
The human brain is arguably the most complex information processing system. It operates by acquiring data from the environment, recognizing patterns of events’ occurrence, anticipating their re-occurrence and in turn generating appropriate behavioral responses. Through the lenses of the free-energy principle any self-organizing system that is at equilibrium with its environment must minimize its free energy either by manipulating the environmental sensory input or by manipulating its internal states thus altering the recognition density of the outside stimuli. However, several sets of challenges interfere with the human brain's ability to learn and adapt in such a theoretically optimal fashion. These may include, and are not limited to, functional inconsistencies related to attention and memory processes, the functions of “fast” and “slow” thinking and responding, and the ability of emotional states to generate unintended behavioral outcomes that are less adaptive or inappropriate. This paper will review literature on the subject of how ideal learning viewed from the free-energy principle perspective may be affected by the above mentioned limitations and will suggest a model of information processing that may have developed as a way of overcoming these challenges. This neurobiological model stipulates that a neuronal network is formed in response to environmental input and is paralleled by at least one and possibly multiple networks that activate intrinsically and represent “virtual responses” to a situation that demands a behavioral response. This model accounts for how the brain generates a multiplicity of potential behavioral responses and may “choose” the one that seems most appropriate and also explains the uncanny ability of humans to socialize and collaborate. Implications for understanding humans’ ability to learn from others, deliberate on opposing constructs and access and utilize information outside of individual minds are also discussed.
Citation: Iliyan Ivanov, Kristin Whiteside. Dyadic Brain - A Biological Model for Deliberative Inference[J]. AIMS Neuroscience, 2017, 4(4): 169-188. doi: 10.3934/Neuroscience.2017.4.169
[1] | Jin Li . Linear barycentric rational collocation method to solve plane elasticity problems. Mathematical Biosciences and Engineering, 2023, 20(5): 8337-8357. doi: 10.3934/mbe.2023365 |
[2] | Wenting Li, Ailing Jiao, Wei Liu, Zhaoying Guo . High-order rational-type solutions of the analogous (3+1)-dimensional Hirota-bilinear-like equation. Mathematical Biosciences and Engineering, 2023, 20(11): 19360-19371. doi: 10.3934/mbe.2023856 |
[3] | Ishtiaq Ali . Bernstein collocation method for neutral type functional differential equation. Mathematical Biosciences and Engineering, 2021, 18(3): 2764-2774. doi: 10.3934/mbe.2021140 |
[4] | Andras Balogh, Tamer Oraby . Stochastic games of parental vaccination decision making and bounded rationality. Mathematical Biosciences and Engineering, 2025, 22(2): 355-388. doi: 10.3934/mbe.2025014 |
[5] | Baojian Hong . Bifurcation analysis and exact solutions for a class of generalized time-space fractional nonlinear Schrödinger equations. Mathematical Biosciences and Engineering, 2023, 20(8): 14377-14394. doi: 10.3934/mbe.2023643 |
[6] | Bruno Buonomo, Alberto d’Onofrio, Deborah Lacitignola . Rational exemption to vaccination for non-fatal SIS diseases: Globally stable and oscillatory endemicity. Mathematical Biosciences and Engineering, 2010, 7(3): 561-578. doi: 10.3934/mbe.2010.7.561 |
[7] | Alessia Andò, Simone De Reggi, Davide Liessi, Francesca Scarabel . A pseudospectral method for investigating the stability of linear population models with two physiological structures. Mathematical Biosciences and Engineering, 2023, 20(3): 4493-4515. doi: 10.3934/mbe.2023208 |
[8] | Max-Olivier Hongler, Roger Filliger, Olivier Gallay . Local versus nonlocal barycentric interactions in 1D agent dynamics. Mathematical Biosciences and Engineering, 2014, 11(2): 303-315. doi: 10.3934/mbe.2014.11.303 |
[9] | Mario Lefebvre . An optimal control problem without control costs. Mathematical Biosciences and Engineering, 2023, 20(3): 5159-5168. doi: 10.3934/mbe.2023239 |
[10] | Dimitri Breda, Davide Liessi . A practical approach to computing Lyapunov exponents of renewal and delay equations. Mathematical Biosciences and Engineering, 2024, 21(1): 1249-1269. doi: 10.3934/mbe.2024053 |
The human brain is arguably the most complex information processing system. It operates by acquiring data from the environment, recognizing patterns of events’ occurrence, anticipating their re-occurrence and in turn generating appropriate behavioral responses. Through the lenses of the free-energy principle any self-organizing system that is at equilibrium with its environment must minimize its free energy either by manipulating the environmental sensory input or by manipulating its internal states thus altering the recognition density of the outside stimuli. However, several sets of challenges interfere with the human brain's ability to learn and adapt in such a theoretically optimal fashion. These may include, and are not limited to, functional inconsistencies related to attention and memory processes, the functions of “fast” and “slow” thinking and responding, and the ability of emotional states to generate unintended behavioral outcomes that are less adaptive or inappropriate. This paper will review literature on the subject of how ideal learning viewed from the free-energy principle perspective may be affected by the above mentioned limitations and will suggest a model of information processing that may have developed as a way of overcoming these challenges. This neurobiological model stipulates that a neuronal network is formed in response to environmental input and is paralleled by at least one and possibly multiple networks that activate intrinsically and represent “virtual responses” to a situation that demands a behavioral response. This model accounts for how the brain generates a multiplicity of potential behavioral responses and may “choose” the one that seems most appropriate and also explains the uncanny ability of humans to socialize and collaborate. Implications for understanding humans’ ability to learn from others, deliberate on opposing constructs and access and utilize information outside of individual minds are also discussed.
The propagation of electromagnetic waves in some materials is usually modeled by the famous Maxwell's equations [4] with various proper medium responses. These significant responses reflect the material's properties, such as the magnetic permeability and electric permittivity with respect to the location and the frequency of the propagating field. When it turns to high intensity radiation situation, not only the medium quantities may depend on the magnitude of the propagating field, but also the response will become nonlinear. In nonlinear optics, one may often focus on the propagation of monochromatic waves, such as continuous high intensity laser beams. In this case, some reasonable assumptions (see [3]) simplify the Maxwell's models to a nonlinear Helmholtz (NLH) equation [5,13]
ΔE(X)+ω20c2n2(X,|E|)E(X)=f(X),n2(X,|E|)=n20(X)+2n0(X)n2(X)|E|2, | (1) |
where
Let the Kerr medium be surrounded by the linear homogeneous medium in which
v(X)=[n0(X)next0]2,ε(X)=2n2(X)n0(X)(next0)2. |
Then, equation (1) can be rewritten as
ΔE(X)+k20[v(X)+ε(X)|E(X)|2]E(X)=f(X). | (2) |
Since the Kerr medium coefficient
v={1,z<0,vint ,0⩽z⩽Zmax,1,z>Zmax,ε={0,z<0,εint ,0⩽z⩽Zmax,0,z>Zmax. |
Remark 1. When the Kerr medium is inhomogeneous which means the whole Kerr medium will be cut into pieces by some other mediums, the linear medium for example. In this case, the discontinuities of the coefficients
When the electric field
d2Edz2+k20(v+ε|E|2)E=f,z∈(0,Zmax). | (3) |
To solve (3) in the interval
E(0+)=E(0−),dEdz(0+)=dEdz(0−),E(Z+max)=E(Z−max),dEdz(Z+max)=dEdz(Z−max). |
According to this fact, the so-called two-way boundary conditions which are also used in [3,14,15] are as follows
(ddz+ik0)E|z=0=2ik0,(ddz−ik0)E|z=Zmax=0, | (4) |
where
Remark 2. According to [3], the above boundary condition (4) is developed from the inhomogeneous Sommerfeld type relation and its complete form is
(ddz+ik0)E|z=0=2ik0E0inc,(ddz−ik0)E|z=Zmax=−2ik0EZmaxinc, |
where
In 2D case, assuming the linear homogeneous medium in
ΔE+k20(v+ε|E|2)E=f,(x,y)∈Ω:=(ax,bx)×(ay,by),∂E∂n+ik0E=g,(x,y)∈Γ:=Γ1∪Γ2∪Γ3∪Γ4, | (5) |
where
Many researches are done for the NLH equation. Through transforming it into a phase-amplitude equation, Chen and Mills proposed an approach to obtain the closed form solution of the NLH equation with a single nonlinear layer [7] and multilayered structures [8]. And by using the multidimensional generalization of the nonlinear Schrödinger equation, the exact solution of the NLH equation in special case was also considered in [18]. Moreover, in [11,12], the existence and asymptotic behavior of the real-valued standing wave solution of the NLH equations were analyzed. On the other hand, numerical methods for the NLH equations are also investigated. Fibich and the collaborators studied the NLH equation in [1,2,3,14,15]: In [14], the authors constructed a two-way artificial boundary condition for the NLH equation to ensure that not only the backscattered waves generate no reflection but also the correct value of the incoming wave can be imposed; the NLH equation was also solved by using nonorthogonal expansions in [15]; by coupling with a new technique of the separation of variables, a fourth order finite difference scheme was developed in [1], and the algorithm was also extended to the three-dimensional axially symmetric problem; in [2,3], the authors solved the NLH equations by developing an efficient Newton's iteration method to deal with the strong nonlinearity. In addition, a finite element method which can approximate the discontinuous coefficient problem is constructed in [22,23]. Recently, Wu and Zou proved the existence and uniqueness of the NLH equation, and also analyzed the stability and the error estimate with explicit wave numbers for the finite element approximation in [29]. And the author proposed a robust modified Newton's method in [31].
There are many difficulties when the NLH equation is approximated by using numerical schemes. Firstly, since the NLH equation is a strong nonlinear problem, we need to search a robust iteration method for solving it. Secondly, similar to the Helmholtz equation, the solution of this problem is highly oscillating with a high number (see [19]). Moreover, it usually contains discontinuous coefficients due to the different propagating mediums. There are also other issues to be solved, such as the strong indefinite linear system generated from this equation and so on. But, in this paper, we mainly focus on the case that it admits highly oscillated solutions. It is well-known that, the FDM is one of the most frequently used numerical methods due to its simple structure. Usually, the FDM is constructed through approximating the derivative terms in the original equation with some difference quotients which are obtained by the Taylor's expansion directly [21,32,33], this may lead the FDM to suffer from some disadvantages, low computation accuracy for example. To simulate the Helmholtz equation with high wave numbers in one dimension, a new finite difference scheme was proposed in [25]. Being different from the classical one directly based on the Taylor expansion, the new finite difference method is constructed by a rearranged formula which can contain more regularity information of the solution, and thus more accurate approximations were achieved. Recently, this kind of schemes were applied to the higher dimensional problems in [16,17,26,27,28]. But all of these problems investigated above are linear problems. In this article, we will extend this idea to solve the NLH equation. Through some iteration methods, the NLH equation will be linearized as a linear one at each iterative step firstly. Several iteration methods are considered, including the classical ones and the error correction iteration method [24] in which the original iteration solution was modified by a residual. Then, based on the above resulted linear problem, the new finite difference scheme is constructed, which is naturally suitable for the problem with discontinuous coefficients to match the different propagating mediums (Kerr and linear mediums).
The rest of this paper is organized as follows. In the next section, we apply several iteration methods including the error correction one to linearize the NLH equations. Then, in Section 3, after constructing the new finite difference scheme for the 1D linearized equation, we extend the scheme to the 2D problem by using the ADI technique. To test the efficiency of the numerical schemes, some numerical experiments are performed in Section 4. We finally make conclusions in Section 5.
To solve the nonlinear equation, an iteration method is needed. In this section, we introduce several kinds of iteration methods for solving the NLH equation. For convenience, we rewrite the problem as,
LE+k20ε|E|2E=f, | (6) |
where
Frozen-nonlinearity iteration may be the simplest iteration method in which the nonlinear term is frozen as a known quantity. For example, by replacing
LEl+1+k20ε|El|2El+1=f, | (7) |
where
λ(E,¯E):=LE+k20εE2¯E−f=0, | (8) |
then there hold,
∂λ∂E(E,¯E)=L+2k20εE¯E,∂λ∂¯E(E,¯E)=k20εE2, | (9) |
where
λ(El,¯El)+∂λ∂E(El,¯El)s+∂λ∂¯E(El,¯El)ˉs=0. | (10) |
Letting
LEl+1+2k20ε|El|2El+1+k20ε(El)2¯El+1=f+2k20ε|El|2El. | (11) |
Furthermore, a modified Newton's method was proposed in [31] by replacing
LEl+1+2k20ε|El|2El+1=f+k20ε|El|2El. | (12) |
In this paper, we also employ the error correction method in [24] for solving the nonlinear Helmholtz equation. Next, we will show the process. For simplicity, the above three iteration methods can be rewritten as a general formula
L˜El+1+N˜El+1+M¯˜El+1=˜f, | (13) |
where
N:={k20ε|El|2,in (7),2k20ε|El|2,in (11),2k20ε|El|2,in (12),M:={0,in (7),k20ε(El)2,in (11),0,in (12),˜f:={f,in (7),f+2k20ε|El|2El,in (11),f+k20ε|El|2El,in (12). |
Assuming that
Lμl+1+k20ε|E|2E−N˜El+1−M¯˜El+1=f−˜f. | (14) |
Since
|E|2E=|˜El+1+μl+1|2(˜El+1+μl+1)=(2|˜El+1|2+|μl+1|2)μl+1+(˜El+1)2¯μl+1+|˜El+1|2˜El+1+2|μl+1|2˜El+1+(μl+1)2¯˜El+1, |
freezing the terms
Lμl+1+Pμl+1+Q¯μl+1=fμ, | (15) |
where
P=2k20ε|˜El+1|2+k20ε|μl|2,Q=k20ε(˜El+1)2,fμ=f−˜f−(2k20ε|μl|2+k20ε|˜El+1|2−N)˜El+1−[k20ε(μl)2−M]¯˜El+1. |
Furthermore, for the error on the boundary, there holds
∂μl+1∂n+ik0μl+1=0. | (16) |
Obviously, by solving (15), the original solution
Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
After applying the iteration methods introduced in the above section, the NLH equation is linearized to linear problems at each iterative step. To implement these methods, a spatial discretization scheme is needed. From the stability analysis in [29], we know that the solution of the NLH equation satisfying
Recalling the iteration formula for the 1D NLH equation
d2˜El+1dz2+(k20v+N)˜El+1+M¯˜El+1=˜f,z∈(0,Zmax), | (17) |
(ddz+ik0)˜El+1|z=0=2ik0,(ddz−ik0)˜El+1|z=Zmax=0. | (18) |
When the frozen-nonlinearity iteration (7) and the modified Newton's method (12) are used, i.e.,
d2˜El+1dz2=τ˜El+1+˜f, | (19) |
where
Since the parameters
τm={τ−m,zm∈[zm−1,zm],τ+m,zm∈[zm,zm+1],˜fm={˜f−m,zm∈[zm−1,zm],˜f+m,zm∈[zm,zm+1]. |
Taylor's expansion tells that
˜El+1m+1=˜El+1m+⋯+(h+m)2k(2k)!(˜El+1m)(2k)+(h+m)2k+1(2k+1)!(˜El+1m)(2k+1)+⋯, | (20) |
˜El+1m−1=˜El+1m+⋯+(h−m)2k(2k)!(˜El+1m)(2k)−(h−m)2k+1(2k+1)!(˜El+1m)(2k+1)+⋯. | (21) |
And according to (19), there hold
(˜El+1m)(2k)=τnm˜El+1m+k∑s=1τk−sm˜f(2s−2)m, | (22) |
(˜El+1m)(2k+1)=τnm(˜El+1m)(1)+k∑s=1τk−sm˜f(2s−1)m. | (23) |
Substituting (22)-(23) into (20) and (21), it yields
˜El+1m+1=G+m˜El+1m+H+m(˜El+1m)(1)++∞∑k=1[L+k;m(˜f+m)(2k−2)+X+k;m(˜f+m)(2k−1)], | (24) |
˜El+1m−1=G−m˜El+1m−H−m(˜El+1m)(1)++∞∑k=1[L−k;m(˜f−m)(2k−2)−X−k;m(˜f−m)(2k−1)], | (25) |
where
G±m=G(τ±m,h±m),H±m=H(τ±m,h±m),L±k;m=Lk(τ±m,h±m),X±k;m=Xk(τ±m,h±m), |
with
G(ρ,h):=eh√ρ+e−h√ρ2,H(ρ,h):=1√ρeh√ρ−e−h√ρ2,Lk(ρ,h):=1ρk[eh√ρ+e−h√ρ2−k−1∑s=0(h√ρ)2s(2s)!],Xk(ρ,h):=1ρk+1/2[eh√ρ−e−h√ρ2−k−1∑s=0(h√ρ)2s+1(2s+1)!]. |
Then, eliminating
H+m˜El+1m−1−(H+mG−m+H−mG+m)˜El+1m+H−m˜El+1m+1=+H−m+∞∑k=1[L+k;m(˜f+m)(2k−2)+X+k;m(˜f+m)(2k−1)]+H+m+∞∑k=1[L−k;m(˜f−m)(2k−2)−X−k;m(˜f−m)(2k−1)]. | (26) |
For the boundary points
˜El+12=G+1˜El+11+H+1(˜El+11)(1)++∞∑k=1[L+k;1(˜f+1)(2k−2)+X+k;1(˜f+1)(2k−1)],˜El+1N−1=G−N˜El+1N−H−N(˜El+1N)(1)++∞∑k=1[L−k;N(˜f−N)(2k−2)−X−k;N(˜f−N)(2k−1)]. |
Then, substituting the boundary condition (18) into the above formulas, we have the numerical schemes for
−(G+1−ik0H+1)˜El+11+˜El+12=2ik0H+1++∞∑k=1[L+k;1(˜f+1)(2k−2)+X+k;1(˜f+1)(2k−1)], | (27) |
˜El+1N−1−(G−N−ik0H−N)˜El+1N=+∞∑k=1[L−k;N(˜f−N)(2k−2)−X−k;N(˜f−N)(2k−1)]. | (28) |
Obviously, taking different
H+m˜El+1m−1−(H+mG−m+H−mG+m)˜El+1m+H−m˜El+1m+1=+H−m[L+1;m˜f+m+2X+1;mh−m+h+m(˜f+m+1−˜f+m−1)]+H+m[L−1;m˜f−m−2X−1;mh−m+h+m(˜f−m+1−˜f−m−1)],m=2,3,⋯,N−1,−(G+1−ik0H+1)˜El+11+˜El+12=2ik0H+1+L+1;1˜f+1+X+1;1h+1(˜f+2−˜f+1), | (29) |
˜El+1N−1−(G−N−ik0H−N)˜El+1N=L−1;N˜f−N−X−1;Nh−N(˜f−N−˜f−N−1). |
Remark 3. In fact, a more accurate numerical scheme could be developed. For example, when the frozen-nonlinearity iteration method is used, (19) is the equation with variable coefficient, it yields,
d2˜El+1dz2=τ(z)˜El+1+˜f, |
where
In this case, like (22)-(23), we can get more precise formulas
(˜El+1m)(2n)=τnm˜El+1m+τn−1m˜fm,(˜El+1m)(2n+1)=τnm(˜El+1m)(1)+(C01+C23+⋯+C2n−22n−1)τn−1mτ(1)m˜El+1m+τn−1m˜f(1)m, |
where
Then, according to the Taylor's series, we get
˜El+1m=A+m˜Em+B+m˜E(1)m+C+m˜fm+D+m˜f(1)m,˜El−1m=A−m˜Em+B−m˜E(1)m+C−m˜fm+D−m˜f(1)m, |
where
A±m=A(τ±m,h±m),B±m=−B(τ±m,h±m),C±m=C(τ±m,h±m),D±m=−D(τ±m,h±m),A(ν,h)=12[e√νh+e−√νh]+ν(1)8{1(√ν)3[e√νh−e−√νh−2√νh]}+ν(1)8{−h(√ν)2[e√νh+e−√νh−2]+h2√ν[e√νh−e−√νh]},B(ν,h)=12√ν[e√νh−e−√νh],C(ν,h)=12ν[e√νh+e−√νh],D(ν,h)=12ν√ν[e√νh−e−√νh]. |
Thus, eliminating the terms
−1B−m˜Em−1−(A+mB+m−A−mB−m)˜Em+1B+m˜Em+1=(C+mB+m−C−mB−m)˜fm+(D+mB+m−D−mB−m)˜f(1)m. | (30) |
Finally, by approximating
When the Newton's iteration method (11) is considered, (17) needs to be separated into the real and imaginary parts due to the existence of
d2Rdz2=ˆRR+ˆII+˜fR, | (31) |
(dRdz−k0I)|z=0=0,(dRdz+k0I)|z=Zmax=0, | (32) |
d2Idz2=⌢II+⌢RR+˜fI, | (33) |
(dIdz+k0R)|z=0=2k0,(dIdz−k0R)|z=Zmax=0, | (34) |
where
R=real(˜El+1),I=imag(˜El+1),˜fR=real(˜f),˜fI=imag(˜f),ˆR=−real(k20v+N+M),ˆI=−imag(M−k20v−N),⌢I=−real(k20v+N−M),⌢R=−imag(M+k20v+N). |
Taking the real part equation (31) for an example, following the same process for (19), we have, at any interior point
R(2k)m=ˆRkmRm+ˆImk∑s=1ˆRk−smI(2s−2)m+k∑s=1ˆRk−sm˜f(2s−2)R;m, | (35) |
R(2k+1)m=ˆRkmR(1)m+ˆImk∑s=1ˆRk−smI(2s−1)m+k∑s=1ˆRk−sm˜f(2s−1)R;m, | (36) |
and
Rm+1=G+mRm+H+mR(1)m++∞∑k=1[L+k;m(ˆI+mIm+˜f+R;m)(2k−2)+X+k;m(ˆI+mIm+˜f+R;m)(2k−1)], | (37) |
Rm−1=G−mRm−H−mR(1)m++∞∑k=1[L−k;m(ˆI−mIm+˜f−R;m)(2k−2)−X−k;m(ˆI−mIm+˜f−R;m)(2k−1)], | (38) |
where
G±m=G(ˆR±m,h±m),H±m=H(ˆR±m,h±m),L±k;m=Lk(ˆR±m,h±m),X±k;m=Xk(ˆR±m,h±m). |
Eliminating
H+mRm−1−(H+mG−+H−mG+)Rm+H−mRm+1=+∞∑k=1(H−mF+k+H+mF−k), | (39) |
where
F+k=ˆI+m(L+k;mI(2k−2)m+X+k;mI(2k−1)m)+L+k;m(˜f+R;m)(2k−2)+X+k;m(˜f+R;m)(2k−1)F−k=ˆI−m(L−k;mI(2k−2)m−X−k;mI(2k−1)m)+L−k;m(˜f−R;m)(2k−2)−X+k;m(˜f−R;m)(2k−1). |
Similarly, setting
−G+1R1+R2−k0H+1I1=+∞∑k=1[ˆI+1(L+k;1I(2k−2)1+X+k;1I(2k−1)1)+L+k;1(˜f+R;1)(2k−2)+X+k;1(˜f+R;1)(2k−1)], | (40) |
RN−1−G−NRN−k0H−NIN=+∞∑k=1[ˆI−N(L−k;NI(2k−2)N−X−k;NI(2k−1)N)+L−k;N(˜f−R;N)(2k−2)−X+k;N(˜f−R;N)(2k−1)]. | (41) |
Obviously, by retaining different terms in the right hand side of (39)-(41), we can also get a series of finite difference schemes for the real part equation (31)-(32). For example, taking
H+mRm−1−(H+mG−+H−mG+)Rm+H−mRm+1=(H−mˆI+mL+1;m+H+mˆI−mL−1;m)Im+(H−mˆI+mX+1;m−H+mˆI−mX−1;m)I(1)m+H−mL+1;m(˜f+R;m)+H+mL−1;m(˜f−R;m)H−mX+1;m(˜f+R;m)(1)−H+mX−1;m(˜f−R;m)(1). |
By approximating
A⋅[R,I]=B⋅FR, | (42) |
where
A=(A1,A2,A3,A4,A5,A6),R=(Rm−1,Rm,Rm+1),I=(Im−1,Im,Im+1),B=(B1,B2,B3,B4,B5,B6),FR=(˜f+R;m−1,˜f+R;m,˜f+R;m+1,˜f−R;m−1,˜f−R;m,˜f−R;m+1), |
with
A1=H+m,A2=−H+mG−m−H−mG+m,A3=H−m,A4=−A6=H−mˆI+mX+1;m−H+mˆI−mX−1;mh+m+h−m,A5=−H−mˆI+mL+1;m−H+mˆI−mL−1;m,B1=−B3=−H−mX+1;mh+m+h−m,B2=H−mL+1;m,B4=−B6=H+mX−1;mh+m+h−m,B5=H+mL−1;m. |
Similarly, letting
−G+1R1+R2−k0H+1I1=ˆI+1(L+1;1I1+X+1;1I(1)1)+L+1;1˜f+R;1+X+1;1(˜f+R;1)(1),RN−1−G−NRN−k0H−NIN=ˆI−N(L−1;NIN−X−1;NI(1)N)+L−1;N˜f−R;N−X+1;N(˜f−R;N)(1). |
By using the boundary condition (34) and approximating
−(G+1−k0ˆI+1X−1;1)R1+R2−(k0H+1+L+1;1ˆI+1)I1=(L+1;1−X+1;1h+1)˜f+R,1+X+1;1h+1˜f+R,2+2k0ˆI+1X−1;1, | (43) |
RN−1−(G−N−k0X−1;NˆI−N)RN−(k0H−N+L−1;NˆI−N)IN=(L−1;N−X−1;Nh−N)˜f−R;N+X−1;Nh−N˜f−R,N−1. | (44) |
For the imaginary part (33)-(34), the same procedure can be applied to develop the new finite difference scheme at any interior point
A⋅[I,R]=B⋅FI, | (45) |
where
FI=(˜f+I;m−1,˜f+I;m,˜f+I;m+1,˜f−I;m−1,˜f−I;m,˜f−I;m+1),A1=T+m,A2=−T+mS−m−T−mS+m,A3=T−m,A4=−A6=T−m⌢R+mY+1;m−T+m⌢R−mY−1;mh+m+h−m,A5=−T−m⌢R+mM+1;m−T+m⌢R−mM−1;m,B1=−B3=−T−mY+1;mh+m+h−m,B2=T−mM+1;m,B4=−B6=T+mY−1;mh+m+h−m,B5=T+mM−1;m, |
with
S±m=G(⌢I±m,h±m),T±m=H(⌢I±m,h±m),M±1;m=L1(⌢I±m,h±m),Y±1;m=X1(⌢I±m,h±m). |
And the schemes for boundary points
−(S+1+k0⌢R+1Y+1;1)I1+I2−(M+1;1⌢R+1−k0T+1)R1=(M+1;1−Y+1;1h+1)˜f+I,1+Y+1;1h+1˜f+I,2+2k0T+1, | (46) |
IN−1−(S−N+k0Y−1;N⌢R−N)IN−(M−1;N⌢R−N−k0T−N)RN=(M−1;N−Y−1;Nh−N)˜f−I;N+Y−1;Nh−N˜f−I,N−1. | (47) |
Obviously, (42)-(47) constitute a finite difference scheme for the system of equations (31)-(34). According to the above process, it can be found that, through translating the high order terms
In this section, we extend the new finite difference scheme to the 2D problem by applying the ADI method [9,10]. Similar to (31)-(34), the 2D equation (5) need to be divided into real and imaginary parts like
ΔR=ˆRR+ˆII+˜fR,(x,y)∈Ω, | (48) |
−∂R∂y−k0I=g1R,(x,y)∈Γ1,∂R∂y−k0I=g2R,(x,y)∈Γ2,−∂R∂x−k0I=g3R,(x,y)∈Γ3,∂R∂x−k0I=g4R,(x,y)∈Γ4, | (49) |
ΔI=⌢II+⌢RR+˜fI,(x,y)∈Ω, | (50) |
−∂I∂y+k0R=g1I,(x,y)∈Γ1,−∂I∂y+k0R=g2I,(x,y)∈Γ2,−∂I∂x+k0R=g3I,(x,y)∈Γ3,∂I∂x+k0R=g4I,(x,y)∈Γ4, | (51) |
where
According to (5) and Fig. 2, the parameters
It is well-known that the ADI method is used to simulate a high-dimensional problem by solving a series of one-dimensional problems. Based on this, by directly separating the real part equation (48) into two 1D equations in
∂2Rm,n∂x2=γxˆRm,nRm,n+γxˆIm,nIm,n+˜fxR;m,n, | (52) |
∂2Rm,n∂y2=γyˆRm,nRm,n+γyˆIm,nIm,n+˜fyR;m,n, | (53) |
where
Similar to (42), the new finite difference schemes for (52) and (53) can be directly got as follows
H+xRm−1,n−(H+xG−x+H−xG+x)Rm,n+H−xRm+1,n−(H−xγxˆI+m,nL+x,1+H+xγxˆI−m,nL−x,1)Im,n−H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1h+m+h−m(Im+1,n−Im−1,n)=(H−xL+x,1+H+xL−x,1)˜fxR;m,n,H+yRm,n−1−(H+yG−y+H−yG+y)Rm,n+H−yRm,n+1−(H−yγyˆI+m,nL+y,1+H+yγyˆI−m,nL−y,1)Im,n | (54) |
−H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1k+n+k−n(Im,n+1−Im,n−1)=(H−yL+y;1+H+yL−y;1)˜fyR;m,n, | (55) |
where
G±x=G(γxˆR±m,n,h±m),H±x=H(γxˆR±m,n,h±m),L±x,1=L1(γxˆR±m,n,h±m),X±x,1=X1(γxˆR±m,n,h±m),G±y=G(γyˆR±m,n,k±n),H±y=H(γyˆR±m,n,k±n),L±y;1=L1(γyˆR±m,n,k±n),X±y;1=X1(γyˆR±m,n,k±n). |
Combining (54) and (55), we get the new finite difference scheme for (48) at the interior point
A⋅[R,I]=˜fR;m,n, | (56) |
where
A=(A1,A2,A3,A4,A5,A6,A7,A8,A9,A10),R=(Rm,n−1,Rm−1,n,Rm,n,Rm+1,n,Rm,n+1),I=(Im,n−1,Im−1,n,Im,n,Im+1,n,Im,n+1), |
with
A1=H+yH−yL+y;1+H+yL−y;1,A2=H+xH−xL+x,1+H+xL−x,1,A3=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−H+yG−y+H−yG+yH−yL+y;1+H+yL−y;1,A4=H−xH−xL+x,1+H+xL−x,1,A5=H−yH−yL+y;1+H+yL−y;1,A6=H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1),A7=H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A8=−H−xγxˆI+m,nL+x,1+H+xγxˆI−m,nL−x,1H−xL+x,1+H+xL−x,1−H−yγyˆI+m,nL+y,1+H+yγyˆI−m,nL−y,1H−yL+y;1+H+yL−y;1,A9=−H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A10=−H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1). |
Similar to the interior points, the new finite difference scheme for each boundary point is also constructed by developing two schemes in
Rm,2=G+yRm,1+H+y∂R(1)m,1∂y+L+y,1˜fyR;m,1+γyˆI+m,1(L+y;1Im,1+X+y,1∂I(1)m,1∂y). |
Then, substituting the corresponding boundary condition into the above formula, we have
−(G+y+k0γyˆI+m,1X+y,1)Rm,1+Rm,2+(k0H+y−γyˆI+m,1L+y,1)Im,1=L+y;1˜fyR;m,1−H+yg1R;m,1−γyˆI+m,1X+y;1g1I;m,1. | (57) |
So, combining (54)(
Ab⋅Rb=Fb, | (58) |
where
Rb=(Rm−1,1,Rm,1,Rm+1,1,Im−1,1,Im,1,Im+1,1,Rm,2),Ab=(A1,A2,A3,A4,A5,A6,A7),A1=H+xH−xL+x,1+H+xL−x,1,A3=H−xH−xL+x,1+H+xL−x,1,A2=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−G+y+k0γyˆI+m,1X+y,1L+y;1,A4=H−xγxˆI+m,1X+x,1+H+xγxˆI−m,1X−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A5=k0H+y−γyˆI+m,1L+y,1L+y;1−H−xγxˆI+m,1L+x,1+H+xγxˆI−m,1L−x,1H−xL+x,1+H+xL−x,1,A6=−H−xγxˆI+m,1X+x,1+H+xγxˆI−m,1X−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A7=1L+y;1,Fb=˜fR;m,1−H+yL+y;1g1R;m,1−γyˆI+m,1X+y;1L+y;1g1I;m,1. |
Similar to (57), on other three boundaries (excluding the vertexes), we also have: for the boundary points on
Rm,Ny−1−(G−y+k0γyˆI−m,NyX−y;1)Rm,Ny+(k0H−y−γyˆI−m,NyL−y;1)Im,Ny=L−y;1˜fyR;m,Ny−H−yg2R;m,Ny−γyˆI−m,NyX−y;1g2I;m,Ny, | (59) |
for the boundary points on
−(G+x+k0γxˆI+1,nX+x;1)R1,n+R2,n+(k0H+x−γxˆI+1,nL+x,1)I1,n=L+x;1˜fxR;1,n−H+xg3R;1,n−γxˆI+1,nX+x;1g3I;1,n, | (60) |
for the boundary points on
RNx−1,n−(G−x+k0γxˆINx,nX−x;1)RNx,n+(k0H−x−γxˆINx,nL−x,1)INx,n=L−x,1˜fxR;Nx,n−H−xg4R;Nx,n−γxˆI−Nx,nX−x;1g4I;Nx,n. | (61) |
Thus, applying the same process, the new finite difference schemes for these boundary points can also be written as (58) with different
Rb=(Rm−1,Ny,Rm,Ny,Rm+1,Ny,Im−1,Ny,Im,Ny,Im+1,Ny,Rm,Ny−1),A1=H+xH−xL+x,1+H+xL−x,1,A3=H−xH−xL+x,1+H+xL−x,1,A2=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−G−y+k0γyˆI−m,NyX−y;1L−y;1,A4=H−xγxˆI+m,NyX+x,1+H+xγxˆI−m,NyX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A5=k0H−y−γyˆI−m,NyL−y;1L−y;1−H−xγxˆI+m,NyL+x,1+H+xγxˆI−m,NyL−x,1H−xL+x,1+H+xL−x,1,A6=−H−xγxˆI+m,NyX+x,1+H+xγxˆI−m,NyX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A7=1L−y;1,Fb=˜fR;m,Ny−H−yg2R;m,NyL−y;1−γyˆI−m,NyX−y;1g2I;m,NyL−y;1, |
and for the points on
Rb=(R1,n−1,R1,n,R1,n+1,I1,n−1,I1,n,I1,n+1,R2,n),A1=H+yH−yL+y;1+H+yL−y;1,A3=H−yH−yL+y;1+H+yL−y;1,A2=−G+x+k0γxˆI+1,nX+x;1L+x;1−H+yG−y+H−yG+yH−yL+y;1+H+yL−y;1,A4=H−yγyˆI+1,nX+y,1+H+yγyˆI−1,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1),A5=k0H+x−γxˆI+1,nL+x,1L+x;1−H−yγyˆI+1,nL+y,1+H+yγyˆI−1,nL−y,1H−yL+y;1+H+yL−y;1,A6=−H−yγyˆI+1,nX+y,1+H+yγyˆI−1,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1),A7=1L+x;1,Fb=˜fR;1,n−H+xg3R;1,nL+x;1−γxˆI+1,nX+x;1g3I;1,nL+x;1, |
and for the points on
\begin{align*} &\mathbf{R_b} = (R_{N_x,n-1},R_{N_x,n},R_{N_x,n+1},I_{N_x,n-1},I_{N_x,n},I_{N_x,n+1},R_{N_x-1,n}),\\ &A_1 = \frac{H_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\quad A_3 = \frac{H_{y}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\\ &A_2 = -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n} X_{x;1}^{-}}{L_{x, 1}^{-}} -\frac{H_{y}^{+} G_{y}^{-}+H_{y}^{-} G_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} , \\ &A_4 = \frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})}, \end{align*} |
\begin{align*} &A_5 = \frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n} L_{x, 1}^{-}}{L_{x, 1}^{-}} -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} L_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} L_{y, 1}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}},\\ &A_6 = -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})}, \\ &A_7 = \frac{1}{L_{x, 1}^{-}},\quad F_b = \widetilde{f}_{R ; N_{x}, n}-\frac{H_{x}^{-} g_{4 R ; N_{x}, n}}{L_{x, 1}^{-}} -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, n}}{L_{x, 1}^{-}}. \end{align*} |
According to (57), (59)-(61), the new finite schemes for four vertexes can also be obtained. For example setting
\begin{align} \mathbf{A_v\cdot R_v} = F_v, \end{align} | (62) |
where
\begin{align*} &\mathbf{A_v} = (A_1,A_2,A_3,A_4),\quad \mathbf{R_v} = (R_{1,1},R_{2,1},R_{1,2},I_{1,1}),\\ &A_1 = -\frac{G_{y}^{+}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} X_{y, 1}^{+}}{L_{y ; 1}^{+} } -\frac{G_{x}^{+}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} X_{x;1}^{+}}{L_{x ; 1}^{+} },\\ &A_2 = \frac{1}{L_{x ; 1}^{+} },\quad A_3 = \frac{1}{L_{y ; 1}^{+} },\quad A_4 = \frac{k_{0} H_{y}^{+}-\gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} L_{y, 1}^{+}}{L_{y ; 1}^{+} } +\frac{k_{0} H_{x}^{+}-\gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} L_{x, 1}^{+}}{L_{x ; 1}^{+} },\\ &F_v = \widetilde{f}_{R ; 1, 1}-\frac{H_{y}^{+} }{L_{y ; 1}^{+} }g_{1 R ; 1, 1} -\frac{\gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} X_{y ; 1}^{+}}{L_{y ; 1}^{+} } g_{1 I ; 1, 1} -\frac{H_{x}^{+} g_{3 R ; 1, 1}}{L_{x ; 1}^{+} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} X_{x ; 1}^{+} g_{3 I ; 1, 1}}{L_{x ; 1}^{+} }. \end{align*} |
Similarly, for the rest three vertexes, their new finite difference schemes can be also concluded in (62) with different
\begin{align*} &\mathbf{R_v} = (R_{N_x-1,1},R_{N_x,1},R_{N_x,2},I_{N_x,1}),\\ &A_1 = \frac{1}{L_{x, 1}^{-} },\quad A_3 = \frac{1}{L_{y ; 1}^{+} },\\ &A_2 = -\frac{G_{y}^{+}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} X_{y, 1}^{+}}{L_{y ; 1}^{+} } -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1} X_{x;1}^{-}}{L_{x, 1}^{-} } ,\\ &A_4 = \frac{k_{0} H_{y}^{+}-\gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} L_{y, 1}^{+}}{L_{y ; 1}^{+} } +\frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1} L_{x, 1}^{-}}{L_{x, 1}^{-} } ,\\ &F_v = \widetilde{f}_{R ; N_x, 1}-\frac{H_{y}^{+} }{L_{y ; 1}^{+} }g_{1 R ; N_x, 1} -\frac{\gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} X_{y ; 1}^{+}}{L_{y ; 1}^{+} } g_{1 I ; N_x, 1}\\ &-\frac{H_{x}^{-} g_{4 R ; N_{x}, 1}}{L_{x, 1}^{-} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, 1}}{L_{x, 1}^{-} }, \end{align*} |
and for the vertex
\begin{align*} &\mathbf{R_v} = (R_{1,N_y-1},R_{1,N_y},R_{2,N_y},I_{1,N_y}),\\ &A_1 = \frac{1}{L_{y ; 1}^{-} },\quad A_3 = \frac{1}{L_{x ; 1}^{+} }, \end{align*} |
\begin{align*} &A_2 = -\frac{G_{y}^{-}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} X_{y;1}^{-}}{L_{y ; 1}^{-} } -\frac{G_{x}^{+}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} X_{x;1}^{+}}{L_{x ; 1}^{+} },\\ &A_4 = \frac{k_{0} H_{y}^{-}-\gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} L_{y ; 1}^{-}}{L_{y ; 1}^{-} } \frac{k_{0} H_{x}^{+}-\gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} L_{x, 1}^{+}}{L_{x ; 1}^{+} } ,\\ &F_v = \widetilde{f}_{R ; 1, N_{y}}-\frac{H_{y}^{-} g_{2 R ; 1, N_{y}}}{L_{y ; 1}^{-} } -\frac{\gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} X_{y ; 1}^{-} g_{2 I ; 1, N_{y}}}{L_{y ; 1}^{-} }\\ &\qquad-\frac{H_{x}^{+} g_{3 R ; 1, N_y}}{L_{x ; 1}^{+} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} X_{x ; 1}^{+} g_{3 I ; 1, N_y}}{L_{x ; 1}^{+} }, \end{align*} |
and for the vertex
\begin{align*} &\mathbf{R_v} = (R_{N_x,N_y-1},R_{N_x-1,N_y},R_{N_x,N_y},I_{N_x,N_y}),\\ &A_1 = \frac{1}{L_{y ; 1}^{-} },\quad A_2 = \frac{1}{L_{x, 1}^{-} },\\ &A_3 = -\frac{G_{y}^{-}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} X_{y;1}^{-}}{L_{y ; 1}^{-} } -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y} X_{x;1}^{-}}{L_{x, 1}^{-} } ,\\ &A_4 = \frac{k_{0} H_{y}^{-}-\gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} L_{y ; 1}^{-}}{L_{y ; 1}^{-} } +\frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y} L_{x, 1}^{-}}{L_{x, 1}^{-} },\\ &F_v = \widetilde{f}_{R ; N_x, N_{y}}-\frac{H_{y}^{-} g_{2 R ; N_x, N_{y}}}{L_{y ; 1}^{-} } -\frac{\gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} X_{y ; 1}^{-} g_{2 I ; N_x, N_{y}}}{L_{y ; 1}^{-} }\\ &\qquad-\frac{H_{x}^{-} g_{4 R ; N_{x}, N_y}}{L_{x, 1}^{-} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, N_y}}{L_{x, 1}^{-} }. \end{align*} |
For the imaginary part (50)-(51), the new finite difference scheme can be produced in the same way. And the details are omitted here.
Remark 4. In fact, for the 2D equation, when the frozen-nonlinearity iteration and the modified Newton's method are used, it is not necessary to separate (5) into real and imaginary parts. In this case, the 2D equation can be divided into two 1D equations directly. Furthermore, since two 1D equations are separated from a 2D equation, we assume that
In this section, we will show some numerical tests to verify the efficiency of the scheme proposed in the above section. And we set
Firstly, let
\begin{align} E& = \lambda(z) e^{i \varphi(z)}, \end{align} | (63) |
where
\begin{align} \varphi(z)& = \varphi(0)+W \int_{0}^{z} \frac{1}{\beta(t)} d t, \end{align} | (64) |
\begin{align} \lambda(z)& = \sqrt{\beta(z)}, \end{align} | (65) |
with
\begin{align} \beta(z)& = \beta_{2}+(\beta_{1}-\beta_{2}) \operatorname{cn}^{2}\left[\sqrt{\frac{\varepsilon}{2v}}\left(\beta_{1}-\beta_{3}\right)^{1 / 2} k_0\sqrt{v}(Z_{\max}-z)\bigg | \frac{\beta_{1}-\beta_{2}}{\beta_{1}-\beta_{3}}\right], \end{align} | (66) |
\begin{align} \beta_{1}& = \frac{W}{k_{0}}, \end{align} | (67) |
\begin{align} \beta_{2}& = \frac{v}{\varepsilon}\left\{\left[\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)^{2}+\frac{2 \varepsilon W}{v^{2} k_{0}}\right]^{1 / 2}-\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)\right\}, \end{align} | (68) |
\begin{align} \beta_{3}& = -\frac{v}{\varepsilon}\left\{\left[\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)^{2}+\frac{2 \varepsilon W}{v^{2} k_{0}}\right]^{1 / 2}+\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)\right\}, \end{align} | (69) |
and
Moreover, at
\begin{align} \frac{1}{2} \varepsilon \beta^{2}(0)+\left(v-1\right) \beta(0)+4-\left(v+3\right) \frac{W}{k_{0}}-\frac{1}{2}\frac{ \varepsilon W^{2}}{k_{0}^{2}} = 0, \end{align} | (70) |
\begin{align} \frac{d\lambda}{d z}\bigg|_{z = 0} = 2 k_{0} \sin \varphi(0), \end{align} | (71) |
with
\begin{align} \left(\frac{d \lambda}{d z}\right)^{2}+\frac{W^{2}}{\lambda^{2}}+k_0^{2}v \lambda^{2}+\frac{1}{2} k_0^{2}\varepsilon \lambda^{4} = A. \end{align} | (72) |
Thus, putting (66) with
To test the accuracy of the proposed scheme, with
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
Then, under the same computational environments, but with
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
Furthermore, we simulate the optical bistability by using the proposed finite difference scheme. Firstly, letting the transmittance
Now, we turn to a 2D problem. Setting
\begin{align*} E = \frac{5 \sqrt{2} e^{\mathrm{i} y \sqrt{k_0^{2}+25}}}{\sqrt{\varepsilon} k_0 \cosh (5 x)}. \end{align*} |
In Fig. 7, we exhibit the exact solution and the numerical solution obtained by the new finite difference scheme with
To simulate the transmission and collision of the nonparaxial solitons which are also considered in [3,29], we solve the NLH equation (5) with two different incident waves
\begin{align*} E_{\text {inc }}^1& = \frac{20 \sqrt{2} e^{i y \sqrt{k_{0}^{2}+400}}}{\cosh (20 x)},\\ E_{\text {inc }}^2& = \frac{20 \sqrt{2} e^{i y \sqrt{k_{0}^{2}+400}}}{\cosh (20 x)}+\frac{20 \sqrt{2} e^{i \sqrt{k_{0}^{2}+400}(y / 2-\sqrt{3} x / 2)}}{\cosh [20(x / 2+\sqrt{3} y / 2)]}. \end{align*} |
And the source term is set as
\begin{align*} f = \left\{\begin{array}{c} -\Delta E_{\text {inc }}^l-k_{0}^{2} E_{\text {inc }}^l,(x, y) \in \Omega \backslash \Omega_{0}, \\ 0,\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, y) \in \Omega_{0}, \end{array}\right. l = 1,2. \end{align*} |
The intensities of the incident field and the total field for different cases are shown in Fig. 8 and Fig. 9. When only one incident wave
In this paper, we construct a kind of new finite difference schemes for solving the nonlinear Helmholtz equation based on some iteration methods. Numerical results indicate that, the proposed scheme not only can approximate the high oscillation solution with better computational accuracy, but also can be used to simulate some physical phenomenons in the Kerr medium, such as the optical bistability and the collision of nonparaxial solitons. Moreover, without any extra consideration, this finite difference scheme also provides a route to deal with the problems with discontinuous coefficients or source terms. Thus, it can be extended to much more complex cases, such as the multi-layered Kerr mediums propagating problem and the nonlinear Maxwell's equations.
[1] |
Friston K (2010) The free-energy principle: a unified brain theory? Nat Rev Neurosci 11: 127-138. doi: 10.1038/nrn2787
![]() |
[2] | Jarzynski C (2012) Nonequilibrium equality for free energy differences. Phys Rev Lett 78: 2690-2693. |
[3] | Landauer R (1996) Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J Res Dev 1: 223-231. |
[4] |
Sengupta B, Stemmler MB, Friston KJ (2013) Information and efficiency in the nervous system--a synthesis. PLoS Comput Biol 9: e1003157. doi: 10.1371/journal.pcbi.1003157
![]() |
[5] |
Simons DJ, Chabris CF (2011) What people believe about how memory works: a representative survey of the U.S. population. PLoS One 6: e22757. doi: 10.1371/journal.pone.0022757
![]() |
[6] |
Friston K, FitzGerald T, Rigoli F, et al. (2016) Active inference and learning. Neurosci Biobehav Rev 68: 862-879. doi: 10.1016/j.neubiorev.2016.06.022
![]() |
[7] |
Friston K, Rigoli F, Ognibele D, et al. (2015) Active inference and epistemic value. Cogn Neurosci 6: 187-214. doi: 10.1080/17588928.2015.1020053
![]() |
[8] |
Ognibene D, Chinellato E, Sarabia M, et al. (2013) Contextual action recognition and target localization with an active allocation of attention on a humanoid robot. Bioinspir Biomim 8: 035002. doi: 10.1088/1748-3182/8/3/035002
![]() |
[9] | Kahneman D (2011) Thinking, fast and slow. New York, NY. |
[10] |
Clark A, Chalmers DJ (1998) The extended mind. Analysis 58: 7-19. doi: 10.1093/analys/58.1.7
![]() |
[11] | Kahneman D, Frederick S (2002) Representativeness revisited: Attribute substitution in intuitive judgment. In Gilovich T, Griffin D, Kahneman, D (Eds), Heuristics and biases: The psychology of intuitive judgment, New York, NY: 49-81. |
[12] |
Kahneman D (2003) Maps of bounded rationality: Psychology for behavioral economics. American Economic Review 93: 1449-1475. doi: 10.1257/000282803322655392
![]() |
[13] |
Simon HA (1955) A behavioral model of rational choice. Quarterly Journal of Economics 69: 99-118. doi: 10.2307/1884852
![]() |
[14] | Lerner RG, Trigg GL (1991) Encyclopedia of Physics (2nd Edition). New York, NY: VCH Publishers. |
[15] | Parker CB (1994) McGraw-Hill Encyclopedia of Physics (2nd Edition). New York, NY: McGraw-Hill. |
[16] | Penrose R (2007) The Road to Reality: A Complete Guide to the Laws of the Universe. New York, NY: Vintage Books. |
[17] |
Panksepp J (2005) Affective consciousness: core emotional feelings in animals and humans. Conscious Cogn 14: 30-80. doi: 10.1016/j.concog.2004.10.004
![]() |
[18] | Boltzmann L (1877) U ̈ er die beziehung zwischen dem zweiten haupt- satz der mechanischen wa ̈rmetheorie und der wahrscheinlichkeitsrech- nung respektive den sa ̈tzen u ̈ber das wa ̈rmegleichgewicht. [On the relationship between the second law of the mechanical theory of heat and the probability calculus]. Wiener Berichte 76: 373-435. |
[19] | Wiener N (1961) Cybernetics-or control and communication in the animal and the machine. New York, NY. |
[20] |
Hirsh JB, Mar RA, Peterson JB (2012) Psychological entropy: A framework for understanding uncertainty-related anxiety. Psychol Rev 119: 304-320. doi: 10.1037/a0026767
![]() |
[21] |
Keysers C (2009) Mirror neurons. Curr Biol 19: 971-973. doi: 10.1016/j.cub.2009.08.026
![]() |
[22] | Molenberghs P, Cunnington R, Mattingley JB (2009) Is the mirror neuron system involved in imitation? A short review and meta-analysis. Neurosci Biobehav Rev 33: 975-980. |
[23] |
Rizzolatti G, Craighero L (2004) The mirror-neuron system. Ann Rev Neurosci 27: 169-192. doi: 10.1146/annurev.neuro.27.070203.144230
![]() |
[24] |
Filimon F, Rieth CA, Sereno MI, et al. (2015) Observed, executed, and imagined action representations can be decoded from ventral and dorsal areas. Cereb Cortex 25: 3144-3158. doi: 10.1093/cercor/bhu110
![]() |
[25] |
Trapp K, Spengler S, Wüstenberg T, et al. (2014) Imagining triadic interactions simultaneously activates mirror and mentalizing systems. Neuroimage 98: 314-323. doi: 10.1016/j.neuroimage.2014.05.003
![]() |
[26] | Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1: 1-47. |
[27] |
von Stein A, Sarnthein J (2000) Different frequencies for different scales of cortical integration: from local gamma to long range alpha/theta synchronization. Int J Psychophysiol 38: 301-313. doi: 10.1016/S0167-8760(00)00172-0
![]() |
[28] |
Réka Albert AB (2002) Statistical mechanics of complex networks. Rev Mod Phys 74: 47-97. doi: 10.1103/RevModPhys.74.47
![]() |
[29] | Amit DJ (1995) The Hebbian paradigm reintegrated: local reverberations as internal representations. Behav Brain Sci 18: 631. |
[30] |
Goldman-Rakic PS (1995) Cellular basis of working memory. Neuron 14: 477-485. doi: 10.1016/0896-6273(95)90304-6
![]() |
[31] |
Wang XJ (2001) Synaptic reverberation underlying mnemonic persistent activity. Trends Neurosci 24: 455-463. doi: 10.1016/S0166-2236(00)01868-3
![]() |
[32] |
Trantham-Davidson H, Neely LC, Lavin A, et al. (2004) Mechanisms underlying differential D1 versus D2 dopamine receptor regulation of inhibition in prefrontal cortex. J Neurosci 24: 10652-10659. doi: 10.1523/JNEUROSCI.3179-04.2004
![]() |
[33] |
Stephens GJ, Silbert LJ, Hasson U (2010) Speaker–listener neural coupling underlies successful communication. Proc Natl Acad Sci U S A 107: 14425-14430. doi: 10.1073/pnas.1008662107
![]() |
[34] |
Hasson U, Ghazanfa AA, Galantucci B, et al. (2012) Brain-to-brain coupling: a mechanism for creating and sharing a social world. Trends Cogn Sci 16: 114-121. doi: 10.1016/j.tics.2011.12.007
![]() |
[35] |
Benoit RG, Gilbert SJ, Frith CD, et al. (2012) Rostral prefrontal cortex and the focus of attention in prospective memory. Cereb Cortex 22: 1876-1886. doi: 10.1093/cercor/bhr264
![]() |
[36] |
Nader K, Schafe GE, Le Doux JE (2000) Fear memories require protein synthesis in the amygdala for reconsolidation after retrieval. Nature 406: 722-726. doi: 10.1038/35021052
![]() |
[37] |
Przybyslawski J, Sara SJ (1997) Reconsolidation of memory after its reactivation. Behav Brain Res 84: 241-246. doi: 10.1016/S0166-4328(96)00153-2
![]() |
[38] |
Sara SJ (2000) Retrieval and reconsolidation: toward a neurobiology of remembering. Learn Mem 7: 73-84. doi: 10.1101/lm.7.2.73
![]() |
[39] | Colarusso CA, Nemiroff RA (1981) Adult Development: A New Dimension in Psychodynamic Theory and Practice. New York, NY: Plenum. |
[40] |
Pennisi E (2010) Conquering by copying. Science 328: 165-167. doi: 10.1126/science.328.5975.165
![]() |
1. | Jin Li, Barycentric rational collocation method for semi-infinite domain problems, 2023, 8, 2473-6988, 8756, 10.3934/math.2023439 | |
2. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving KPP equation, 2023, 31, 2688-1594, 3014, 10.3934/era.2023152 | |
3. | Jin Li, Kaiyan Zhao, Xiaoning Su, Selma Gulyaz, Barycentric Interpolation Collocation Method for Solving Fractional Linear Fredholm-Volterra Integro-Differential Equation, 2023, 2023, 2314-8888, 1, 10.1155/2023/7918713 | |
4. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving fractional cable equation, 2023, 31, 2688-1594, 3649, 10.3934/era.2023185 | |
5. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving 3 dimensional convection–diffusion equation, 2024, 304, 00219045, 106106, 10.1016/j.jat.2024.106106 | |
6. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation, 2023, 31, 2688-1594, 4034, 10.3934/era.2023205 | |
7. | Jin Li, Yongling Cheng, Spectral collocation method for convection-diffusion equation, 2024, 57, 2391-4661, 10.1515/dema-2023-0110 | |
8. | Jin Li, Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation, 2023, 8, 2473-6988, 16494, 10.3934/math.2023843 |
Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |