PDE-constrained Optimization Problems Research Articles

Differentiable solver for time-dependent deformation problems with contact

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.

An iDCA with Sieving Strategy for PDE-Constrained Optimization Problems with $$L^{1-2}$$-Control Cost

An iDCA with Sieving Strategy for PDE-Constrained Optimization Problems with $$L^{1-2}$$-Control Cost

A Discretize-then-Optimize Approach to PDE-Constrained Shape Optimization

We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see R. Herzog and E. Loayza-Romero, Math. Comput. 92 (2022) 1-50. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of a globally optimal solution for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.

On fast convergence rates for generalized conditional gradient methods with backtracking stepsize

<p style='text-indent:20px;'>A generalized conditional gradient method for minimizing the sum of two convex functions, one of them differentiable, is presented. This iterative method relies on two main ingredients: First, the minimization of a partially linearized objective functional to compute a descent direction and, second, a stepsize choice based on an Armijo-like condition to ensure sufficient descent in every iteration. We provide several convergence results. Under mild assumptions, the method generates sequences of iterates which converge, on subsequences, towards minimizers. Moreover, a sublinear rate of convergence for the objective functional values is derived. Second, we show that the method enjoys improved rates of convergence if the partially linearized problem fulfills certain growth estimates. Most notably these results do not require strong convexity of the objective functional. Numerical tests on a variety of challenging PDE-constrained optimization problems confirm the practical efficiency of the proposed algorithm.</p>

Version 2.0 - cashocs: A Computational, Adjoint-Based Shape Optimization and Optimal Control Software

In this paper, we present version 2.0 of cashocs. Our software automates the solution of PDE constrained optimization problems for shape optimization and optimal control. Since its inception, many new features and useful tools have been added to cashocs, making it even more flexible and efficient. The most significant additions are a framework for space mapping, the ability to solve topology optimization problems with a level-set approach, the support for parallelism via MPI, and the ability to handle additional (state) constraints. In this software update, we describe the key additions to cashocs, which is now even better-suited for solving complex PDE constrained optimization problems.

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.

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.

A reordering-based preconditioner for elliptic PDE-constrained optimization problems with small Tikhonov parameters

A reordering-based preconditioner for elliptic PDE-constrained optimization problems with small Tikhonov parameters

Direct approaches for PDE-constrained dynamic optimization based on space–time orthogonal collocation on finite elements

Dynamic optimization constrained by partial differential equations (PDE) involves the state distribution on space, and can generate improved operational policies compared to ODE-constrained dynamic optimization. With the help of space–time orthogonal collocation on finite elements (ST-OCFE) which can guarantee relatively high accuracy in both space and time, PDE models are transformed into a set of algebraic equations. For solving PDE-constrained optimization problems, three direct approaches, namely ST-OCFE based single shooting, ST-OCFE based multiple shooting, and ST-OCFE based simultaneous collocation are proposed. The similarities and differences among these approaches are analyzed, where the discretized nonlinear programming problems to be solved in each algorithm are described. Furthermore, the first-order and second-order sensitivity computation for ST-OCFE based shooting approaches is introduced with algorithmic differentiation to efficiently evaluate the exact Hessian of the Lagrangian. Three cases including different dynamics and constraints are investigated. The optimal solution given by these ST-OCFE based direct approaches is consistent, demonstrating the effectiveness of these algorithms.

Parameters identification for a nonlinear partial differential equation in image denoising

Abstract In this work and in the context of PDE constrained optimization problems, we are interested in identification of a parameter in the diffusion equation proposed in [1]. We propose to identify this parameter automatically by a gradient descent algorithm to improve the restoration of a noisy image. Finally, we give numerical results to illustrate the performance of the automatic selection of this parameter and compare our numerical results with other image denoising approaches or algorithms.

Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs

We make connections between complexity of training of physics-informed neural networks (PINNs) and Kolmogorov n-width of the solution. Leveraging this connection, we then propose Lagrangian PINNs (LPINNs) as a partial differential equation (PDE)-informed solution for convection-dominated problems. PINNs employ neural-networks to find the solutions of PDE-constrained optimization problems with initial conditions and boundary conditions as soft or hard constraints. These soft constraints are often blamed to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the complexity of training (i) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts, and therefore becomes apparent in convection-dominated flows, and (ii) persists even when the boundary conditions are strictly enforced. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum learning, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection–diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain, i.e., on the characteristics. This approach is unique as it reduces the complexity of convection-dominated PINNs at the PDE level, instead of optimization strategies and/or schedulers. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called “complexity” of the problems, i.e., convection, compared to those in the traditional PINNs in the Eulerian framework.

A Locally Adapted Reduced-Basis Method for Solving Risk-Averse PDE-Constrained Optimization Problems

A Locally Adapted Reduced-Basis Method for Solving Risk-Averse PDE-Constrained Optimization Problems

Parallel generalized Lagrange–Newton method for fully coupled solution of PDE-constrained optimization problems with bound-constraints

In large-scale simulations of optimization problems constrained by partial differential equations (PDEs), the class of fully coupled methods is attracting great attention due to their impressive robustness and scalability. In this study, we introduce and investigate a parallel generalized Lagrange–Newton solver, which is based on the Lagrange active–set reduced–space (LASRS) method and the domain decomposition technique, for solving a family of PDE-constrained optimization problems with inequality constraints, i.e., the distributed control problem with bound-constraints. In the approach, a Lagrangian functional is constructed and the corresponding first-order optimality conditions are derived. After that, the resultant nonlinear algebraic system is solved by the active–set reduced–space method to guarantee the nonlinear consistency of the fully coupled system in a monolithic way. For the linear system arising at each generalized Newton iteration, the overlapping additive Schwarz preconditioners are employed to enhance the convergence of the linear iterations and the scalability of the large-scale simulations. Numerical results on the Tianhe supercomputer show that the fully coupled methods are robust and effective for some two- and three-dimensional distributed control problems with bound-constraints.

Automated computation of topological derivatives with application to nonlinear elasticity and reaction–diffusion problems

While topological derivatives have proven useful in applications of topology optimization and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the numerical computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction–diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behavior also for other, previously unknown problems, indicating the correctness of the procedure.

A class of nonlinear parabolic systems having standard growth and [formula omitted] data

In this paper, we are interested in the mathematical and numerical study of a reaction–diffusion system arising in the PDE-constrained optimization problems and image processing. The proposed model extends the recent work (Afraites et al., 2016) by taking L1 data. We prove the existence and uniqueness of entropy solutions on the Orlicz–Sobolev space. Then, we present numerical experiments to illustrate the performance of the L1 data choice and we discuss its efficiency compared to other image denoising algorithms.