Underwater Sensor Network Node Locator Based on Improved Whale Optimization Iterative Algorithm
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摘要:
针对水下无线传感器网络中锚节点较少、迭代误差大导致节点定位精度低的问题，文章提出了一种基于改进的鲸鱼优化-牛顿迭代的水下三维节点定位算法(Improved Whale Optimization-Newton Iteration，IWONI)。该算法首先使用牛顿迭代算法对节点距离远近关系建立对应法则，并利用目标位置估计值和修正因子为改进的鲸鱼优化算法提供动态搜索区域；其次，建立以测量误差为权重的适应度函数作为判断基准，采用改进的鲸鱼优化算法进行迭代求解，以获得最优解；最后，利用定位方程得到网络节点位置。为了验证IWONI算法的性能，将IWONI算法与时间差定位算法(TDOA-CHAN、TDOA-Taylor)、测距定位算法(最小二乘法、高斯牛顿迭代法)和牛顿迭代算法进行定位误差、收敛性能和定位成功率对比实验，并验证了节点数量对定位精度的影响。实验结果表明：(1)IWONI算法的定位误差和收敛速度明显优于其他对比算法。(2)IWONI算法在测量噪声大时的定位成功率高达92%，明显优于其他对比算法。(3)在通信半径不变的情况下，选择5~7个传感器节点可以在IWONI算法中实现定位精度与成本开销的平衡。
Abstract:To address the issues of low node localization accuracy caused by the limited number of anchor nodes and large iteration errors in underwater wireless sensor networks, an improved whale optimization-Newton iteration (IWONI) algorithm for underwater three-dimensional node localization was proposed. IWONI first uses the Newton iteration algorithm to establish a corresponding rule for the distance relationship between nodes, and utilizes the estimated target position and correction factor to provide a dynamic search area for the improved whale optimization algorithm. Secondly, a fitness function weighted by measurement error is established as the judgment criterion, and the improved whale optimization algorithm is used for iterative solution to obtain the optimal solution. Finally, the network node positions are calculated through the localization equation. To validate the performance of the IWONI algorithm, comparative experiments were conducted on localization error, convergence performance, and localization success rate against time difference of arrival algorithms (TDOA-Taylor, TDOA-CHAN), ranging algorithms (least squares method, Gauss-Newton iteration), and Newton iteration algorithm. The impact of the number of nodes on localization accuracy was also investigated. The comparison results show that: (1)The IWONI algorithm has significantly lower localization error and faster convergence speed than other compared algorithms. (2)The IWONI algorithm has a high localization success rate of 92% even in the presence of high measurement noise, which is significantly better than other compared algorithms. (3)In the case of a constant communication radius, employing 5 to 7 sensor nodes can achieve a balance between localization accuracy and cost effectiveness in the IWONI algorithm.
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分数阶导数具有非局部性, 相比整数阶导数，它能更精确地描述现实世界中具有记忆和遗传性质的问题.因此，分数阶微分方程已被广泛应用于众多领域，如半导体物理、弹性力学、生物学异速生长、金融学期权定价、信号处理和分形理论等[1-2].但分数阶导数的非局部性使得这类方程难以获得解析解, 或者解析解需要用特殊函数来表达, 这不利于实际应用.所以, 研究其数值解显得尤为重要.
本文研究如下时间分数阶次扩散方程的初边值问题:
{C0Dαtu(x,t)=μ∂xxu(x,t)+f(x,t)((x,t)∈[0,1]×(0,T]),u(x,0)=u0(x)(x∈[0,1]),u(0,t)=u(1,t)=0(t∈(0,T]), (1) 其中, 0 < α < 1, μ>0, 0CDtα是Caputo分数阶微分算子, 其定义为
C0Dαtu(x,t)=1Γ(1−α)∫t0(t−s)−α∂su(x,s)ds, 这里
Γ(·)表示Gamma函数.
时间分数阶次扩散方程是一类重要的数学模型，常被用于描述运输中的反常扩散行为，能很好地刻画长时记忆的流通过程.已有很多学者研究了有关时间分数阶次扩散方程的数值方法[3-12]，如：给出了一种隐式差分格式, 讨论了格式的收敛阶，证明了该格式无条件稳定[3]；研究了二维问题的一种高阶紧致差分格式[5]；给出了一种有限元方法，并证明了所得全离散格式具有超收敛性[4]；给出了一种时间方向具有二阶收敛阶的有限元计算格式，证明了该格式无条件稳定且空间方向具有最优收敛阶[6]；利用经典有限元方法讨论了时间分数阶对流扩散方程和电报方程的数值解[7-8]；给出了一种最小二乘谱方法[9]；运用时空谱方法讨论了方程(1)的数值解，并获得了先验误差估计[10].
但上述方法并未涉及全离散所得线性方程组的快速求解问题.事实上，对于大尺度、高精度的实际问题而言，全离散格式将导出一个大规模的线性方程组，而求解该方程组需要消耗巨大的计算量.
为了高效求解全离散所得线性方程组，本文研究了方程(1)的一种多尺度快速方法.首先，基于时间方向采用L1逼近和空间方向采用多尺度Galerkin逼近建立全离散格式，给出了全离散格式的误差估计；然后，利用矩阵分裂策略设计快速求解该全离散格式的多层扩充算法, 并证明了该算法具有最优收敛阶.
1. 预备知识
本节介绍一些相关记号和引理, 以方便后文引用.没有特别说明, 后文中出现的字母c表示与时间和空间步长无关的常数, 不同的位置可能表示不一样的常数.
记I=(a, b), (·, ·)和‖·‖0分别表示空间L2(I)上的内积和范数. H01(I)是满足零边界条件的函数构成的Sobolev空间, 配备内积及其相应的范数如下:
⟨u,v⟩1:=∫Iu′(x)v′(x)dx,‖ 设Xn是H01(I)的有限维子空间, 其元素是以a+\frac{b-a}{2^{n}}(j=1, 2, …, 2n-1)为节点的次数不超过r的分片多项式.则易知, 序列{Xn}是嵌套的, 即Xn⊂Xn+1(n \in \mathbb{N}).于是，Xn具有如下多尺度分解:
{X_n} = {X_{n - 1}}{ \oplus ^ \bot }{W_n} = {X_0}{ \oplus ^ \bot }{W_1}{ \oplus ^ \bot }{W_2}{ \oplus ^ \bot } \cdots { \oplus ^ \bot }{W_n}, 其中，Wn是Xn-1在Xn中的正交补空间, 此处正交指的是在内积〈·，·〉1意义下.定义正交投影算子(Ritz投影)\mathcal{P}n:H01(I)→Xn如下:
{\langle u - {\mathcal{P}_n}u,v\rangle _1} = ({\partial _x}(u - {\mathcal{P}_n}u),{\partial _x}v) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (u{\kern 1pt} \in {\kern 1pt} H_0^1(I),v{\kern 1pt} \in {\kern 1pt} {X_n}). 引理1[5] 设m, r, r≥1是正整数, u∈Hm(I)∩H01(I).如果1≤m≤r+1, 则
{\left\| {{\kern 1pt} u - {\mathcal{P}_n}u{\kern 1pt} } \right\|_p} \le c{2^{ - (m - p)n}}{\left\| {{\kern 1pt} u{\kern 1pt} } \right\|_m}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (p = 0,1). 2. L1-多尺度Galerkin全离散格式
本节给出时间分数阶次扩散方程(1)的L1-多尺度Galerkin全离散格式, 并推导其误差估计.
设τ为时间步长, 记tk=kτ(k=0, 1, …, NT; NTτ=T), uk:= u(·, tk).定义离散算子
D_\tau ^\alpha {u^k} = \frac{{{\tau ^{ - \alpha }}}}{{\Gamma (1 - \alpha )}}\left[ {{u^k} - \sum\limits_{j = 1}^{k - 1} {({a_{k - j - 1}} - {a_{k - j}})} {u^k} - {a_{k - 1}}{u^0}} \right], 其中, aj=(j+1)1-α-j1-α (j≥0).根据文献[2], 如果u∈C2([0, T]; L2(I)), 则用Dταuk逼近分数阶导数0CDαtku的截断误差为：
{\left\| {{\kern 1pt} _0^{\rm{C}}D_{{t_k}}^\alpha u - D_\tau ^\alpha {u^k}{\kern 1pt} } \right\|_0} \le c{\tau ^{2 - \alpha }}. (2) 取多尺度空间Xn作为空间方向的逼近子空间, 结合上述记号, 得到L1-多尺度Galerkin全离散格式：对每个时间层t=tk(k=1, 2, …, NT), 求unk∈Xn, 使得
\left\{ {\begin{array}{*{20}{l}} {(D_\tau ^\alpha u_n^k,{v_n}) + \mu ({\partial _x}u_n^k,{\partial _x}{v_n}) = ({f^k},{v_n}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({v_n}{\kern 1pt} \in {\kern 1pt} {X_n}),}\\ {u_n^0 = {\mathcal{P}_n}{u_0},} \end{array}} \right. (3) 其中, fk=f(x, tk).
定理1 对每个时间层t=tk (k=1, 2, …, NT), 全离散格式(3)都存在唯一解unk.如果问题(1)的解析解u∈C2([0, T]; Hr(I)∩H01(I)), 0CDtαu∈Hr-2(I), 且u0∈Hr(I), 则全离散格式具有如下误差估计
{\left\| {{\kern 1pt} {u^k} - u_n^k{\kern 1pt} } \right\|_0} \le c({2^{ - nr}} + {\tau ^{2 - \alpha }})\quad (1 \le k \le {N_T}). (4) 证明 先证解的存在唯一性.为此, 只需要考虑对应的齐次方程只有零解.根据离散导数Dτα的定义可知, 方程(3)对应的齐次方程为
(u_n^k,{v_n}) + \mu ({\partial _x}u_n^k,{\partial _x}{v_n}) = 0\quad ({v_n}{\kern 1pt} \in {\kern 1pt} {X_n}). 令vn=unk, 则
0 = (u_n^k,u_n^k) + \mu ({\partial _x}u_n^k,{\partial _x}u_n^k) = \left\| {{\kern 1pt} u_n^k{\kern 1pt} } \right\|_0^2 + \left\| {{\kern 1pt} \mu u_n^k{\kern 1pt} } \right\|_1^2, (5) 从而unk=0.解的存在唯一性获证.
接下来论证误差估计.由方程(1)可知, 真解uk满足方程
\begin{array}{l} (D_\tau ^\alpha {u^k},{v_n}) + \mu ({\partial _x}{u^k},{\partial _x}{v_n}) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({f^k},{v_n}) + (D_\tau ^\alpha {u^k} - _0^{\rm{C}}D_{{t_k}}^\alpha u,{v_n}). \end{array} (6) 令enk=\mathcal{P}nuk-unk (k=0, 1, 2, …, NT).将方程(6)减方程(3), 并取vn=enk，则enk满足方程
\begin{array}{l} (D_\tau ^\alpha e_n^k,e_n^k) + \mu ({\partial _x}e_n^k,{\partial _x}e_n^k) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (D_\tau ^\alpha ({\mathcal{P}_n}{u^k} - {u^k}),e_n^k) - (D_\tau ^\alpha {u^k} - _0^{\rm{C}}D_{{t_k}}^\alpha u,e_n^k). \end{array} (7) 由于1=a0>a1>a2>…>an>0, 则由Cauchy-Schwarz不等式、三角不等式及引理2, 有
\begin{array}{l} (D_\tau ^\alpha e_n^k,e_n^k) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\tau ^{ - \alpha }}}}{{\Gamma (2 - \alpha )}}\left( {e_n^k - \sum\limits_{i = 1}^{k - 1} {({a_{k - i - 1}} - {a_{k - i}})} e_n^i - {a_{k - 1}}e_n^0,e_n^k} \right) \ge \\ {\kern 1pt} \left. \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\tau ^{ - \alpha }}}}{{\Gamma (2 - \alpha )}}\left\| {e_n^k} \right\|_0^2 - \left( {\sum\limits_{i = 1}^{k - 1} {\left( {({a_{k - i - 1}} - {a_{k - i}})} \right. \times } } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\frac{{\left\| {e_n^i} \right\|_0^2 + \left\| {e_n^k} \right\|_0^2}}{2}} \right) - {a_{k - 1}}\frac{{\left\| {e_n^0} \right\|_0^2 + \left\| {e_n^k} \right\|_0^2}}{2} \end{array} \right) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{\tau ^{ - \alpha }}}}{{\Gamma (2 - \alpha )}}\left( {\left\| {e_n^k} \right\|_0^2 - \sum\limits_{i = 1}^{k - 1} {({a_{k - i - 1}} - {a_{k - i}})} \left\| {e_n^i} \right\|_0^2 - } \right.\\ {\kern 1pt} \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_{k - 1}}\left\| {e_n^0} \right\|_0^2} \right), \end{array} \begin{array}{l} \begin{array}{*{20}{l}} {(D_\tau ^\alpha ({\mathcal{P}_n}{u^k} - {u^k}),e_n^k)| = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {(D_\tau ^\alpha {\mathcal{P}_n}{u^k} - _0^{\rm{C}}D_{{t_k}}^\alpha {\mathcal{P}_n}u + _0^{\rm{C}}D_{{t_k}}^\alpha {\mathcal{P}_n}u - _0^{\rm{C}}D_{{t_k}}^\alpha u + _0^{\rm{C}}D_{{t_k}}^\alpha u - } \right.}\\ {\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D_\tau ^\alpha {u^k},e_n^k)} \right| \le ({{\left\| {D_\tau ^\alpha {\mathcal{P}_n}{u^k} - _0^{\rm{C}}D_{{t_k}}^\alpha {\mathcal{P}_n}u} \right\|}_0} + } \end{array}\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left\| {_0^{\rm{C}}D_{{t_k}}^\alpha {\mathcal{P}_n}u - _0^{\rm{C}}D_{{t_k}}^\alpha u} \right\|}_0} + {{\left\| {_0^{\rm{C}}D_{{t_k}}^\alpha u - D_\tau ^\alpha {u^k}} \right\|}_0}){{\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|}_0} \le }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (c{\tau ^{2 - \alpha }} + c{2^{ - nr}}){{\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|}_0} \le {{(c{\tau ^{2 - \alpha }} + c{2^{ - nr}})}^2} + \frac{1}{2}\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|_0^2.} \end{array} \end{array} 再注意到式(7)左端第二项非负, 则有
\begin{array}{*{20}{l}} {\left( {\frac{{{\tau ^{2 - \alpha }}}}{{\Gamma (2 - \alpha )}} - \frac{1}{2}} \right)\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|_0^2 \le }\\ {\quad c{{({\tau ^{2 - \alpha }} + {2^{ - nr}})}^2} + \sum\limits_{k = 1}^{k - 1} {({a_{k - i - 1}} - {a_{k - i}})} \left\| {{\kern 1pt} e_n^i{\kern 1pt} } \right\|_0^2 + {a_{k - 1}}\left\| {{\kern 1pt} e_n^0{\kern 1pt} } \right\|_0^2,} \end{array} 适当选取步长τ, 使得\frac{\tau^{2-\alpha}}{\Gamma(2-\alpha)}-\frac{1}{2}>0, 由Gronwall's不等式, 有
{\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|_0} \le c({\tau ^{2 - \alpha }} + {2^{ - nr}}). 从而由三角不等式及引理2，有
{\left\| {{\kern 1pt} {u^k} - u_n^k{\kern 1pt} } \right\|_0} \le {\left\| {{\kern 1pt} {u^k} - {\mathcal{P}_n}{u^k}{\kern 1pt} } \right\|_0} + {\left\| {{\kern 1pt} e_n^k{\kern 1pt} } \right\|_0} \le c({\tau ^{2 - \alpha }} + {2^{ - nr}}). 证毕.
注1 当取vn=Dατenk时, 可类似地证明全离散格式的H1收敛阶为
{\left\| {{\kern 1pt} {u^k} - u_n^k{\kern 1pt} } \right\|_1} \le c({\tau ^{2 - \alpha }} + {2^{ - n(r - 1)}}). 3. 求解全离散格式的多层扩充算法
基底的多尺度特性使得全离散格式(3)对应的线性方程组的系数矩阵具有高低频层次结构, 因此, 本文利用多层扩充法进行高效求解.为了方便叙述, 先将全离散格式改写成:
\begin{array}{*{20}{l}} {({\partial _x}u_n^k,{\partial _x}v) + \lambda (u_n^k,v) = }\\ {\quad \left( {\lambda \left( {\sum\limits_{j = 1}^{k - 1} {({a_{k - j - 1}} - {a_{k - j}})} u_n^j + {a_{k - 1}}u_n^0} \right) + {f^k},v} \right),} \end{array} (8) 其中，\lambda:=\frac{\tau^{-\alpha}}{\mu \Gamma(1-\alpha)}，un0=\mathcal{P}nu0.记Zn={0, 1, 2, …, n-1}，w(i)表示空间Wi的维数, 指标集Jn:= {(i, j):j∈Zw(i), i∈Zn}.在t=tk处, 将unk=\sum\limits_{(i, j) \in J_{n}}cijkwij(x)代入式(8), 得矩阵形式的线性方程组
({\mathit{\boldsymbol{E}}_n} + {\mathit{\boldsymbol{K}}_n})\mathit{\boldsymbol{C}}_n^k = \mathit{\boldsymbol{F}}_n^k, (9) 其中，En=[(w′ij, w′i′j′):(i, j), (i′, j′)∈Jn], Kn=λ[(wij, wi′j′):(i, j), (i′, j′) ∈Jn], Cnk=[cijk]Τ(i, j)∈Jn, Fnk=λ(\sum\limits_{j=1}^{k-1}(ak-j-1-ak-j)Cnj+ak-1Cn0)+[(fk, wij)]Τ(i, j)∈Jn.由基底的正交性可知En是单位矩阵.
为使用多层扩充算法, 先将Kn进行分块.对m0, p, p′∈\mathbb{N}_{+}, 记
{\mathit{\boldsymbol{K}}_{0,0}^{{m_0}}: = {\mathit{\boldsymbol{K}}_{{m_0}}},\mathit{\boldsymbol{K}}_{{p^\prime },p}^{{m_0}}: = {\mathit{\boldsymbol{K}}_{{m_0} + {p^\prime },{m_0} + p}},} {\mathit{\boldsymbol{K}}_{0,p}^{{m_0}}: = [{\mathit{\boldsymbol{K}}_{i_1^\prime ,i}}:{i^\prime } \in {Z_{{m_0} + 1}},i = {m_0} + p],} {\mathit{\boldsymbol{K}}_{p,0}^{{m_0}}: = [{\mathit{\boldsymbol{K}}_{{i^\prime },i}}:{i^\prime } = {m_0} + p,i \in {Z_{{m_0} + 1}}],} 于是, 当n=m0+m时, Kn可表示为分块矩阵Km0+m=[Ki′, im0:i′, i ∈Zm+1].根据矩阵高低频层次特点, 把Km0+m分解成2个矩阵之和, 即
{\mathit{\boldsymbol{K}}_{{m_0} + m}} = \mathit{\boldsymbol{K}}_{{m_0},m}^L + \mathit{\boldsymbol{K}}_{{m_0},m}^H, 其中
\mathit{\boldsymbol{K}}_{{m_0},m}^L: = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{K}}_{0,0}^{{m_0}}}&{\mathit{\boldsymbol{K}}_{0,1}^{{m_0}}}& \cdots &{\mathit{\boldsymbol{K}}_{0,m}^{{m_0}}}\\ \mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}& \cdots &\mathit{\boldsymbol{0}}\\ \vdots & \vdots &{}& \vdots \\ \mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}& \cdots &\mathit{\boldsymbol{0}} \end{array}} \right], \mathit{\boldsymbol{K}}_{{m_0},m}^H: = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}& \cdots &\mathit{\boldsymbol{0}}\\ {\mathit{\boldsymbol{K}}_{1,0}^{{m_0}}}&{\mathit{\boldsymbol{K}}_{1,1}^{{m_0}}}& \cdots &{\mathit{\boldsymbol{K}}_{1,m}^{{m_0}}}\\ \vdots & \vdots &{}& \vdots \\ {\mathit{\boldsymbol{K}}_{m,0}^{{m_0}}}&{\mathit{\boldsymbol{K}}_{m,1}^{{m_0}}}& \cdots &{\mathit{\boldsymbol{K}}_{m,m}^{{m_0}}} \end{array}} \right]. 记m0和m是2个固定的正整数, m0 < <n,则多层扩充算法可描述为：
算法1 求解线性方程组(8)的多层扩充算法
Step 1.对n=m0, 解方程(Em0+Km0)Ckm0= Fkm0, 得到Ckm0.
Step 2.令Ckm0, 0:=Ckm0，l=1.执行以下步骤：
Step 2.1 分别将KLm0, l-1和KHm0, l-1扩充为KLm0, l和KHm0, l;
Step 2.2 将Ckm0, l-1扩充为{\mathit{\boldsymbol{\bar C}}_{{m_0}, l}}: = \left[{\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{C}}_{{m_0}, l - 1}}}\\ \mathit{\boldsymbol{0}} \end{array}} \right];
Step 2.3 解方程(Em0+l+KLm0, l)Ckm0, l=fkm0+l+KHm0, lCm0, l, 获得Ckm0, l∈Rx(m0+l).
Step 3.令l=l+1, 返回执行Step 2.1, 直到l=m，从而得到多层扩充解ukm0, m=\sum\limits_{(i, j) \in J_{n}} C_{i j}^{k} w_{i j}.
该算法先在一个维数较低的初始层上求得一个较粗的近似解, 然后逐层扩充, 每次扩充只需要解一个系数矩阵(Em0+K0, 0m0)相同而仅右端项不同的低维线性方程组, 获得解的低频部分, 而高频部分通过矩阵向量乘积获得.由此可见, 该算法比直接在高维逼近子空间上解线性方程组要简单高效.文献[13-15]证明了多层扩充解ukm0, m与直接求解线性方程组得到的解ukm0+m有相同的收敛阶, 即保持最优收敛阶：
{\left\| {{\kern 1pt} u_{{m_0},m}^k - {u^k}{\kern 1pt} } \right\|_p} \le c({\tau ^{2 - \alpha }} + {2^{ - ({m_0} + m)(r - p)}})\quad (p = 0,1). 4. 数值算例
本节以数值算例来验证本文的理论估计和算法1的计算效果.考虑方程(1)，其中
\begin{array}{*{20}{c}} {{u_0}(x) = 2{\rm{sin}}(2\pi x),}\\ {f(x,t) = \left[ {\frac{{\Gamma (3 + \alpha )}}{2}{t^2} + \frac{{{t^{1 - \alpha }}}}{{\Gamma (2 - \alpha )}} + 4{\pi ^2}({t^{2 + \alpha }} + t + 2)} \right]{\rm{sin}}(2\pi x),} \end{array} 其真解u=(tα+2+t+2)sin(2πx).
为验证算法1的时间收敛阶, 选取二次多尺度正交基(基底具体构造可参看文献[14]), 并在多层扩充求解时取(m0, m)=(2, 5), 时间步长τ分别取为1/4、1/8、1/16、1/32、1/64、1/128, 对不同的α进行测试.由所得结果(图 1)可以看到：数值结果与理论的时间收敛阶2-α相吻合.
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图 1 u2, 5(x, t)在t=1时刻的时间收敛阶Figure 1. The temporal convergence order of u2, 5(x, t) at t=1在空间方向, 分别取线性基底和二次基底进行数值计算, 用算法1时，初始层m0分别取为4和2, m表示扩充的层数, x(n)表示相应逼近子空间的维数, n=m0+m.收敛阶为：
{\rm{order}} = {\rm{lo}}{{\rm{g}}_2}\frac{{{{\left\| {u - {u_{{m_0},m}}} \right\|}_p}}}{{{{\left\| {u - {u_{{m_0},m + 1}}} \right\|}_p}}}, 这里p可取1和0, 分别对应H1和L2范数.对于线性和二次基底, 其H1范数的理论收敛阶分别为1和2;其L2范数的理论收敛阶分别为2和3.由表 1和表 2可以看到：算法1所得的数值结果与理论收敛阶非常吻合.
表 1 线性基底的数值结果Table 1. The numerical results for linear basism x(n) p=1 p=0 条件数 ‖u-u4, m‖1 Order ‖u-u4, m‖0 Order 0 15 2.009 5e+00 — 3.925 7e-02 — 1.216 4 1 31 1.006 7e+00 0.997 2 9.823 9e-03 1.998 6 1.217 6 2 63 5.035 8e-01 0.999 3 2.457 4e-03 1.999 2 1.217 9 3 127 2.518 2e-01 0.999 8 6.143 7e-04 1.999 9 1.217 9 4 255 1.259 1e-01 1.000 0 1.535 3e-04 2.000 6 1.217 9 5 511 6.295 7e-02 1.000 0 3.831 5e-05 2.002 5 1.218 0 6 1 023 3.147 8e-02 1.000 0 9.509 9e-06 2.010 4 1.218 0 7 2 047 1.573 9e-02 1.000 0 2.316 5e-06 2.037 5 1.218 0 利用Matlab中的Cond命令, 计算了系数矩阵En+Kn的条件数, 从表 1和表 2可知条件数一致有界.
表 2 二次基底的数值结果Table 2. The numerical results for quadratic basism x(n) p=1 p=0 条件数 ‖u-u2, m‖1 Order ‖u-u2, m‖0 Order 0 7 1.577 5e+00 — 6.062 9e-02 — 1.215 1 1 15 4.049 8e-01 1.961 7 7.804 1e-03 2.957 7 1.217 4 2 31 1.019 1e-01 1.990 5 9.825 6e-04 2.989 6 1.217 8 3 63 2.552 0e-02 1.997 6 1.230 5e-04 2.997 4 1.217 9 4 127 6.382 6e-03 1.999 4 1.539 5e-05 2.998 7 1.217 9 5 255 1.595 8e-03 1.999 9 1.990 7e-06 2.951 1 1.218 0 在计算效率方面, 对比算法1与Gauss迭代法在线性基底下的数值结果.由对比结果(图 2)可知：算法1具有更高的效率, 而且问题规模越大, 其计算优势越明显.
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图 2 算法1与Gauss迭代法的计算时间对比Figure 2. The comparison of computational time between algorithm 1 and the Gauss iteration method5. 结束语
本文利用L1-多尺度Galerkin方法离散时间分数阶次扩散方程, 对所得全离散格式建立了多层扩充算法, 理论分析和数值算例证明了该算法具有最优收敛阶，提高了求解效率.对于含有非线性源项f(u, t)的非线性时间分数阶次扩散方程，可先用f(unk-1, t)来逼近f(unk, t)，以将问题线性化，然后利用多层扩充算法快速求解.此外, 对时间分数阶导数, 若能用更高效的方法来离散(比如SOE法[13]), 或者用更高阶的逼近格式, 将有助于进一步提高时间分数阶次扩散方程的求解效率.
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图 3 水下传感器网络节点邻居关系模型
Figure 3. Neighbor relationship model of underwater sensor networks
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图 5 不同噪声干扰值下6种定位算法的定位效果
Figure 5. The positioning effects of 6 positioning algorithms under different noise interference values
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图 6 同一目标定位6种定位算法估计解与真实值的分布
Figure 6. The distribution of estimated positioning solutions and actual values of 6 positioning algorithms for the same target
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图 7 100个随机目标下6种定位算法的定位效果
Figure 7. The positioning effect of 6 positioning algorithms for 100 random targets
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图 8 不同传感器节点数量下IWONI算法的定位效果
Figure 8. The positioning effect of IWONI algorithm under different sensor node number
表 1 6种定位算法的定位成功率
Table 1 Localization success rate of 6 positioning algorithms
对比算法 IWONI NM T-CHAN T-TAYLOR LS GNM 定位成功率/% 92 79 83 89 84 85 
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