arXiv subjects

Liu-Di Lu

Publications and source records attributed to Liu-Di Lu.

12 recordsLinked to original sources

arXiv2026

From PDEs constrained optimization to controllability problems via time domain decomposition

This paper focuses on the application of time domain decomposition to solve partial differential equations constrained optimization problems and controllability problems. After clarifying the link between these two types of problems, we show that applying time domain decomposition to both problems leads to the same convergence behavior. Our numerical experiments also confirm these theoretical findings.

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arXiv2026

Schwarz Modulus Based Matrix Splittings with Minimal Polynomial Extrapolation Acceleration for linear complementarity problems arising from American option pricing

Pricing American options is more complicated than pricing European options, because they can be exercised at any time, and one thus needs to solve a linear complementarity problem instead of simply doing time stepping for computing European options. We introduce a new Schwarz modulus-based splitting method for solving such linear complementarity problems, and further accelerate them using Modified Polynomial Extrapolation, a non-linear vector sequence acceleration technique, which is very much related to Krylov methods in the linear case. Numerical experiments on a model problem show that our new solver can have close to an order of magnitude lower iteration counts than the classically used modulus-based matrix splitting technique.

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arXiv2026

Weak Scalability of time parallel Schwarz methods for parabolic optimal control problems

Parabolic optimal control problems arise in numerous scientific and engineering applications. They typically lead to large-scale coupled forward-backward systems that cannot be treated with classical time-stepping schemes and are computationally expensive to solve. Therefore, parallel methods are essential to reduce the computational time required. In this work, we investigate a time domain decomposition approach, namely the time parallel Schwarz method, applied to parabolic optimal control problems. We analyze the convergence behavior and focus on the weak scalability property of this method as the number of time intervals increases. To characterize the spectral radius of the iteration matrix, we present two analysis techniques: the construction of a tailored matrix norm and the application of block Toeplitz matrix theory. Our analyses yield both nonasymptotic bounds on the spectral radius and an asymptotic characterization of the eigenvalues as the number of time intervals tends to infinity. Numerical experiments further confirm our theoretical findings and demonstrate the weak scalability of the time parallel Schwarz method. This work introduces the first theoretical tool for analyzing the weak scalability of time domain decomposition methods, and our results shed light on the suitability of our algorithm for large-scale simulations on modern high-performance computing architectures.

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arXiv2025

Optimized Schwarz methods for heterogeneous heat transfer problems

We present here nonoverlapping optimized Schwarz methods applied to heat transfer problems with heterogeneous diffusion coefficients. After a Laplace transform in time, we derive the error equation and obtain the convergence factor. The optimal transmission operators are nonlocal, and thus inconvenient to use in practice. We introduce three versions of local approximations for the transmission parameter, and provide a detailed analysis at the continuous level in each case to identify the best local transmission conditions. Numerical experiments are presented to illustrate the performance of each local transmission condition. As shown in our analysis, local transmission conditions, which are scaled appropriately with respect to the heterogeneous diffusion coefficients, are more efficient and robust especially when the discontinuity of the diffusion coefficient is large.

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arXiv2024

Non-overlapping Schwarz methods in time for parabolic optimal control problems

We present here the classical Schwarz method with a time domain decomposition applied to unconstrained parabolic optimal control problems. Unlike Dirichlet-Neumann and Neumann-Neumann algorithms, we find different properties based on the forward-backward structure of the optimality system. Variants can be found using only Dirichlet and Neumann transmission conditions. Some of these variants are only good smoothers, while others could lead to efficient solvers.

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arXiv2024

New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms

We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

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arXiv2023

Dirichlet-Neumann and Neumann-Neumann Methods for Elliptic Control Problems

We present the Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to the optimal control problems arising from elliptic partial differential equations (PDEs) under the $H^{-1}$ regularization. We use the Lagrange multiplier approach to derive a forward-backward optimality system with the $L^2$ regularization, and a singular perturbed Poisson equation with the $H^{-1}$ regularization. The $H^{-1}$ regularization thus avoids solving a coupled bi-Laplacian problem, yet the solutions are less regular. The singular perturbed Poisson equation is then solved by using the DN and NN methods, and a detailed analysis is given both in the one-dimensional and two-dimensional case. Finally, we provide some numerical experiments with conclusions.

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arXiv2023

New time domain decomposition methods for parabolic control problems I: Dirichlet-Neumann and Neumann-Dirichlet algorithms

We present new Dirichlet-Neumann and Neumann-Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet-Neumann and Neumann-Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

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arXiv2022

Optimal periodic resource allocation in reactive dynamical systems: Application to microalgal production

In this article, we focus on a periodic resource allocation problem applied to a dynamical system which comes from a biological system. More precisely, we consider a system with $N$ resources and $N$ activities, each activity use the allocated resource to evolve up to a given time $T > 0$ where a control (represented by a given permutation) will be applied on the system to reallocate the resources. The goal is to find the optimal control strategies which optimize the cost or the benefit of the system. This problem can be illustrated by an industrial biological application, namely, the optimization of a mixing strategy to enhance the growth rate in a microalgal raceway system. A mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures, hence the light perceived at each lap. We prove that if the dynamics of the system is periodic, then the period corresponds to one reallocation whatever the order of the involved permutation matrix is. A nonlinear optimization problem for one reallocation process is then introduced. Since $N!$ permutations need to be tested in the general case, it can be numerically solved only for a limited number of $N$. To overcome this difficulty, we introduce a second optimization problem which provides a suboptimal solution of the initial problem, but whose solution can be determined explicitly. A sufficient condition to characterize cases where the two problems have the same solution is given. Some numerical experiments are performed to assess the benefit of optimal strategies in various settings.

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arXiv2021

Optimal optical conditions for Microalgal production in photobioreactors

The potential of industrial applications for microalgae has motivated their recent fast development. Their growth dynamics depends on different factors that must be optimized. Since they get their energy from photosynthesis, light is a key factor that strongly influences their productivity. Light is absorbed and scattered in the liquid medium, and irradiance exponentially decreases towards the darkest part of the photobioreactor at a rate nonlinearly depending on the biomass concentration. Maximizing productivity is then a tricky problem, especially when the growth rate is inhibited by an excess of light. Productivity optimization turns out to be highly dependent on how light is distributed along the reactor, and is therefore related to the extinction rate and the background turbidity. The concept of optical depth productivity is introduced for systems where background turbidity must be accounted for and a global optimum maximizing productivity is proposed, extending the concept of the compensation condition. This optimal condition consists in compensating the algal growth rate at the bottom of the reactor by the respiration. This condition can drive the optimization of the surface biomass productivity depending on the minimum reachable depth. We develop a nonlinear controller and prove the global asymptotic stability of the biomass concentration towards the desired optimal value.

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arXiv2020

Mixing strategies combined with shape design to enhance productivity of a raceway pond

This paper focuses on mixing strategies and designing shape of the bottom topographies to enhance the growth of the microalgae in raceway ponds. A physical-biological coupled model is used to describe the growth of the algae. A simple model of a mixing device such as a paddle wheel is also considered. The complete process model was then included in an optimization problem associated with the maximization of the biomass production. The results show that non-trivial topographies can be coupled with some specific mixing strategies to improve the microalgal productivity.

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arXiv2020

Topography optimization for enhancing microalgal growth in raceway ponds

Modelling the evolution process for the growth of microalgae in an artificial pond is a huge challenge, given the complex interaction between hydrodynamics and biological processes occurring across various timescales. In this paper, we consider a raceway, i.e., an oval pond where the water is set in motion by a paddle wheel. Our aim is to investigate theoretically and numerically the impact of bottom topography in such raceway ponds on microalgae growth. To achieve this goal, we consider a biological model based on the Han model, coupled with the Saint--Venant systems that model the fluid. We then formulate an optimization problem, for which we apply the weak maximum principle to characterize optimal topographies that maximize biomass production over one lap of the raceway pond or multiple laps with a paddle wheel. In contrast to a widespread belief in the field of microalgae, we show that a flat topography in a periodic regime satisfies the necessary optimality condition, and observe in the numerical experiments that the flat topography is actually optimal in this case. However, non-trivial topographies may be more advantageous in alternative scenarios, such as when considering the effects of mixing devices within the model. This study sheds light on the intricate relationship between bottom topography, fluid dynamics, and microalgae growth in raceway ponds, offering valuable insights into optimizing biomass production.

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