Fast tensor product solvers for optimization problems with fractional differential equations as constraints
Fast tensor product solvers for optimization problems with fractional differential equations as constraints
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DOI：
10.1016/j.amc.2015.09.042
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发表时间：
2016-01
期刊：
影响因子：
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通讯作者：
S. Dolgov;J. Pearson;D. Savostyanov;M. Stoll
中科院分区：
文献类型：
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作者：
S. Dolgov;J. Pearson;D. Savostyanov;M. Stoll
分数阶微分方程最近在计算数学和应用科学领域受到广泛关注，其数值处理是一个重要的研究领域，因为此类方程对现有算法提出了重大挑战。考虑由分数阶微分方程给出的约束的优化问题，其离散形式导致高维张量方程。为了减少计算时间和存储空间，以张量序列格式寻求解决方案。我们比较了三种类型的解决方案策略，这些策略采用复杂的迭代技术，使用预处理的 Krylov 求解器或定制的交替方案。这些方法的竞争力通过几个具有常数和可变系数的例子来展示。
Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.