In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models for pattern evolution within biological processes. Upon a Sequential Quadratic Programming linearization of the optimization problem, we devise appropriate time-stepping schemes and discrete approximations of the cost functionals such that the discretization and optimization operations are commutative, a highly desirable property of a discretization of such problems. We formulate the large-scale, coupled linear systems in such a way that efficient preconditioned iterative methods can be applied within a Krylov subspace solver. Numerical experiments demonstrate the viability and efficiency of our approach.

翻译：本文针对生物过程模式演化的非线性偏微分方程约束优化模型，研究了有效的离散化策略与迭代求解器。通过对优化问题进行序列二次规划线性化，我们设计了合适的时间步进方案与代价泛函的离散近似，使得离散化与优化操作可交换——这是此类问题离散化中极为理想的性质。我们以能够应用高效预条件迭代方法于Krylov子空间求解器的方式，构建了大规模耦合线性系统。数值实验验证了本方法的可行性与高效性。