数据驱动下的个性化自适应学习研究综述

朱佳; 张丽君; 梁婉莹

doi:10.6054/j.jscnun.2020055

数据驱动下的个性化自适应学习研究综述

朱佳^{1, 2, ,},
张丽君¹,
梁婉莹¹

1.
华南师范大学计算机学院，广州 510631
2.
广州市大数据智能教育重点实验室，广州 510631

基金项目:

国家自然科学基金项目 61877020

国家自然科学基金项目 U1811263

广州市大数据智能教育重点实验室 201905010009

详细信息

通讯作者:
朱佳，教授，Email：jzhu@m.scnu.edu.cn

中图分类号: TP18;G434
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- 文章访问数: 01944
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出版历程
- 收稿日期: 2020-02-12
- 网络出版日期: 2020-09-03
- 刊出日期: 2020-08-24

A Review of Data-Driven Personalized Adaptive Learning

1.
School of Computer Science, South China Normal University, Guangzhou 510631, China
2.
Guangzhou Key Laboratory of Big Data and Intelligent Education, Guangzhou 510631, China

摘要

摘要: 智能教育环境下的教学更加关注学习者的个性化诉求，而自适应学习能够为实现个性化教育提供技术和方法支持.文章从数据驱动的视角出发，通过开展国内外个性化自适应学习研究的综述分析，对其系统框架和相关组件进行阐述和解读.其中，重点从领域知识模型、学习者特征模型和教学模型三方面对其实现机制进行探析，提出当前研究存在的问题和不足，并在此基础上介绍了近年来可促进解释性提升的相关组件技术研究，奠定进一步深入个性化自适应学习研究的基础.
- 个性化自适应学习 /
- 教育知识图谱及其表示学习 /
- 知识追踪 /
- 个性化学习路径推荐
Abstract: Teaching in a smart education environment pays more attention to the individual demands of learners, and adaptive learning can provide technical and methodological support for achieving personalized education. A review of the domestic and foreign research on personalized adaptive learning research is conducted to interpret its system framework and related components. Its implementation mechanism is given at three aspects, i.e., the domain knowledge model, the learner characteristic model, and the teaching model. After a comprehensive analysis, the problems and deficiencies of current research are pointed out. On that basis, the research on related component technologies in recent years that can promote interpretability is introduced to provide references for the next step of personalized adaptive learning research.
- personalized adaptive learning /
- educational knowledge graph and representation learning /
- knowledge tracing /
- personalized learning path recommendation

HTML全文

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:

$\left\{ \begin{array}{l} _0^{\rm{C}}D_t^\alpha u(x,t) = \mu {\partial _{xx}}u(x,t) + f(x,t)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ((x,t){\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} [0,1] \times (0,T]),\\ u(x,0) = {u_0}(x){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} [0,1]),\\ u(0,t) = u(1,t) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (t{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} (0,T]), \end{array} \right.$

(1)

其中, 0 < α < 1, μ>0, ₀^CD_t^α是Caputo分数阶微分算子, 其定义为

$_0^{\rm{C}}D_t^\alpha u(x,t) = \frac{1}{{\Gamma (1 - \alpha )}}\int_0^t {{{(t - s)}^{ - \alpha }}} {\partial _s}u(x,s){\rm{d}}s,$

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:

${\langle u,v\rangle _1}: = \int_I {{u^\prime }} (x){v^\prime }(x){\rm{d}}x,{\left\| {{\kern 1pt} u{\kern 1pt} } \right\|_1} = \sqrt {{{\langle u,u\rangle }_1}} {\kern 1pt} {\kern 1pt} {\kern 1pt} (u,v{\kern 1pt} \in {\kern 1pt} H_0^1(I)).$

设X_n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{X_n}是嵌套的, 即X_n⊂X_n+1 $(n \in \mathbb{N})$ .于是，X_n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，W_n是X_n-1在X_n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)→X_n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∈H^m(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t_k=kτ(k=0, 1, …, N_T; N_Tτ=T), u^k:= u(·, t_k).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, a_j=(j+1)^1-α-j^1-α (j≥0).根据文献[2], 如果u∈C²([0, T]; L²(I)), 则用D_τ^αu^k逼近分数阶导数₀^CD^α_{t_k}u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X_n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=t_k(k=1, 2, …, N_T), 求u_n^k∈X_n, 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, f^k=f(x, t_k).

定理1 对每个时间层t=t_k (k=1, 2, …, N_T), 全离散格式(3)都存在唯一解u_n^k.如果问题(1)的解析解u∈C²([0, T]; H^r(I)∩H₀¹(I)), ₀^CD_t^αu∈H^r-2(I), 且u₀∈H^r(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D_τ^α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v_n=u_n^k, 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u_n^k=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u^k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e_n^k= $\mathcal{P}$ _nu^k-u_n^k (k=0, 1, 2, …, N_T).将方程(6)减方程(3), 并取v_n=e_n^k，则e_n^k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>…>a_n>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v_n=D^α_τe_n^k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， $\lambda:=\frac{\tau^{-\alpha}}{\mu \Gamma(1-\alpha)}$ ，u_n⁰= $\mathcal{P}$ _nu₀.记Z_n={0, 1, 2, …, n-1}，w(i)表示空间W_i的维数, 指标集J_n:= {(i, j):j∈Z_w(i), i∈Z_n}.在t=t_k处, 将u_n^k= $\sum\limits_{(i, j) \in J_{n}}$ c_ij^kw_ij(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，E_n=[(w′_ij, w′_i′j′):(i, j), (i′, j′)∈J_n], K_n=λ[(w_ij, w_i′j′):(i, j), (i′, j′) ∈J_n], C_n^k=[c_ij^k]^Τ_{(i, j)∈J_n}, F_n^k=λ( $\sum\limits_{j=1}^{k-1}$ (a_k-j-1-a_k-j)C_n^j+a_k-1C_n⁰)+[(f^k, w_ij)]^Τ_{(i, j)∈J_n}.由基底的正交性可知E_n是单位矩阵.

为使用多层扩充算法, 先将K_n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, K_n可表示为分块矩阵K_m₀+m=[K_{i′, i}^m₀:i′, i ∈Z_m+1].根据矩阵高低频层次特点, 把K_m₀+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(E_m₀+K_m₀)C^k_m₀= F^k_m₀, 得到C^k_m₀.

Step 2.令C^k_{m₀, 0}:=C^k_m₀，l=1.执行以下步骤：

Step 2.1 分别将K^L_{m₀, l-1}和K^H_{m₀, l-1}扩充为K^L_{m₀, l}和K^H_{m₀, l};

Step 2.2 将C^k_{m₀, 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(E_m₀+l+K^L_{m₀, l})C^k_{m₀, l}=f^k_m₀+l+K^H_{m₀, l}C_{m₀, l}, 获得C^k_{m₀, l}∈R^x(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u^k_{m₀, 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(E_m₀+K_{0, 0}^m₀)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u^k_{m₀, m}与直接求解线性方程组得到的解u^k_m₀+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=(t^α+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.

图 1 u_{2, 5}(x, t)在t=1时刻的时间收敛阶

Figure 1. The temporal convergence order of u_{2, 5}(x, t) at t=1

下载: 全尺寸图片幻灯片

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 basis

m	x(n)	p=1		p=0		条件数
m	x(n)	‖u-u_{4, m}‖₁	Order	‖u-u_{4, m}‖₀	Order	条件数
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

下载: 导出CSV

| 显示表格

利用Matlab中的Cond命令, 计算了系数矩阵E_n+K_n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 basis

m	x(n)	p=1		p=0		条件数
m	x(n)	‖u-u_{2, m}‖₁	Order	‖u-u_{2, m}‖₀	Order	条件数
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

下载: 导出CSV

| 显示表格

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.

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 method

下载: 全尺寸图片幻灯片

5. 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(u_n^k-1, t)来逼近f(u_n^k, 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.

图 1 个性化自适应学习基本框架

Figure 1. The basic framework of personalized adaptive learning

下载: 全尺寸图片幻灯片

图 2 个性化自适应学习框架的相关组件及对应关键技术

Figure 2. The related components and the key technologies of the personalized adaptive learning framework

下载: 全尺寸图片幻灯片

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1. 预备知识
2. L1-多尺度Galerkin全离散格式
3. 求解全离散格式的多层扩充算法
4. 数值算例
5. 结束语

数据驱动下的个性化自适应学习研究综述

通讯作者: 朱佳，教授，Email：jzhu@m.scnu.edu.cn

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