Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics

Abstract

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.

Analysis of soliton interactions of modified Korteweg-de Vries equation using conserved quantities

“…The comparison between the maximum merged wave profiles obtained by the conserved quantities and those obtained by the pseudo-spectral method [46] (numerical method) are shown in figures 3 and 4, and good agreement has been found. The numerical error and CUP time (Lenovo P720) required for solving the mKdV equation via the numerical method are shown in table 2. the two waves cannot merge completely, and a single solitary wave is not sufficient to describe the merged waveform.…”

Dynamic density functional theory for sedimentation processes on complex domains: Modelling, spectral elements, and control problems

“…Therefore, we now consider applying the DDFT model as a PDE constraint within a control problem. We refer to [61] for a general discussion of optimal control problems with PDE constraints, and to [2,3,41] for optimal control with mean-field DDFT models as constraints. We note that the mathematical analysis of such non-linear problems is highly challenging; see, e.g., [11].…”

“…Recently, advances have been made in solving DDFT models for soft particles on complicated domains 41 , as well as in solving associated control problems 2,41 . However, until now, these methods had not been applied to more complicated DDFTs, such as those that model volume exclusion.…”

“…However, until now, these methods had not been applied to more complicated DDFTs, such as those that model volume exclusion. This paper demonstrates how the framework developed in [2] and [41] can be extended to these more complicated DDFT models, which lays a crucial foundation for computing even more sophisticated DDFTs on complex domains, as well as solving control problems involving such models.…”

Perspective: New directions in dynamical density functional theory

“…Recent work has considered stochastic DDFT (the Dean-Kawasaki equation) [177][178][179][180][181][182][183] and the McKean-Vlasov equation (a DDFT-type model) [184][185][186][187] from a mathematical perspective. Moreover, numerical methods were developed for DDFT [188][189][190][191][192][193][194][195][196] and PFC models [197][198][199][200]. DDFT was also used to test a new Brownian dynamics simulation method [201].…”