PDE-constrained Optimization Research Articles

Suppressing instability in a Vlasov–Poisson system by an external electric field through constrained optimization

Maintaining the stability and shape of a plasma is a crucial task in many technological applications ranging from beam shaping to fusion energy. This is often challenging as plasma systems tend to be naturally unstable and kinetic effects can play an important role in the behavior of the instabilities. Due to the large energies involved, the primary way to control plasma systems is the application of external electromagnetic fields.In this work, we deploy a PDE-constrained optimization formulation that uses a kinetic description for plasma dynamics as the constraint. This is to optimize, over all possible controllable external electric fields, the stability of the plasma dynamics under the condition that the Vlasov–Poisson (VP) equation is satisfied. Mathematically we formulate it as a PDE-constrained optimization. For computing the functional derivative with respect to the external field in the optimization updates, the adjoint equation is derived. Furthermore, in the discrete setting, where we employ the semi-Lagrangian method as the forward solver, we also explicitly formulate the corresponding adjoint solver and the gradient as the discrete analogy to the adjoint equation and the Fréchet derivative. A distinct feature we observed of this constrained optimization is the complex landscape of the objective function and the existence of numerous local minima, largely due to the hyperbolic nature of the VP system. To overcome this issue, we utilize a gradient-accelerated genetic algorithm, leveraging the advantages of the genetic algorithm's exploration feature to cover a broader search of the solution space and the fast local convergence aided by the gradient information. We show that our algorithm obtains good electric fields that are able to maintain a prescribed profile in a beam shaping problem and uses nonlinear effects to suppress plasma instability in a two-stream configuration.

Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations

In the past decades, multiple shooting methods have proven to be a promising direction to speed up the optimization process, especially in the context of ODE-based optimization. Very recently, Fang et al. (Journal of Computational Physics, vol. 452, 110926, 2022) proposed a multiple shooting algorithm for large-scale PDE-constrained optimization. The current paper continues this line of work and explores the potential of multiple shooting methods for optimal control problems governed by the three-dimensional Navier–Stokes equations. Similar to Fang et al., the augmented Lagrangian (AL) method is used to solve the resulting equality-constrained optimization problem, and we employ the classical limited-memory BFGS method for the unconstrained subproblems inside the AL loop. In the current work, we exploit the multiple shooting paradigm in full by processing the shooting windows parallel-in-time, allowing for significant parallel speed-ups compared to single shooting. The proposed method is validated on a velocity tracking case, using up to 100 windows. Our analysis shows that the multiple shooting algorithm allows for considerable algorithmic and parallel speed-ups. While algorithmic speed-up depends on the exact tracking case and initialization of the shooting windows, the multiple shooting algorithm always outperforms single shooting in terms of computational time (if the number of windows is sufficiently high) due to the parallel-in-time implementation. For a given amount of resources, we also show that the proposed parallel-in-time strategy can outperform spatial parallelization alone, especially when the spatial parallelization is saturated.

A scalable framework for multi-objective PDE-constrained design of building insulation under uncertainty

This paper introduces a scalable computational framework for optimal design under high-dimensional uncertainty, with application to thermal insulation components. The thermal and mechanical behaviors are described by continuum multi-phase models of porous materials governed by partial differential equations (PDEs), and the design parameter, material porosity, is an uncertain and spatially correlated field. After finite element discretization, these factors lead to a high-dimensional PDE-constrained optimization problem. The framework employs a risk-averse formulation that accounts for both the mean and variance of the design objectives. It incorporates two regularization techniques, the ℓ0-norm and phase field functionals, implemented using continuation numerical schemes to promote spatial sparsity in the design parameters. To ensure efficiency, the framework utilizes a second-order Taylor approximation for the mean and variance and exploits the low-rank structure of the preconditioned Hessian of the design objective. This results in computational costs that are determined by the rank of preconditioned Hessian, remaining independent of the number of uncertain parameters. The accuracy, scalability with respect to the parameter dimension, and sparsity-promoting abilities of the framework are assessed through numerical examples involving various building insulation components.

An optimization based 3D-1D coupling strategy for tissue perfusion and chemical transport during tumor-induced angiogenesis

The application of an optimization based 3D-1D coupling strategy is proposed, for the first time, for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. A well posed mathematical model is worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). The problems are then solved in a PDE-constrained optimization framework, under which no mesh conformity is required. This makes the method particularly suitable for this kind of application, since no remeshing is required as the capillary network grows. In order to handle both the evolution of the quantities of interest and the changes in the geometry, a discrete-hybrid strategy is adopted, combining a continuous modeling of the tissue and of the chemicals with a discrete tip-tracking model to account for the vascular network growth. The tip-tracking strategy, together with some proper rules for branching and anastomosis, is able to provide a realistic representation of the capillary network.

Optimization Problems for PDEs in Weak Space-Time Form

Optimization problems constrained by time-dependent Partial Differential Equations (PDEs) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous amount of computer memory allocations for the respective other variables. In contrast, adaptive-in-time-and-space methods aim at distributing the available degrees of freedom in an a-posteriori fashion to capture singularities and are, therefore, most promising. Recently, well-posed weak variational formulations have been introduced for time-dependent PDEs such as the heat equation, linear transport and the wave equation. Those formulations also allow for a sharp relation between the approximation error and the residual, which is particularly relevant for model reduction. Moreover, for those tensor-basis formulations, advanced algebraic solvers designed to take into account these multiarray (tensorial) formulations appear to be particularly competitive with respect to time-marching schemes, especially in higher dimensions. We plan to discuss whether these techniques can be extended to nonlinear PDEs like Hamilton-Jacobi-Bellman equations, or stochastic PDEs and variational inequalities. Another topic will be adaptive schemes which, when properly designed, inherit the stability of the continuous formulation. The central goals of the workshop are the analysis, fast solvers and model reduction for PDE-constrained control and optimization problems based on weak formulations of the underlying PDE(s).

Nonlinear Variational Inequalities in PDE-Constrained Optimization

Nonlinear Variational Inequalities (NVI) have emerged as a powerful mathematical framework for addressing complex optimization problems constrained by Partial Differential Equations (PDEs). This article explores the integration of NVI into PDE-constrained optimization, providing insights into the mathematical foundations, numerical techniques, and practical applications of this synergy. By understanding the role of NVI in PDE-constrained optimization, researchers and practitioners can tackle challenging optimization problems arising in fields such as engineering, physics, and computational science.

A new learning space-variant anisotropic constrained-PDE for image denoising

In this paper, we propose an improved enhancement space-variant anisotropic PDE-constrained for image denoising, based on a learning optimization procedure. Since the tensor structure of the Weickert-type operators encodes three critical parameters: λ1,λ2 and θ, which define the local direction geometry in the image and also controls smoothing intensity along image features. We have then adopted an automatic estimation of these parameters based on a PDE-constrained optimization with a learning step of the additional information about the clean image. For the numerical solution of the computed free-noise image, we introduce a non-smooth Alternating Direction Method of Multipliers (ADMM) algorithm. The numerous tests in the experiments part incense, visually and quantitatively, the incomes of this new spatial tensor for PDE smoothing over the other denoising approaches.

Preconditioners based on matrix splitting for the structured systems from elliptic PDE-constrained optimization problems

With regard to the structured systems of linear equations, which arises from the Galerkin finite element discretizations of elliptic PDE-constrained optimization problems, some new preconditioners based on coefficient matrix splitting are established to speed up the convergence rate of Krylov subspace methods such as GMRES. Additionally, eigenvalue distribution of the corresponding preconditioned matrices is deeply discussed. Numerical simulations are carried out, the results of which exhibit that the corresponding preconditioned GMRES methods perform very well with the theoretical analysis results and are superior to other newly devised preconditioners.

Quasi-Newton methods for topology optimization using a level-set method

The ability to efficiently solve topology optimization problems is of great importance for many practical applications. Hence, there is a demand for efficient solution algorithms. In this paper, we propose novel quasi-Newton methods for solving PDE-constrained topology optimization problems. Our approach is based on and extends the popular solution algorithm of Amstutz and Andrä (J Comput Phys 216: 573–588, 2006). To do so, we introduce a new perspective on the commonly used evolution equation for the level-set method, which allows us to derive our quasi-Newton methods for topology optimization. We investigate the performance of the proposed methods numerically for the following examples: Inverse topology optimization problems constrained by linear and semilinear elliptic Poisson problems, compliance minimization in linear elasticity, and the optimization of fluids in Navier–Stokes flow, where we compare them to current state-of-the-art methods. Our results show that the proposed solution algorithms significantly outperform the other considered methods: They require substantially less iterations to find a optimizer while demanding only slightly more resources per iteration. This shows that our proposed methods are highly attractive solution methods in the field of topology optimization.

Weighted sparsity regularization for source identification for elliptic PDEs

Abstract This investigation is motivated by PDE-constrained optimization problems arising in connection with electrocardiograms (ECGs) and electroencephalography (EEG). Standard sparsity regularization does not necessarily produce adequate results for these applications because only boundary data/observations are available for the identification of the unknown source, which may be interior. We therefore study a weighted ℓ 1 \ell^{1} -regularization technique for solving inverse problems when the forward operator has a significant null space. In particular, we prove that a sparse source, regardless of whether it is interior or located at the boundary, can be exactly recovered with this weighting procedure as the regularization parameter 𝛼 tends to zero. Our analysis is supported by numerical experiments for cases with one and several local sources. The theory is developed in terms of Euclidean spaces, and our results can therefore be applied to many problems.

A new free-parameter double splitting iteration method for PDE-constrained optimization problems

We present a new free-parameter double splitting iteration method for solving the system of linear equations arisen from finite element discretization of a PDE-constrained optimization problem. We prove that the eigenvalues of the iteration matrix of the method are included in the interval (0,12]. So the spectral radius of the iteration matrix is less than or equal to 12, which shows that the method is unconditionally convergent. Numerical comparison with two known methods is given to show the effectiveness of the method.

Space Mapping for PDE Constrained Shape Optimization

Space Mapping for PDE Constrained Shape Optimization

Performance Bounds for PDE-Constrained Optimization under Uncertainty

Performance Bounds for PDE-Constrained Optimization under Uncertainty

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

AbstractA numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient solving strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

Maximum load potential of hinged plates with non-homogeneous thickness

In this paper, a PDE-constrained optimization problem corresponding to the bi-Laplacian operator is investigated. The optimization problem arises when we search for the maximum load potential in a uniform elastic plate hinged at its boundary with a given total amount of force. It is established that there is a bang–bang solution to the optimization problem, and an optimality condition is derived. Based on the optimality condition, we obtain some topological and geometrical properties of the maximizer. Although analytical solutions for PDE-constrained problems are very rare, an analytical solution is determined if the plate is circular and its thickness has a particular form. In those cases, it is proved that the solution is unique. For plates of general shape, an efficient numerical algorithm is developed to solve the PDE-constrained optimization problem, and its convergence is studied. According to the results, one can determine the profile of the regions in a plate where the largest and smallest external force should be exerted to have the maximum load potential. According to the research results, the thickness of the plate plays a crucial role in determining the shape of those regions. Therefore, disregarding the impact of plate thickness in our studies for the sake of simplicity would be impractical and might result in inaccurate outcomes. By utilizing our findings, engineers and designers can determine the optimal external force pattern that yields the maximum load potential for plates. This, in turn, enables them to create plates that are more efficient and effective, tailored to meet their specific needs and requirements.

A Unified Approach to Shape and Topological Sensitivity Analysis of Discretized Optimal Design Problems

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction–diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.

Learning dynamics on invariant measures using PDE-constrained optimization.

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Sensitivity Analysis of the Data Assimilation-Driven Decomposition in Space and Time to Solve PDE-Constrained Optimization Problems

This paper is presented in the context of sensitivity analysis (SA) of large-scale data assimilation (DA) models. We studied consistency, convergence, stability and roundoff error propagation of the reduced-space optimization technique arising in parallel 4D Variational DA problems. The results are helpful to understand the reliability of DA, to assess what confidence one can have that the simulation results are correct and to determine its configuration in any application. The main contributions of the present work are as follows. By using forward error analysis, we derived the number of conditions of the parallel approach. We found that the parallel approach reduces the number of conditions, revealing that it is more appropriate than the standard approach usually implemented in most operative software. As the background values are used as initial conditions of local PDE models, we analyzed stability with respect to time direction. Finally, we proved consistency of the proposed approach by analyzing local truncation errors of each computational kernel.

JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science

This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented with pure Python while scalable to efficiently solve problems with moderate to large sizes. For example, in a 3D tensile loading problem with 7.7 million degrees of freedom, JAX-FEM with GPU achieves around 10× acceleration compared to a commercial FEM code depending on platform. Beyond efficiently solving forward problems, JAX-FEM employs the automatic differentiation technique so that inverse problems are solved in a fully automatic manner without the need to manually derive sensitivities. Examples of 3D topology optimization of nonlinear materials are shown to achieve optimal compliance. Finally, JAX-FEM is an integrated platform for machine learning-aided computational mechanics. We show an example of data-driven multi-scale computations of a composite material where JAX-FEM provides an all-in-one solution from microscopic data generation and model training to macroscopic FE computations. The source code of the library and these examples are shared with the community to facilitate computational mechanics research. Program summaryProgram Title: JAX-FEMCPC Library link to program files:https://doi.org/10.17632/hgwshjbcw6.1Developer's repository link:https://github.com/tianjuxue/jax-am/tree/main/jax_am/femLicensing provisions: GPLv3Programming language: PythonNature of problem: Implementation of the finite element method (FEM) with several appealing features that classic FEM software typically does not have: realized with pure Python; running on CPU/GPU; differentiable for solving PDE-constrained optimization problems; seamless integration with machine learning.Solution method: Our framework JAX-FEM is based on Google JAX, a high-performance numerical computing library with automatic differentiation features and supporting both CPU/GPU. Unlike classic FEM software written in Fortran or C/C++, JAX-FEM is implemented with pure Python and can easily be installed as a Python package. We demonstrate our software by solving problems including forward PDE prediction, inverse design/optimization, and data-driven analysis.

Optimization of methanol synthesis under forced periodic operation in isothermal fixed-bed reactors

Methanol is usually produced in large amounts in fixed-bed reactors under steady-state conditions using Cu/ZnO/Al2O3-catalyst. In this paper, possible improvements through forced periodic operation compared to optimal steady-state solution are investigated. The numerical study is done with a simplified model which describes a hypothetical 7-meter-long isothermal and isobaric fixed-bed reactor. The forced periodic operation is implemented using harmonic forcing and square-waves. Multi-objective optimization is performed for different operating conditions quantifying the achievable molar flow rate of methanol at the reactor outlet and the yield of methanol based on total carbon. The resulting PDE-constrained optimization problems are solved using a simultaneous approach, with orthogonal collocation as discretization. The results show most significant improvements through forced periodic operation with square-waves. The periodic operation can provide theoretical improvements of up to 50% for the methanol mean outlet molar flow rate and improvements of up to 10% for the yield based on total carbon.