Non-linear multi-solution branch search method and device based on compression continuation algorithm

Abstract

The embodiment of the invention provides a nonlinear multi-solution branch search method and device based on a compression continuation algorithm, and the method comprises the steps: S1, determining a key parameter and a value range in a nonlinear problem, and representing the key parameter and the value range as a tensor function Q for solving a nonlinear minimization problem; s2, dividing the two-dimensional rectangle to obtain a discrete grid, and linearizing a nonlinear problem by using a Newton iteration method; s3, taking the lower limit value of the value range of the epsilon as the previous epsilon, providing an initial guess solution for the previous epsilon, solving a nonlinear minimization problem by using a Newton iteration method, and calculating to obtain all solutions under the previous epsilon; s4, by utilizing arc length continuation, continuating all solutions of the previous epsilon to serve as initial guess solutions of a nonlinear minimization problem of the next epsilon adjacent to the previous epsilon, and obtaining all solutions corresponding to the next epsilon; and S5, sequentially moving to the next epsilon to perform branch search until the next epsilon reaches the upper limit value of the epsilon value range. And searching and continuation of non-connected branches are supported.

Description

Technical Field

[0001] The present invention relates to the field of optimization branch calculation methods, and in particular to a nonlinear multi-solution branch search method and device based on a compression and extension algorithm. Background Art

[0002] Nonlinear problems usually have multiple solutions, which can form multiple branches during the calculation process. The systematic search for multiple solutions is very challenging. It is necessary not only to find as many solutions as possible, but also to enhance the credibility and accuracy of the search process to ensure the integrity of the branch graph. In the past, the more classic method was to use a combination of arc length extension and branch switching to complete the branches and search for new branches respectively, or to directly construct a variety of different initial guesses in order to converge to different solutions during the calculation process. The former combination of arc length extension and branch switching is more suitable for problems with connected branches (see Figure 4 ) is quite effective, however, in the presence of disconnected branches (see Figure 4 ), this combination cannot systematically and effectively calculate non-connected branches. One possible remedy is to study the symmetry of the computational problem, first use branch switching to find as many branches as possible, then introduce the asymmetry that appears in the computational problem during the calculation process, and extend the existing branch solutions found from the symmetric structure to the asymmetric structure. Another possible remedy is to introduce asymmetry by changing special parameters in the computational problem (for example, the asymmetry of the pressure distribution can be reflected as a parameter in the computational problem). This is a straightforward approach, but if the asymmetry is introduced in other ways (such as considering the geometric structure), this process is difficult to apply, making it impossible to systematically and effectively complete the search for as many branches as possible.

[0003] In order to systematically search for multiple solutions, some related technical research progress has been made in recent years. For example, [Jianyuan Yin, Yiwei Wang, et al. Construction of a pathway map on a complicated energy landscape, Physical Review Letters, 124 (090601), 2020] searches for solutions by calculating the dynamic system of the equilibrium point, and finds the balance index of different solutions to determine the connection relationship between the solutions, thereby determining different branches. However, this method tends to find the distribution relationship between high energy and low energy, and does not distinguish the connectivity of the branches; [Lidong Fang, Apala Majumdar, Lei Zhang. Surface, size and topological effects for some nematic equilibria on rectangular domains, Mathematics and Mechanics of Solids, 25 (1847), 20220] uses a combination of classic arc length extension and branch switching to search for branches, and designs different initial guesses to find branches, but this method is more difficult to systematically find non-connected branches, which easily leads to the loss of branches.

[0004] Therefore, for nonlinear problems, given that the branches formed by the calculated solutions mostly include non-connected branches, it is necessary to design more effective and systematic methods to search for multiple solutions, and it is also necessary to provide stability calculations for each branch. Summary of the invention

[0005] The embodiment of the present invention provides a nonlinear multi-solution branch search method and device based on a compression and extension algorithm, which can solve the technical problem in the prior art that it is difficult to systematically find non-connected branches and easily cause the loss of branches.

[0006] To achieve the above-mentioned purpose, on the one hand, an embodiment of the present invention provides a nonlinear multi-solution branch search method based on a compression and extension algorithm, comprising:

[0007] S1: determining a key parameter in a nonlinear problem, giving a value range of the key parameter, and based on the variation range of the key parameter, specifically expressing the nonlinear problem as a tensor function Q in solving a nonlinear minimization problem, wherein the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range;

[0008] S2: dividing the value range of the weighting coefficient ε into sub-value ranges to obtain a discrete grid, approximating the function on each discrete grid unit by a polynomial, and linearizing the nonlinear problem by using a Newton iteration method;

[0009] S3: Take the lower limit of the range of ε as the previous ε, provide an initial guess solution for the previous ε, use the Newton iteration method to solve the nonlinear minimization problem, and calculate all solutions under the previous ε;

[0010] S4: Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next ε;

[0011] S5: Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, repeat S4 to obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

[0012] On the other hand, an embodiment of the present invention provides a nonlinear multi-solution branch search device based on a compression and extension algorithm, comprising:

[0013] A problem conversion unit is used to determine a key parameter in a nonlinear problem, and given a value range of the key parameter, based on a variation range of the key parameter, the nonlinear problem is specifically expressed as a tensor function Q in solving a nonlinear minimization problem, wherein the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range;

[0014] A linearization unit, used for dividing the two-dimensional rectangle into discrete grids by dividing the value range of the weighting coefficient ε into sub-value ranges, approximating the function on each discrete grid unit by a polynomial, and linearizing the nonlinear problem by using a Newton iteration method;

[0015] The arc length extension unit is used to take the lower limit of the value range of ε as the previous ε, provide an initial guess solution for the previous ε, solve the nonlinear minimization problem using the Newton iteration method, and calculate all solutions under the previous ε;

[0016] Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next ε;

[0017] Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, solve the corresponding nonlinear minimization problem, obtain all solutions corresponding to the next ε, and obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

[0018] The above technical solution has the following beneficial effects: the multi-solution branch search method based on the compression and extension algorithm does not need to detect the location of the branch point, nor does it need to design different initial guesses. It is used to calculate nonlinear problems with multiple solutions, which is conducive to strengthening the understanding of the specific branch structure of nonlinear problems and supporting the search and extension of non-connected branches. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0020] Figure 1 It is a flow chart of a nonlinear multi-solution branch search method based on a compression and extension algorithm according to an embodiment of the present invention;

[0021] Figure 2 It is a structural diagram of a nonlinear multi-solution branch search device based on a compression and extension algorithm according to an embodiment of the present invention;

[0022] Figure 3 is an overall flow chart of an embodiment of the present invention;

[0023] Figure 4 It is a diagram for explaining the difference between connected and non-connected branches in the prior art;

[0024] Figure 5 It is a branch searched by using the nonlinear multi-solution branch search method based on the compression and extension algorithm of an embodiment of the present invention;

[0025] Figure 6 These are examples of multiple solutions searched when ε=0.08 in the example of the present invention. DETAILED DESCRIPTION

[0026] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0027] like Figure 1 As shown, in combination with an embodiment of the present invention, a nonlinear multi-solution branch search method based on a compression and extension algorithm is provided, comprising:

[0028] S1: determining a key parameter in a nonlinear problem, giving a value range of the key parameter, and based on the variation range of the key parameter, specifically expressing the nonlinear problem as a tensor function Q in solving a nonlinear minimization problem, wherein the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range;

[0029] S2: dividing the value range of the weighting coefficient ε into sub-value ranges to obtain a discrete grid, approximating the function on each discrete grid unit by a polynomial, and linearizing the nonlinear problem by using a Newton iteration method;

[0030] S3: Take the lower limit of the range of ε as the previous ε, provide an initial guess solution for the previous ε, use the Newton iteration method to solve the nonlinear minimization problem, and calculate all solutions under the previous ε;

[0031] S4: Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next ε;

[0032] S5: Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, repeat S4 to obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

[0033] Preferably, S1 includes:

[0034] For nonlinear problems, a two-dimensional rectangle Ω = [0, a] × [0, b] is marked as the calculation area of ​​the nonlinear problem. a and b are the length and width of the two-dimensional rectangle respectively. The nonlinear problem is specifically expressed as the tensor function Q in solving the nonlinear minimization problem. The nonlinear minimization problem is expressed as:

[0035]

[0036] Where J(Q) represents the energy functional, W(Q) represents the energy density, and H 1 In mathematics, it represents the Hilbert function space. The functions belonging to this Hilbert function space have integral boundedness and integral boundedness of first-order derivatives. This Hilbert function space is to require that the tensor function Q can make the value obtained by equation (1) bounded in a mathematical sense, and a specific expression of W(Q) is expressed as:

[0037]

[0038] Among them, the weighting coefficient ε is a key parameter, which is a variable positive real number with a value range of [0.05, 0.15]. represents the gradient of the tensor Q, The specific meaning of the gradient of the tensor Q is: if Q = [n 1 ,n 2 ,n 3 ] T , n 1 ,n 2 ,n 3 Represent the components of the tensor function Q in the x, y, and z directions respectively, then we have Q is symmetric and rank zero, Q is represented by (q 1 ,q 2 ) is expressed as: Q∈H 1 (Ω) means Q satisfies the following conditions:

[0039]

[0040] (q 1 ,q 2 ) satisfies the following boundary conditions:

[0041] On the upper and lower boundaries y=0 and y=1,

[0042] On the left and right boundaries x=0 and x=1,

[0043] x, y are the horizontal and vertical coordinates in the corresponding rectangular coordinate system respectively.

[0044] Preferably, S2 includes:

[0045] The value range of the weighting coefficient ε is discretized with equal distances by using a set step size Δε to obtain multiple sub-value ranges, and the two-dimensional rectangle Ω is discretized by using the upper limit value and the lower limit value of each sub-value range to obtain multiple discrete networks. The discrete grid is specifically to use the upper limit value and the lower limit value of each sub-value range as the upper and lower limits of each sub-value range to evenly divide the two-dimensional rectangle Ω horizontally and vertically to obtain multiple small rectangular blocks, and then each small rectangle is diagonally divided into triangles, and all the obtained small triangles are used as discrete grids;

[0046] Choose an appropriate polynomial to approximate the function on each discrete grid cell;

[0047] The Newton iteration method is used to linearize the function on each discrete grid element.

[0048] Preferably, S3 includes:

[0049] The lower limit of the range of ε is ε=0.05 as the previous ε: ε=ε - , is ε - Provide an initial guess solution Among them, x and y correspond to the horizontal axis coordinate and the vertical axis coordinate in the rectangular coordinate system respectively;

[0050] According to each initial guess solution, the Newton iteration method is used to solve the equation (1) corresponding to the nonlinear minimization problem to obtain ε - All the solutions below.

[0051] 5. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 4, characterized in that S4 comprises:

[0052] Using arc length extension, ε - A solution of ε is extended and used as - Next adjacent ε: ε = ε + An initial guess solution to the nonlinear minimization problem of + The branch search solves the corresponding nonlinear minimization problem and obtains ε + The solution should be

[0053] Until ε - All solutions of are extended as ε + The initial guess solution of , and solve the corresponding nonlinear minimization problem to obtain ε + All solutions of ;

[0054] The specific values ​​of the previous ε and the next ε are determined by dividing the value range of the weighted coefficient ε into sub-value ranges: ε + =ε - +Δε, the upper limit and lower limit of each sub-range are used as the previous ε and the next ε adjacent to the previous ε, and the upper limit of the adjacent previous sub-range is equal to the lower limit of the next sub-range.

[0055] Preferably, S4 includes:

[0056] Introduce a compression operator: intensity parameter p and shift parameter α, take intensity parameter p = 2 and shift parameter α = 1, so as to ensure that the solution Q of the nonlinear minimization problem will not converge to the initial guess solution Q through the compression operator * , define the compression operator D(Q;Q * )for:

[0057]

[0058] The intensity parameter p is used to control Q to approach Q *The shift parameter α ensures that when ‖QQ * When ‖→∞, the compression operator can maintain the properties of the original nonlinear problem;

[0059] To ensure that the solution Q of the nonlinear minimization problem does not converge to the initial guess solution Q * , then a new problem needs to be defined in the solution process:

[0060] G(Q)≡D(Q;Q * )=0

[0061] When the new problem G is defined to satisfy the following two properties, the guaranteed solution Q is the same as the initial guess solution Q * different:

[0062] (1) The solution of G(Q) = 0 is the same as the solution of F(Q) = 0, where F is the Euler-Lagrange equation of equation (1), which can be expressed as follows: * ,

[0063] (2) For a known solution Q where F(Q) = 0 * , G will no longer converge to Q * , specifically expressed as: given any sequence Q i →Q * , Q i represents the ith solution.

[0064] Preferably, the upper limit of the value range of the weighting coefficient ε is ε=0.15.

[0065] Preferably, the nonlinear multi-solution branch search method based on the compression and extension algorithm further includes S6:

[0066] Use Cholesky decomposition to calculate the Hessian matrix corresponding to each Q to determine the stability of Q;

[0067] When the corresponding Hessian matrix is ​​positive definite, Q is determined to be a stable solution, and when the corresponding Hessian matrix has a non-zero negative eigenvalue, Q is determined to be an unstable solution.

[0068] Preferably, a and b are respectively selected as 1.

[0069] like Figure 2 As shown, in combination with an embodiment of the present invention, a nonlinear multi-solution branch search device based on a compression and extension algorithm is provided, comprising:

[0070] A problem conversion unit 21 is used to determine a key parameter in a nonlinear problem, and given a value range of the key parameter, based on a variation range of the key parameter, the nonlinear problem is specifically expressed as a tensor function Q in solving a nonlinear minimization problem, where the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range;

[0071] A linearization unit 22, configured to divide the two-dimensional rectangle into discrete grids by dividing the value range of the weight coefficient ε into sub-value ranges, approximate the function on each discrete grid unit by a polynomial, and linearize the nonlinear problem by using a Newton iteration method;

[0072] The arc length extension unit 23 is used to take the lower limit of the value range of ε as the previous ε, provide an initial guess solution for the previous ε, solve the nonlinear minimization problem by using the Newton iteration method, and calculate all solutions under the previous ε;

[0073] Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next ε;

[0074] Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, solve the corresponding nonlinear minimization problem, obtain all solutions corresponding to the next ε, and obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

[0075] Preferably, the question conversion unit 21 is specifically used for:

[0076] For nonlinear problems, a two-dimensional rectangle Ω = [0, a] × [0, b] is marked as the calculation area of ​​the nonlinear problem. a and b are the length and width of the two-dimensional rectangle respectively. The nonlinear problem is specifically expressed as the tensor function Q in solving the nonlinear minimization problem. The nonlinear minimization problem is expressed as:

[0077]

[0078] Where J(Q) represents the energy functional, W(Q) represents the energy density, and H 1 In mathematics, it represents the Hilbert function space. The functions belonging to this Hilbert function space have integral boundedness and integral boundedness of first-order derivatives. This Hilbert function space is to require that the tensor function Q can make the value obtained by equation (1) bounded in a mathematical sense, and a specific expression of W(Q) is expressed as:

[0079]

[0080] Among them, the weighting coefficient ε is a key parameter, which is a variable positive real number with a value range of [0.05, 0.15]. represents the gradient of the tensor Q, The specific meaning of the gradient of the tensor Q is: if Q = [n 1 ,n 2 ,n 3 ] T , n 1 ,n 2 ,n 3 Represent the components of the tensor function Q in the x, y, and z directions respectively, then we have Q is symmetric and rank zero, Q is represented by (q 1 ,q 2 ) is expressed as: Q∈H 1 (Ω) means Q satisfies the following conditions:

[0081]

[0082] (q 1 ,q 2 ) satisfies the following boundary conditions:

[0083] On the upper and lower boundaries y=0 and y=1,

[0084] On the left and right boundaries x=0 and x=1,

[0085] x, y are the horizontal and vertical coordinates in the corresponding rectangular coordinate system respectively.

[0086] Preferably, the linearization unit 22 is specifically used for:

[0087] The value range of the weighting coefficient ε is discretized with equal distances by using a set step size Δε to obtain multiple sub-value ranges, and the two-dimensional rectangle Ω is discretized by using the upper limit value and the lower limit value of each sub-value range to obtain multiple discrete networks. The discrete grid is specifically to use the upper limit value and the lower limit value of each sub-value range as the upper and lower limits of each sub-value range to evenly divide the two-dimensional rectangle Ω horizontally and vertically to obtain multiple small rectangular blocks, and then each small rectangle is diagonally divided into triangles, and all the obtained small triangles are used as discrete grids;

[0088] Choose an appropriate polynomial to approximate the function on each discrete grid cell;

[0089] The Newton iteration method is used to linearize the function on each discrete grid element.

[0090] Preferably, the arc length extension unit 23 is specifically used for:

[0091] The lower limit of the range of ε is ε=0.05 as the previous ε: ε=ε - , is ε - Provide an initial guess solution Among them, x and y correspond to the horizontal axis coordinate and the vertical axis coordinate in the rectangular coordinate system respectively;

[0092] According to each initial guess solution, the Newton iteration method is used to solve the equation (1) corresponding to the nonlinear minimization problem to obtain ε - All the solutions below.

[0093] Preferably, the arc length extension unit 23 is specifically used for:

[0094] Using arc length extension, ε - A solution of ε is extended and used as - Next adjacent ε: ε = ε + An initial guess solution to the nonlinear minimization problem of + The branch search solves the corresponding nonlinear minimization problem and obtains ε + The solution should be

[0095] Until ε - All solutions of are extended as ε + The initial guess solution of , and solve the corresponding nonlinear minimization problem to obtain ε + All solutions of ;

[0096] The specific values ​​of the previous ε and the next ε are determined by dividing the value range of the weighted coefficient ε into sub-value ranges: ε + =ε - +Δε, the upper limit and lower limit of each sub-range are used as the previous ε and the next ε adjacent to the previous ε, and the upper limit of the adjacent previous sub-range is equal to the lower limit of the next sub-range.

[0097] The arc length extension unit 23 is specifically used for:

[0098] Introduce a compression operator: intensity parameter p and shift parameter α, take intensity parameter p = 2 and shift parameter α = 1, so as to ensure that the solution Q of the nonlinear minimization problem will not converge to the initial guess solution Q through the compression operator * , define the compression operator D(Q;Q * )for:

[0099]

[0100] The intensity parameter p is used to control Q to approach Q * The shift parameter α ensures that when ‖QQ *When ‖→∞, the compression operator can maintain the properties of the original nonlinear problem;

[0101] To ensure that the solution Q of the nonlinear minimization problem does not converge to the initial guess solution Q * , then a new problem needs to be defined in the solution process:

[0102] G(Q)≡D(Q;Q * )=0

[0103] When the new problem G is defined to satisfy the following two properties, the guaranteed solution Q is the same as the initial guess solution Q * different:

[0104] (1) The solution of G(Q) = 0 is the same as the solution of F(Q) = 0, where F is the Euler-Lagrange equation of equation (1), which can be expressed as follows: * ,

[0105] (2) For a known solution Q where F(Q) = 0 * , G will no longer converge to Q * , specifically expressed as: given any sequence Q i →Q * , Q i represents the ith solution.

[0106] Preferably, the upper limit of the value range of the weighting coefficient ε is ε=0.15.

[0107] Preferably, the method further comprises a stability determination unit, configured to:

[0108] Use Cholesky decomposition to calculate the Hessian matrix corresponding to each Q to determine the stability of Q;

[0109] When the corresponding Hessian matrix is ​​positive definite, Q is determined to be a stable solution, and when the corresponding Hessian matrix has a non-zero negative eigenvalue, Q is determined to be an unstable solution.

[0110] Preferably, a and b are respectively selected as 1.

[0111] The above technical solution of the embodiment of the present invention is described in detail below in conjunction with specific application examples. For technical details not introduced during the implementation process, please refer to the relevant description in the previous text.

[0112] A nonlinear multi-solution branch search method and device based on a compression and extension algorithm mainly involves multi-solution search for nonlinear problems. Specifically, a multi-solution branch search method based on a compression and extension algorithm is provided for nonlinear problems using a compression and extension algorithm, and the stability of each branch is calculated. There is no need to detect the location of the branch point, nor is there a need to design different initial guesses. Therefore, it can be widely used to calculate nonlinear problems with multiple solutions, for example, solving multiple paths of an aircraft at a certain time, which is conducive to strengthening the understanding of the specific branch structure of nonlinear problems, and can also be combined with MPI technology for parallel computing to improve computing efficiency. This method meets the following characteristics:

[0113] 1. The calculation solution that has been searched will not be searched again;

[0114] 2. Ability to systematically search for as many solutions as possible in nonlinear problems;

[0115] 3. It will not converge to a known solution even with the same initial guess.

[0116] like Figure 3 As shown, a nonlinear multi-solution branch search method based on a compression and extension algorithm is divided into the following steps:

[0117] S1: Determine the key parameters (weighting coefficient ε) in the nonlinear problem and give the range of variation of the key parameters.

[0118] For nonlinear problems, a two-dimensional rectangle Ω = [0, a] × [0, b] is marked as the calculation area, where a and b are the length and width of the two-dimensional rectangle respectively. Based on the range of variation of key parameters, the nonlinear problem at this time is specifically expressed as solving the tensor function Q in the following nonlinear minimization problem:

[0119]

[0120] Where J(Q) represents the energy functional, W(Q) represents the energy density, and H 1 In mathematics, it represents the Hilbert function space, and the functions belonging to this space have integral boundedness and integral boundedness of first-order derivatives. This function space is to require that the tensor function Q can make the value obtained in Problem 1 bounded in a mathematical sense, and a specific expression of W(Q) can be:

[0121]

[0122] Among them, ε is the weighting coefficient, which is a positive real number with a value range of [0.05, 0.15]. Represents the gradient of the tensor function Q (i.e. The specific meaning of the gradient of the tensor Q is: if Q = [n 1 ,n2 ,n 3 ] T , n 1 ,n 2 ,n 3 Represent the components of the tensor function Q in the x, y, and z directions respectively. If Q is an unknown tensor function on a two-dimensional rectangle Ω, it is symmetric and has rank zero, that is, Q can be expressed as (q 1 ,q 2 ) is expressed as the following matrix form: Q∈H 1 (Ω) means Q satisfies the following conditions:

[0123]

[0124] (q 1 ,q 2 ) satisfies the following boundary conditions:

[0125] On the upper and lower boundaries y=0 and y=1,

[0126] On the left and right boundaries x=0 and x=1,

[0127] x, y are the horizontal and vertical coordinates of the corresponding rectangular coordinate system.

[0128] S2: The two-dimensional rectangle is divided into discrete grids by dividing the weighted coefficient ε, and a suitable polynomial is selected to approximate the function on each discrete grid unit. Finally, the nonlinear problem is linearized using the Newton iteration method.

[0129] The two-dimensional rectangle (area) Ω is divided into discrete grids using a set step size Δε. Specifically, the value range of the key parameter (weighting coefficient ε) is discretized at equal distances using the set step size Δε to obtain multiple sub-value ranges. The two-dimensional rectangle (area) Ω is discretized by the upper and lower limits of each sub-value range to obtain multiple discrete networks. The discrete grid specifically divides a complete two-dimensional rectangle horizontally and vertically using the extreme values ​​of each sub-value range as an average to obtain small rectangular blocks, and then diagonalizes each small rectangle into triangles. The final set of all small triangles is the discrete grid. For example, the set step size is Δε=10 -3 , using Δε=10 -3 The value range of ε is [0.05, 0.15].

[0130] Select a suitable polynomial (such as a first-order or second-order polynomial, etc.) to approximate the function on each discrete grid unit.

[0131] Finally, we use Newton's iteration method to linearize the nonlinear problem.

[0132] S3: Provide an initial guess solution Wherein, x and y correspond to the horizontal and vertical axis coordinates in the rectangular coordinate system, respectively. Starting from the lower limit value of ε (ε=0.05), the Newton iteration method is used to solve the equation (1) corresponding to the nonlinear problem, and all solutions under the current ε parameter value are calculated; so that in S4, according to the equal division of the value range of the weighted parameter ε, the equation (1) corresponding to the nonlinear problem under different values ​​of ε is cyclically calculated in turn, and all known solutions under the current parameter value are extended to the nonlinear problem of the next value of ε by using arc length extension, until all known solutions under the current ε are extended, thereby entering the next stage;

[0133] In order to illustrate the universality of the algorithm, the current value ε = ε - For example, ε - Indicates the previous value of ε, and explains how to move to the next value ε = ε + =ε - +Δε steps, ε + Indicates the next value of ε, Δε indicates the step size (interval) of movement; at this time, the value range of ε is [0.05, 0.15] with a step size (interval) of Δε=10 -3 It is divided equally to facilitate the subsequent calculation of the parameters. - Get n known solutions

[0134] S4: Start branching search for the next value of ε (i.e., calculate and solve G(Q)), use the compression operator to use each known solution of the previous parameter value of ε as an initial guess for the new nonlinear problem, and solve the new nonlinear problem until all known solutions have been used as initial guesses;

[0135] Specifically, a compression operator needs to be introduced in S4 and the relevant parameter values ​​of the compression operator (the intensity parameter p and the shift parameter α in Equation 2) need to be given to ensure that the solution of the nonlinear minimization problem does not converge to the known solution Q * , define the compression operator D(Q;Q * )for:

[0136]

[0137] The intensity parameter p is used to control Q to approach Q * The shift parameter α ensures that when ‖QQ * ‖→∞ (i.e., the target solution Q and the known solution Q * The distance is very far, among which ‖QQ * ‖ represents Q and Q *The compression operator can maintain the properties of the original nonlinear problem when the distance between the two is greater than 2. Usually, p=2 and α=1 are taken.

[0138] In S4, in order to ensure that it does not converge to a known solution, a new problem needs to be defined during the solution process:

[0139] G(Q)≡D(Q;Q * )=0

[0140] Among them, D(Q;Q * ) is a given compression operator. Obviously, the new problem G defined satisfies the following two properties, which ensures that the solution to the new problem is different from the known solution:

[0141] 1. The solution of G(Q) = 0 is the same as the solution of F(Q) = 0, where F is the Euler-Lagrange equation of equation (1); that is, for all Q ≠ Q * ,

[0142] 2. For a known solution Q of F(Q) = 0 * , G will no longer converge to Q * ; That is, given any sequence Q i →Q * , Q i is the i-th new solution.

[0143] S5: Then, move to the next value of ε and repeat S3-S4 until ε is moved to the upper target value of the value range, completing the branch search of the overall problem;

[0144] Once from ε - All initial guesses of have been exhausted, and the algorithm moves ε to the next consecutive value. Repeat process S3-4 until ε reaches the desired target value of 0.15. The search process needs to be applied to all parameter values.

[0145] S6: Calculate the stability of the searched solution and determine the stable branch.

[0146] Specifically, the stability is calculated by using Cholesky decomposition to calculate the Hessian matrix with respect to Q. When the Hessian matrix is ​​positive definite, it is determined to be a stable solution. When the Hessian matrix has a non-zero negative eigenvalue, it is determined to be an unstable solution.

[0147] The following are some examples:

[0148] S1: Determine the key parameters in the nonlinear problem and give the range of variation of the key parameters.

[0149] Select a and b as 1 respectively, mark a two-dimensional square Ω = [0, 1] × [0, 1] as the calculation area, and the problem at this time is specifically expressed as solving the tensor function Q in the following nonlinear minimization problem, which is specifically expressed as:

[0150]

[0151] Among them, a specific expression of W(Q) can be:

[0152]

[0153] Among them, ε is a positive real number, ranging from [0.05, 0.15], represents the gradient of the tensor Q (i.e., if Q = [n 1 ,n 2 ,n 3 ] T , Q is an unknown tensor function on the two-dimensional rectangle Ω, which is symmetric and has rank zero, that is, Q can be expressed as (q 1 ,q 2 ) is expressed as Q∈H 1 (Ω) means Q satisfies the following conditions:

[0154]

[0155] (q 1 ,q 2 ) satisfies the following boundary conditions:

[0156] On the upper and lower boundaries y=0 and y=1,

[0157] On the left and right boundaries x=0 and x=1,

[0158] S2: Divide the discrete grid and select the finite element space, and use the Newton iteration method to linearize the nonlinear problem.

[0159] Divide the two-dimensional rectangle (region) Ω into discrete grids, use Newton iteration method to linearize the nonlinear problem, select appropriate finite element space for discretization, and discretize the continuous variation range of parameters; the step size of the selected finite element space Δε=10 -3 And divide the value range of ε into [0.05,0.15] equally.

[0160] S3: Provide an initial guess solution Where x and y correspond to the horizontal and vertical coordinates of the rectangular coordinate system, respectively. Starting from ε=0.05, the Newton iteration method is used to solve the equation (1) corresponding to the nonlinear problem, and all solutions under the current ε parameter value are calculated. According to the equal division of the parameter value range, the nonlinear problem (1) under different values ​​of ε is calculated in turn in a loop until all known solutions are extended, thus entering the next stage.

[0161] Using arc length continuation, all known solutions under the current parameter value are extended to the nonlinear problem of the next parameter value until all known solutions are extended, thus entering the next stage;

[0162] In order to illustrate the universality of the algorithm, the current value ε = ε - As an example, how to move to the next value ε = ε + =ε - +Δε step, assuming that ε=ε - Get n known solutions

[0163] Specifically, S3 uses the classic arc length extension method. In short, at a certain parameter value ε * , given a known solution Q * , taking it as a starting point, the arc length extension technique will calculate this parameter value ε * As the step size Δε increases until the maximum value of ε (that is, 0.15 in the embodiment of the present invention) is reached, the solution Q is obtained, so that a set of solutions under all ε values ​​can be obtained from the starting point, forming a complete branch (that is, a set of solutions) within the range of parameter ε values. In the embodiment of the present invention, the arc length extension method is selected to continuously extend the known solution under a certain parameter to the solution of the next adjacent parameter.

[0164] S4: Start branch search for the next parameter value, use compression operators to use each known solution of the previous parameter value as an initial guess for the new nonlinear problem, and solve the new nonlinear problem until all known solutions have been used as initial guesses;

[0165] Specifically, a compression operator needs to be introduced in S4 and the relevant parameter values ​​of the compression operator need to be given to ensure that it does not converge to the known solution Q * , define the compression operator D(Q;Q * )for

[0166]

[0167] The intensity parameter p is used to control Q to approach Q * The shift parameter α ensures that when ‖QQ * ‖→∞ (i.e., the target solution Q and the known solution Q * The distance is very far, among which ‖QQ* ‖ represents Q and Q * The compression operator can maintain the properties of the original nonlinear problem when the distance between the two is . Usually, p = 2 and α = 1.

[0168] In S4, in order to ensure that it does not converge to a known solution, a new problem is defined during the solution process:

[0169] G(Q)≡D(Q;Q * )=0.

[0170] Among them, D(Q;Q * ) is a given compression operator. Obviously, the new problem G defined satisfies the following two properties, which ensures that the solution to the new problem is different from the known solution:

[0171] 1. The solution for G(Q) = 0 is the same as the solution for F(Q) = 0; that is, for all Q ≠ Q * ,

[0172] 2. For a known solution Q of F(Q) = 0 * , G will no longer converge to Q * ; That is, given any sequence Q i →Q * ,

[0173] S5: Move to the next parameter value and repeat S3-S4 until the parameter value reaches the upper target value of the value range, completing the branch search of the overall problem;

[0174] Once from ε - All initial guesses of have been exhausted and the algorithm moves to the next continuous parameter value. Process S2 is repeated until the continuous parameter ε reaches the desired target value of 0.15. The search process is applied to all continuous parameter values, i.e. no prior knowledge of the location of the disconnected branches is assumed.

[0175] S6: Calculate the stability of the searched solution and determine the stable branch.

[0176] Specifically, the stability is calculated by using Cholesky decomposition to calculate the Hessian matrix with respect to Q. When the Hessian matrix is ​​positive definite, it is determined to be a stable solution. When the Hessian matrix has a non-zero negative eigenvalue, it is determined to be an unstable solution.

[0177] The present invention tests the algorithm for a two-dimensional rectangular area and evaluates the effectiveness of the method. All numerical experiments are based on continuous piecewise linear discretization of the unknown function space. All calculations are performed on eight cores and parallelized using MPI. Figure 5As shown in the figure, the calculation results include both non-connected branches and connected branches, which fully demonstrates the effectiveness of this method for multi-solution branches. Figure 6 The results show that when ε = 0.08, multiple solutions are searched, and the stability analysis of the solutions shows that Figure 6 The first two solutions of are stable.

[0178] It can be seen that the embodiment of the present invention performs algorithm testing and branch analysis on nonlinear tensor problems, obtains reliable and effective analysis results, searches for multiple solutions, provides effective calculation of non-connected branches, and provides stability calculation of each solution, further enhancing the credibility and effectiveness of the algorithm.

[0179] The beneficial effects achieved by the embodiments of the present invention are as follows:

[0180] 1. Strong scalability. The compression and extension algorithm is applicable to various complex multi-solution mining problems, and can obtain faster and more systematic branch graphs, which is conducive to further dynamic analysis of multiple solutions.

[0181] 2. Support the search and extension of non-connected branches. Previous studies often focused on the systematic search of connected branches.

[0182] It should be understood that the specific order or hierarchy of steps in the disclosed process is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of protection of the present disclosure. The attached method claims present the elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0183] In the above detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that the embodiments of the claimed subject matter require more features than are clearly stated in each claim. On the contrary, as reflected in the appended claims, the invention is in a state of having less than all the features of the disclosed individual embodiments. Therefore, the appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0184] The disclosed embodiments are described above to enable any person skilled in the art to implement or use the present invention. Various modifications of these embodiments are obvious to those skilled in the art, and the general principles defined herein may also be applied to other embodiments without departing from the spirit and scope of the present disclosure. Therefore, the present disclosure is not limited to the embodiments given herein, but is consistent with the broadest scope of the principles and novel features disclosed in this application.

[0185] The above description includes examples of one or more embodiments. Of course, it is impossible to describe all possible combinations of components or methods for the purpose of describing the above embodiments, but it should be recognized by those skilled in the art that the various embodiments may be further combined and arranged. Therefore, the embodiments described herein are intended to cover all such changes, modifications and variations that fall within the scope of protection of the appended claims. In addition, with respect to the term "comprising" used in the specification or claims, the word is covered in a manner similar to the term "including", just as "including," is explained as a transitional word in the claims. In addition, any term "or" used in the specification of the claims is intended to mean "non-exclusive or".

[0186] Those skilled in the art may also understand that the various illustrative logical blocks, units, and steps listed in the embodiments of the present invention may be implemented by electronic hardware, computer software, or a combination of the two. In order to clearly demonstrate the interchangeability of hardware and software, the various illustrative components, units, and steps described above have generally described their functions. Whether such functions are implemented by hardware or software depends on the specific application and the design requirements of the entire system. Those skilled in the art may use various methods to implement the described functions for each specific application, but such implementation should not be understood as exceeding the scope of protection of the embodiments of the present invention.

[0187] The various illustrative logic blocks or units described in the embodiments of the present invention can be implemented or operated by a general-purpose processor, a digital signal processor, an application-specific integrated circuit (ASIC), a field programmable gate array or other programmable logic device, a discrete gate or transistor logic, a discrete hardware component, or any combination of the above. The general-purpose processor can be a microprocessor, and optionally, the general-purpose processor can also be any conventional processor, controller, microcontroller or state machine. The processor can also be implemented by a combination of computing devices, such as a digital signal processor and a microprocessor, a plurality of microprocessors, one or more microprocessors combined with a digital signal processor core, or any other similar configuration.

[0188] The steps of the method or algorithm described in the embodiments of the present invention can be directly embedded in hardware, a software module executed by a processor, or a combination of the two. The software module can be stored in a RAM memory, a flash memory, a ROM memory, an EPROM memory, an EEPROM memory, a register, a hard disk, a removable disk, a CD-ROM, or other storage media of any form in the art. Exemplarily, the storage medium can be connected to the processor so that the processor can read information from the storage medium and can write information to the storage medium. Optionally, the storage medium can also be integrated into the processor. The processor and the storage medium can be arranged in an ASIC, and the ASIC can be arranged in a user terminal. Optionally, the processor and the storage medium can also be arranged in different components in the user terminal.

[0189] In one or more exemplary designs, the above functions described in the embodiments of the present invention can be implemented in hardware, software, firmware or any combination of the three. If implemented in software, these functions can be stored on a computer-readable medium, or transmitted in the form of one or more instructions or codes on a computer-readable medium. Computer-readable media include computer storage media and communication media that facilitate the transfer of computer programs from one place to another. The storage medium can be any available medium that can be accessed by any general or special computer. For example, such computer-readable media can include but are not limited to RAM, ROM, EEPROM, CD-ROM or other optical disk storage, disk storage or other magnetic storage devices, or any other medium that can be used to carry or store program codes in the form of instructions or data structures and other forms that can be read by general or special computers, or general or special processors. In addition, any connection can be appropriately defined as a computer-readable medium, for example, if the software is transmitted from a website site, server or other remote resource through a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL) or wirelessly, such as infrared, wireless and microwave, it is also included in the defined computer-readable medium. The disk and disc include compact disk, laser disk, optical disk, DVD, floppy disk and blue-ray disk. Disks usually copy data magnetically, while discs usually copy data optically with lasers. The above combination can also be included in computer readable media.

[0190] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A nonlinear multi-solution branch search method based on a compression and extension algorithm, characterized in that: include: S1: determining a key parameter in a nonlinear problem, giving a value range of the key parameter, and based on the variation range of the key parameter, specifically expressing the nonlinear problem as a tensor function Q in solving a nonlinear minimization problem, wherein the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range; S2: dividing the value range of the weighting coefficient ε into sub-value ranges to obtain a discrete grid, approximating the function on each discrete grid unit by a polynomial, and linearizing the nonlinear problem by using a Newton iteration method; S3: Take the lower limit of the range of ε as the previous ε, provide an initial guess solution for the previous ε, use the Newton iteration method to solve the nonlinear minimization problem, and calculate all solutions under the previous ε; S4: Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next v; S5: Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, repeat S4 to obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

2. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 1, characterized in that: S1 includes: For nonlinear problems, a two-dimensional rectangle Ω = [0, a] × [0, b] is marked as the calculation area of ​​the nonlinear problem. a and b are the length and width of the two-dimensional rectangle respectively. The nonlinear problem is specifically expressed as the tensor function Q in solving the nonlinear minimization problem. The nonlinear minimization problem is expressed as: Where J(Q) represents the energy functional, W(Q) represents the energy density, and H 1 In mathematics, it represents the Hilbert function space. The functions belonging to this Hilbert function space have integral boundedness and integral boundedness of first-order derivatives. This Hilbert function space is to require that the tensor function Q can make the value obtained by equation (1) bounded in a mathematical sense, and a specific expression of W(Q) is expressed as: Among them, the weighting coefficient ε is a key parameter, which is a variable positive real number with a value range of [0.05, 0.15]. represents the gradient of the tensor Q, The specific meaning of the gradient of the tensor Q is: if Q = [n1, n2, n3] T , n1, n2, n3 represent the components of the tensor function Q in the x, y, and z directions respectively, Q is symmetric and has rank zero. Q is represented by (q1,q2) as: Q∈H 1 (Ω) means Q satisfies the following conditions: (q1,q2) satisfies the following boundary conditions: On the upper and lower boundaries y=0 and y=1, On the left and right boundaries x=0 and x=1, x, y are the horizontal and vertical coordinates in the corresponding rectangular coordinate system respectively.

3. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 2, characterized in that S2 include: The value range of the weighting coefficient ε is discretized with equal distances by using a set step size Δε to obtain multiple sub-value ranges, and the two-dimensional rectangle Ω is discretized by using the upper limit value and the lower limit value of each sub-value range to obtain multiple discrete networks. The discrete grid is specifically to use the upper limit value and the lower limit value of each sub-value range as the upper and lower limits of each sub-value range to evenly divide the two-dimensional rectangle Ω horizontally and vertically to obtain multiple small rectangular blocks, and then each small rectangle is diagonally divided into triangles, and all the obtained small triangles are used as discrete grids; Choose an appropriate polynomial to approximate the function on each discrete grid cell; The Newton iteration method is used to linearize the function on each discrete grid element.

4. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 3, characterized in that S3 include: The lower limit of the range of ε is ε=0.05 as the previous ε: ε=ε - , is ε - Provide an initial guess solution Among them, x and y correspond to the horizontal axis coordinate and the vertical axis coordinate in the rectangular coordinate system respectively; According to each initial guess solution, the Newton iteration method is used to solve the equation (1) corresponding to the nonlinear minimization problem to obtain ε - All the solutions below.

5. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 4, characterized in that S4 include: Using arc length extension, ε - A solution of ε is extended and used as - Next adjacent ε: ε = ε + An initial guess solution to the nonlinear minimization problem of + The branch search solves the corresponding nonlinear minimization problem and obtains ε + The solution should be Until ε - All solutions of are extended as ε + The initial guess solution of , and solve the corresponding nonlinear minimization problem to obtain ε + All solutions of ; The specific values ​​of the previous ε and the next ε are determined by dividing the value range of the weighted coefficient ε into sub-value ranges: ε + =ε - +Δε, the upper limit and lower limit of each sub-range are used as the previous ε and the next ε adjacent to the previous ε, and the upper limit of the adjacent previous sub-range is equal to the lower limit of the next sub-range.

6. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 5, characterized in that S4 include: Introduce a compression operator: intensity parameter p and shift parameter α, take intensity parameter p = 2 and shift parameter α = 1, so as to ensure that the solution Q of the nonlinear minimization problem will not converge to the initial guess solution Q through the compression operator * , define the compression operator D(Q;Q * )for: The intensity parameter p is used to control Q to approach Q * The shift parameter α ensures that when ‖QQ * When ‖→∞, the compression operator can maintain the properties of the original nonlinear problem; To ensure that the solution Q of the nonlinear minimization problem does not converge to the initial guess solution Q * , then a new problem needs to be defined in the solution process: G(Q)≡D(Q;Q * )=0 When the new problem G is defined to satisfy the following two properties, the guaranteed solution Q is the same as the initial guess solution Q * different: (1) The solution of G(Q) = 0 is the same as the solution of F(Q) = 0, where F is the Euler-Lagrange equation of equation (1), which can be expressed as follows: * , (2) For a known solution Q where F(Q) = 0 * , G will no longer converge to Q * , specifically expressed as: given any sequence Q i →Q * , Q i represents the ith solution.

7. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 6, characterized in that: The upper limit of the value range of the weighting coefficient ε is ε=0.

15.

8. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 7, characterized in that: Also includes S6: Use Cholesky decomposition to calculate the Hessian matrix corresponding to each Q to determine the stability of Q; When the corresponding Hessian matrix is ​​positive definite, Q is determined to be a stable solution, and when the corresponding Hessian matrix has a non-zero negative eigenvalue, Q is determined to be an unstable solution.

9. The nonlinear multi-solution branch search method based on the compression and extension algorithm according to claim 2, characterized in that: Set a and b to 1 respectively.

10. A nonlinear multi-solution branch search device based on a compression and extension algorithm, characterized in that: include: A problem conversion unit is used to determine a key parameter in a nonlinear problem, and given a value range of the key parameter, based on a variation range of the key parameter, the nonlinear problem is specifically expressed as a tensor function Q in solving a nonlinear minimization problem, wherein the key parameter is a weighting coefficient ε in the nonlinear problem, and the weighting coefficient ε has a value range; A linearization unit, used for dividing the two-dimensional rectangle into discrete grids by dividing the value range of the weighting coefficient ε into sub-value ranges, approximating the function on each discrete grid unit by a polynomial, and linearizing the nonlinear problem by using a Newton iteration method; The arc length extension unit is used to take the lower limit of the value range of ε as the previous ε, provide an initial guess solution for the previous ε, solve the nonlinear minimization problem using the Newton iteration method, and calculate all solutions under the previous ε; Using arc length extension, all solutions of the previous ε are extended and used as the initial guess solution of the nonlinear minimization problem of the next ε adjacent to the previous ε, and the branch search of the next ε is started to solve the corresponding nonlinear minimization problem and obtain all solutions corresponding to the next ε; Move to the next ε in sequence for branch search until the next ε reaches the upper limit of the ε value range. For each next ε, solve the corresponding nonlinear minimization problem, obtain all solutions corresponding to the next ε, and obtain all solutions under each next ε, thereby completing all branch searches for nonlinear problems.

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