbidity of the transfer function matrix during the inversion process, reduce the particle size spectrum inversion error caused by response errors, address the non-negativity problem existing in the inversion process, and restore a sharp size distribution.

Description

Technical Field

[0001] The present invention relates to the technical field of ultrafine particle size distribution identification in an environment, and in particular to an ultrafine particle size spectrum inversion method under non-negative constraints. Background Art

[0002] The main measurement principles for studying aerosol particle size spectra include aerodynamics, light scattering, and electromigration. Due to these limitations, particle size spectrometers based on aerodynamics and light scattering exhibit poor accuracy when measuring smaller particles. However, particle size spectrometers based on electromigration exhibit the opposite performance. Generally speaking, for particles smaller than 300 nm, an electromigration-based spectrometer is preferred for obtaining an accurate size spectrum.

[0003] In particle size spectrum inversion, there are two widely used calculation methods:

[0004] The first method is a multi-charge correction algorithm developed based on the zero-order method. Not all particles in the entire particle size range have single charges. Particles with larger sizes carry a lot of charges. In this case, it is impossible to know the exact value of the largest particle entering the DMA.

[0005] The second type of method is the particle size spectrum inversion method, which uses certain mathematical methods to solve the inversion model. This is generally an ill-posed problem, resulting in an ill-posed solution. The more complex the particle charge situation, the more serious the overlap of the shell functions in the inversion model, and the more difficult the solution. To overcome this problem, the difficulty of solving can be reduced by reducing the overlap of the shell functions. For example, the least squares method with smoothing constraints is used. By adding "smoothing matrix" and "smoothing coefficient" constraints, the oscillation solution problem caused by directly using the least squares method to solve the quasi-singular matrix is ​​overcome. This solution is very simple, but the solution may have negative values ​​and has no actual physical meaning. In addition, the NNLS non-negative least squares algorithm can also be used. This solution algorithm only considers the physical constraint of no non-negative values, but does not consider the smoothing constraint. There are no meaningless values ​​in the solution, which easily leads to oscillatory solutions. Summary of the Invention

[0006] In order to solve the defects of pathological conditions existing in the particle size spectrum inversion process, the purpose of the present invention is to provide a method for inverting the particle size spectrum of ultrafine particles under non-negative constraints, which reduces the pathological conditions existing in the particle size spectrum inversion process, avoids oscillatory solutions while ensuring the non-negative property of the particle size spectrum, and reduces the particle size spectrum inversion error caused by response error.

[0007] To achieve the above object, the present invention adopts the following technical solution: a method for inverting the particle size spectrum of ultrafine particles under non-negative constraints, the method comprising the following steps in sequence:

[0008] (1) Set up N particle size channels, and the charged particles pass through the graded voltage. Use the microcurrent detection module to obtain the transfer function matrix and the corresponding response matrix;

[0009] (2) Establish the relationship between the transfer function matrix and the response matrix;

[0010] (3) A regularization scheme is introduced, using a logarithmic regularization term matrix to provide a fixed positive size distribution;

[0011] (4) Fitting the logarithmic regularization term matrix using a rectangular fitting solution, obtaining an initial number concentration matrix using a pseudo-inverse method, and then obtaining an initial coefficient matrix. Substituting the initial number concentration matrix and the initial coefficient matrix into the regularization solution formula, iteratively obtaining N regularization solutions, residual terms, and regularization terms;

[0012] (5) Using the L-curve method, multiple differentiations are used to obtain the optimal regularization parameter λ, and then the optimal coefficient matrix is ​​obtained to draw the particle size spectrum.

[0013] In step (2), the relationship between the transfer function matrix and the response matrix is:

[0014] G=Af+e

[0015] Where G is the response matrix of the particle size spectrometer, A is the transfer function matrix, f is the particle number concentration obtained in each particle size channel, and e is the system error.

[0016] The step (3) specifically refers to:

[0017] The regularization scheme is:

[0018]

[0019] Where R is the residual matrix, J is the logarithmic regularization matrix, λ is the regularization parameter, R and J are functions of λ, a and b are the lower and upper limits of the particle size, respectively, and g is i is the response corresponding to each particle size channel in the response matrix, K i (x) is the transfer function value corresponding to each particle size channel; the choice of regularization parameter λ determines the smoothness and fit; E(ε i ) is the systematic error;

[0020] A separate non-negative constraint is introduced in the regularization term, and the logarithmic regularization term matrix J is used to provide a fixed positive size distribution and restore the sharp size distribution. The formula of the logarithmic regularization term matrix J is as follows:

[0021]

[0022] In the above formula, x is the particle size, and f is the particle number concentration obtained in each particle size channel.

[0023] The step (4) specifically refers to: discretely fitting the logarithmic regularization term matrix J by using a rectangular fitting solution, and obtaining the diagonal coefficient matrix L from it:

[0024]

[0025] Where x i is the particle size of item i, f(x i ) is the number concentration corresponding to the i-th particle size channel, and ΔX is the particle size classification interval in logarithmic coordinates;

[0026] By ignoring the systematic error e, the initial number concentration matrix N is obtained by using the pseudo-inverse method. x0 :

[0027]

[0028] Where G is the response matrix of the particle size spectrometer, and A is the transfer function matrix;

[0029] Then obtain the initial coefficient matrix L0:

[0030]

[0031] Use the initial coefficient matrix L0 to obtain N regularized solutions. The regularized solution formula is:

[0032] N X1 =(A T A+λ 2 L0 T L0) -1 (A T G+λ 2 L0 T L0N X0 )

[0033] Where N X1 The first regularized solution, λ is the regularization parameter, each λ corresponds to a regularized solution, and so on. After N-1 iterations, a total of N regularized solutions are obtained, corresponding to N residual terms and regularization terms. The calculation expression is as follows:

[0034] R=AN X -G

[0035] J=LN X

[0036] In the above formula, N X is the regularization solution matrix, R is the residual term matrix, and J is the logarithmic regularization term matrix.

[0037] The step (5) specifically refers to: using the L-curve method, taking the square of the residual term and the regularization term as the horizontal axis and vertical axis of the L-curve respectively, and the calculation expression is as follows:

[0038] H1=||AN X -G||2 2

[0039] H2=||LN X ||2

[0040] Where G is the response matrix of the particle size spectrometer; A is the transfer function matrix; H1 and H2 are the square value of the second norm of the residual term and the second norm of the regularization term, respectively;

[0041] The regularization parameter λ at the maximum curvature position of the L curve is obtained by multiple differentiation, which is the optimal regularization parameter λ. The coefficient matrix L corresponding to the λ value is introduced into the particle size spectrum calculation expression:

[0042] N inv =(A T A+λ 2 L T L) -1 ((A T G+λ 2 L T LN Xinv ) / ΔX

[0043] Where G is the response matrix of the particle size spectrometer, N inv is the particle size spectrum information, ΔX is the particle size channel interval, L is the coefficient matrix corresponding to the maximum curvature position, that is, the optimal coefficient matrix, N Xinv is the regularized solution matrix corresponding to the position of maximum curvature.

[0044] It can be seen from the above technical solution that the beneficial effects of the present invention are: first, the L-curve and regularization methods are used in the inversion process of the particle size spectrum to solve the morbidity problem of the transfer function matrix in the inversion process, thereby reducing the particle size spectrum inversion error caused by the response error; second, the logarithmic regularization term matrix is ​​used to provide a fixed positive size distribution, which solves the non-negativity problem in the inversion process and can restore the sharp size distribution. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 is a flow chart of the method of the present invention;

[0046] Figure 2 Schematic diagram of the device structure of the present invention. DETAILED DESCRIPTION

[0047] like Figure 1As shown in FIG, a method for inverting the particle size spectrum of ultrafine particles under non-negativity constraints includes the following steps in sequence:

[0048] (1) Set up N particle size channels, and the charged particles pass through the graded voltage. Use the microcurrent detection module to obtain the transfer function matrix and the corresponding response matrix;

[0049] (2) Establish the relationship between the transfer function matrix and the response matrix;

[0050] (3) A regularization scheme is introduced, using a logarithmic regularization term matrix to provide a fixed positive size distribution;

[0051] (4) Fitting the logarithmic regularization term matrix using a rectangular fitting solution, obtaining an initial number concentration matrix using a pseudo-inverse method, and then obtaining an initial coefficient matrix. Substituting the initial number concentration matrix and the initial coefficient matrix into the regularization solution formula, iteratively obtaining N regularization solutions, residual terms, and regularization terms;

[0052] (5) Using the L-curve method, multiple differentiations are used to obtain the optimal regularization parameter λ, and then the optimal coefficient matrix is ​​obtained to draw the particle size spectrum.

[0053] In step (2), the relationship between the transfer function matrix and the response matrix is:

[0054] G=Af+e

[0055] Where G is the response matrix of the particle size spectrometer, A is the transfer function matrix, f is the particle number concentration obtained in each particle size channel, and e is the system error.

[0056] The step (3) specifically refers to:

[0057] The regularization scheme is:

[0058]

[0059] Where R is the residual matrix, J is the logarithmic regularization matrix, λ is the regularization parameter, R and J are functions of λ, a and b are the lower and upper limits of the particle size, respectively, and g is i is the response corresponding to each particle size channel in the response matrix, K i (x) is the transfer function value corresponding to each particle size channel; the choice of regularization parameter λ determines the smoothness and fit; E(ε i ) is the systematic error;

[0060] A separate non-negative constraint is introduced in the regularization term, and the logarithmic regularization term matrix J is used to provide a fixed positive size distribution and restore the sharp size distribution. The formula of the logarithmic regularization term matrix J is as follows:

[0061]

[0062] In the above formula, x is the particle size, and f is the particle number concentration obtained in each particle size channel.

[0063] The step (4) specifically refers to: discretely fitting the logarithmic regularization term matrix J by using a rectangular fitting solution, and obtaining the diagonal coefficient matrix L from it:

[0064]

[0065] Where x i is the particle size of item i, f(x i ) is the number concentration corresponding to the i-th particle size channel, and ΔX is the particle size classification interval in logarithmic coordinates;

[0066] By ignoring the systematic error e, the initial number concentration matrix N is obtained by using the pseudo-inverse method. x0 :

[0067]

[0068] Where G is the response matrix of the particle size spectrometer, and A is the transfer function matrix;

[0069] Then obtain the initial coefficient matrix L0:

[0070]

[0071] Use the initial coefficient matrix L0 to obtain N regularized solutions. The regularized solution formula is:

[0072] N X1 =(A T A+λ 2 L0 T L0) -1 (A T G+λ 2 L0 T L0N X0 )

[0073] Where N X1 The first regularized solution, λ is the regularization parameter, each λ corresponds to a regularized solution, and so on. After N-1 iterations, a total of N regularized solutions are obtained, corresponding to N residual terms and regularization terms. The calculation expression is as follows:

[0074] R=AN X -G

[0075] J=LN X

[0076] In the above formula, N Xis the regularization solution matrix, R is the residual term matrix, and J is the logarithmic regularization term matrix.

[0077] The step (5) specifically refers to: using the L-curve method, taking the square of the residual term and the regularization term as the horizontal axis and vertical axis of the L-curve respectively, and the calculation expression is as follows:

[0078] H1=||AN X -G||2 2

[0079] H2=||LN X ||2

[0080] Where G is the response matrix of the particle size spectrometer; A is the transfer function matrix; H1 and H2 are the square value of the second norm of the residual term and the second norm of the regularization term, respectively;

[0081] The regularization parameter λ at the maximum curvature position of the L curve is obtained by multiple differentiation, which is the optimal regularization parameter λ. The coefficient matrix L corresponding to the λ value is introduced into the particle size spectrum calculation expression:

[0082] N inv =(A T A+λ 2 L T L) -1 ((A T G+λ 2 L T LN Xinv ) / ΔX

[0083] Where G is the response matrix of the particle size spectrometer, N inv is the particle size spectrum information, ΔX is the particle size channel interval, L is the coefficient matrix corresponding to the maximum curvature position, that is, the optimal coefficient matrix, N Xinv is the regularized solution matrix corresponding to the position of maximum curvature.

[0084] like Figure 2 As shown, the device includes:

[0085] Particle cutter, i.e. Figure 2 The PM1.0 cutter is used to cut particles with a diameter of 0.1 microns or more, and control the particle size range of the detected particles;

[0086] Particle charge unit, i.e. Figure 2 The particle neutralization source is used to neutralize the initial charge of the detected particles and charge the particles after neutralization, and transmit the charged particles to the particle classification unit;

[0087] The particle classification unit is used to classify particles of different sizes. By applying different high voltages to the differential electric migration particle size classifier, charged particles of different sizes are divided into corresponding particle size channels and transmitted to the particle concentration inversion unit.

[0088] The particle concentration inversion unit is used to invert the particle size spectrum of ultrafine particles. It includes a Faraday cup and a microcurrent detection module. The Faraday cup collects charged particles in different particle size channels, and the microcurrent detection module identifies the microcurrent in different particle size channels.

[0089] In summary, the present invention adopts L-curve and regularization methods in the inversion process of particle size spectrum to solve the morbidity problem of the transfer function matrix in the inversion process, thereby reducing the particle size spectrum inversion error caused by the response error; the logarithmic regularization term matrix is ​​used to provide a fixed positive size distribution, which solves the non-negativity problem in the inversion process and can restore the sharp size distribution.

Claims

1. A method for inverting the particle size spectrum of ultrafine particles under non-negativity constraints, characterized by: The method comprises the following steps in sequence: (1) Set up N particle size channels, and the charged particles pass through the graded voltage. Use the microcurrent detection module to obtain the transfer function matrix and the corresponding response matrix; (2) Establish the relationship between the transfer function matrix and the response matrix; (3) A regularization scheme is introduced, using a logarithmic regularization term matrix to provide a fixed positive size distribution; (4) Fitting the logarithmic regularization term matrix using a rectangular fitting solution, obtaining an initial number concentration matrix using a pseudo-inverse method, and then obtaining an initial coefficient matrix. Substituting the initial number concentration matrix and the initial coefficient matrix into the regularization solution formula, iteratively obtaining N regularization solutions, residual terms, and regularization terms; (5) Using the L-curve method, multiple differentiations are used to obtain the optimal regularization parameter λ, and then the optimal coefficient matrix is ​​obtained to draw the particle size spectrum; In step (2), the relationship between the transfer function matrix and the response matrix is: G=Af+e Where G is the response matrix of the particle size spectrometer, A is the transfer function matrix, f is the particle number concentration obtained in each particle size channel, and e is the system error.

2. The method for inverting the ultrafine particle size spectrum under non-negativity constraints according to claim 1 is characterized by: The step (3) specifically refers to: The regularization scheme is: Where R is the residual matrix, J is the logarithmic regularization matrix, λ is the regularization parameter, R and J are functions of λ, a and b are the lower and upper limits of the particle size, respectively, and g is i is the response corresponding to each particle size channel in the response matrix, K i (x) is the transfer function value corresponding to each particle size channel; the choice of regularization parameter λ determines the smoothness and fit; E(ε i ) is the systematic error; A separate non-negative constraint is introduced in the regularization term, and the logarithmic regularization term matrix J is used to provide a fixed positive size distribution and restore the sharp size distribution. The formula of the logarithmic regularization term matrix J is as follows: In the above formula, x is the particle size, and f is the particle number concentration obtained in each particle size channel.

3. The method for inverting the ultrafine particle size spectrum under non-negativity constraints according to claim 1 is characterized by: The step (4) specifically refers to: discretely fitting the logarithmic regularization term matrix J by using a rectangular fitting solution, and obtaining the diagonal coefficient matrix L from it: Where x i is the particle size of item i, f(x i ) is the number concentration corresponding to the i-th particle size channel, and ΔX is the particle size classification interval in logarithmic coordinates; By ignoring the systematic error e, the initial number concentration matrix N is obtained by using the pseudo-inverse method. x0 : Where G is the response matrix of the particle size spectrometer, and A is the transfer function matrix; Then obtain the initial coefficient matrix L0: Use the initial coefficient matrix L0 to obtain N regularized solutions. The regularized solution formula is: N X1 =(A T a+λ 2 L0 T L0) -1 (A T G+λ 2 L0 T L0N X0 ) Where N X1 The first regularized solution, λ is the regularization parameter, each λ corresponds to a regularized solution, and so on. After N-1 iterations, a total of N regularized solutions are obtained, corresponding to N residual terms and regularization terms. The calculation expression is as follows: R=AN X -G J=LN X In the above formula, N X is the regularization solution matrix, R is the residual term matrix, and J is the logarithmic regularization term matrix.

4. The method for inverting the ultrafine particle size spectrum under non-negativity constraints according to claim 1 is characterized by: The step (5) specifically refers to: using the L-curve method, taking the square of the residual term and the regularization term as the horizontal axis and vertical axis of the L-curve respectively, and the calculation expression is as follows: H1=‖AN X -G‖2 2 H2=‖LN X ‖2 Where G is the response matrix of the particle size spectrometer; A is the transfer function matrix; H1 and H2 are the square value of the second norm of the residual term and the second norm of the regularization term, respectively; The regularization parameter λ at the maximum curvature position of the L curve is obtained by multiple differentiation, which is the optimal regularization parameter λ. The coefficient matrix L corresponding to the λ value is introduced into the particle size spectrum calculation expression: N inv =(A T A+λ 2 L T L) -1 ((A T G+λ 2 L T LN Xinv ) / ΔX Where G is the response matrix of the particle size spectrometer, N inv is the particle size spectrum information, ΔX is the particle size channel interval, L is the coefficient matrix corresponding to the maximum curvature position, that is, the optimal coefficient matrix, N Xinv is the regularized solution matrix corresponding to the position of maximum curvature.

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