PDE-constrained Optimization Problem Research Articles

Inverse design of a grating metasurface for enhancing spontaneous emission through hyperbolic metamaterials

This work is concerned with inverse design of the grating metasurface over hyperbolic metamaterials (HMMs) in order to enhance spontaneous emission (SE). We formulate the design problem as a PDE-constrained optimization problem and employ the gradient descent method to solve the underlying optimization problem. The adjoint-state method is applied to compute the gradient of the objective function efficiently. Computational results show that the SE efficiency of the optical structure with the optimized metasurface increases by 600% in the near field compared to the bare HMM layer. In particular, an optimized double-slot metasurface obtained by this design method enhances the SE intensity by a factor of over 100 in the observation region.

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.

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.

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.

Mean Field Games for Diel Vertical Migration with Diffusion

We present a theoretical framework, based on differential mean field games, for expressing diel vertical migration in the ocean as a game with a continuum of players. In such a game, each agent partially controls its own state by adjusting its vertical velocity but the vertical position in a water column is also subject to random fluctuations. A representative player has to make decisions based on aggregated information about the states of the other players. For this vertical differential game, we derive a mean field system of partial differential equations for finding a Nash equilibrium for the whole population. It turns out that finding Nash equilibria in the game is equivalent to solving a PDE-constrained optimization problem. We detail this equivalence when the expected fitness of the representative player can be approximated with a constant and solve both formulations numerically. We illustrate the results on simple numerical examples and construct several test cases to compare the two analytical approaches.

Optimal checkpointing for adjoint multistage time-stepping schemes

We consider checkpointing strategies that minimize the number of recomputations needed when performing discrete adjoint computations using multistage time-stepping schemes that require computing several substeps within one complete time step. Specifically, we propose two algorithms that can generate optimal checkpointing schedules under weak assumptions. The first is an extension of the seminal Revolve algorithm adapted to multistage schemes. The second algorithm, named CAMS, is developed based on dynamic programming, and it requires the least number of recomputations when compared with other algorithms. The CAMS algorithm is made publicly available in a library with bindings to C and Python. Numerical results show that the proposed algorithms can deliver up to two times the speedup compared with that of classical Revolve. Moreover, we discuss the utilization of the CAMS library in mature scientific computing libraries and demonstrate the ease of using it in an adjoint workflow. The proposed algorithms have been adopted by the PETScTSAdjoint library. Their performance has been demonstrated with a large-scale PDE-constrained optimization problem on a leadership-class supercomputer. This work is a significant extension of the authors’ conference paper (Zhang and Constantinescu, 2021).

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.

A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform

This study is a review of recent applications to seismic imaging of optimal transport based numerical tools. Modern seismic imaging methods used in the industry rely on the interpretation of the full signal. The characterization of the subsurface mechanical properties is formulated as a PDE-constrained optimization problem, solved through local optimization strategies. The choice of the misfit function used to measure the distance between actual seismic recordings and those synthetized by the solution of wave propagation PDE is crucial. Indeed, the conventional least-squares distance function leads to a non-convex optimization problem whose solution through local optimization then strongly depends on the initial guess. Using an optimal transport distance is an interesting alternative from its convexity properties with respect to translation and dilation. Specific strategies need however to be implemented as seismic data are oscillatory, while the optimal transport theory has been developed for the comparison of positive measures. In this study we review two optimal transport based misfit functions, from their mathematical formulation to their application to field data through their numerical implementation. Advantages and drawbacks of both strategies are discussed. Numerical experiments show that they represent two interesting and complementary alternative to the classical least-squares misfit function, mitigating the dependency to the choice of the initial guess.

Robust preconditioned one-shot methods and direct-adjoint-looping for optimizing Reynolds-averaged turbulent flows

We compare the performance of direct-adjoint-looping (DAL) and one-shot methods in a design optimization task involving turbulent flow modeled using Reynolds-Averaged-Navier–Stokes equations. Two preconditioned variants of the one-shot algorithm are proposed and tested. The role of an approximate Hessian as a preconditioner for the one-shot method iterations is highlighted. We find that the preconditioned one-shot methods can solve the PDE-constrained optimization problem with the cost of computation comparable (about fourfold) to that of the simulation run alone. This cost is substantially less than that of DAL, which requires O(10) direct-adjoint loops to converge. The optimization results arising from the one-shot method can be used for optimal sensor/actuator placement tasks, or to provide a reference trajectory to be used for online feedback control applications.

An efficient two-level overlapping domain decomposition method for recovering unsteady sources of 3D parabolic problems

We develop a parallel two-level domain decomposition method for the 3D unsteady source identification problem governed by a parabolic partial differential equation (PDE). The domain of the PDE is firstly decomposed into several overlapping subdomains and the original inverse source identification problem is then transformed into smaller independent subproblems defined on these subdomains. Each subproblem is formulated as a PDE-constrained optimization problem with appropriate conditions prescribed on the inner boundaries and discretized by finite element method. The resulting coupled algebraic systems are solved simultaneously by restarted GMRES method with a space-time restricted additive Schwarz preconditioner. When forming the preconditioner, a second level of domain decomposition is introduced for each subdomain. The solutions of these subproblems are combined together to form an approximated global solution to the original inverse problem by discarding the overlapping parts of the solution. Since all the subproblems are solved independently, the two-level domain decomposition method provides higher degree of parallelism and saves much computing time. Numerical experiments conducted on a supercomputer with thousands of processor cores validate the efficiency and robustness of the proposed approach.

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI is often implemented by solving for the frequency-domain representation of the wavefield using the finite-difference method. The approach requires matrix inversions and affords limited flexibility to accommodate irregular model geometries. On the other hand, the physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN) to provide flexible continuous functional approximations of the solutions without matrix inversions. By including a data-constrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network, and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

An optimal variable exponent model for Magnetic Resonance Images denoising

• A novel PDE-constrained optimization problem is proposed for MRI denoising task. • The proposed optimization procedure computes the variable exponent and the clean image for MRI images. • The comparative results demonstrates the performance on two MRI datasets This paper investigates a novel PDE-constrained optimization model with discontinuous variable exponent p ( x ) identification. Since the parameter p is always related to a better approximation of the image gradient, its computation plays a critical role in preserving the image texture. Analytically, we include results on the approximation of this parameter as well as the resolution of the encountered PDE in a well posed framework. In addition, to resolve the PDE-constrained minimization problem, we proposed a modified primal-dual algorithm. Finally, numerical results are provided to compute the parameter p and also to remove high intensity of noise. The proposed algorithm simultaneously keep safe fine details and important features in medical image applications (Magnetic Resonance Images (MRI)) with numerous comparisons to show the performance of the proposed approach.

Learning viscoelasticity models from indirect data using deep neural networks

We propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks. We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

A Novel Reconstruction Method to Increase Spatial Resolution in Electron Probe Microanalysis

The spatial resolution of electron probe microanalysis (EPMA), a non-destructive method to determine the chemical composition of materials, is currently restricted to a pixel size larger than the volume of interaction between beam electrons and the material, as a result of limitations on the underlying k-ratio model. Using more sophisticated models to predict k-ratios while solving the inverse problem of reconstruction offers a possibility to increase the spatial resolution. Here, a k-ratio model based on the deterministic M1-model in Boltzmann Continuous Slowing-Down approximation (BCSD) will be utilized to present a reconstruction method for EPMA which is implemented as a PDE-constrained optimization problem. Iterative gradient-based optimization techniques are used in combination with the adjoint state method to calculate the gradient in order to solve the optimization problem efficiently. The accuracy of the spatial resolution still depends on the number and quality of the measured data, but in contrast to conventional reconstruction methods, an overlapping of the interaction volumes of different measurements is permissible without ambiguous solutions. The combination of k-ratios measured with various electron beam configurations is necessary for a high resolution. Attempts to reconstruct materials with synthetic data show challenges that occur with small reconstruction pixels, but also indicate the potential to improve the spatial resolution in EPMA using the presented method.

A simple PDE-constrained Optimization Problem to Evaluate the Strategy for Fishery Resource Transportation

A simple PDE-constrained Optimization Problem to Evaluate the Strategy for Fishery Resource Transportation

A parallel multilevel domain decomposition method for source identification problems governed by elliptic equations

In this paper we develop a parallel multilevel domain decomposition method for large-scale source identification problems governed by elliptic equations. A popular approach is to formulate the inverse problem as a PDE-constrained optimization problem. The minima satisfies a Karush–Kuhn–Tucker (KKT) system consisting of the state, adjoint and source equations which is rather difficult to solve on parallel computers. We propose and study a parallel method that decomposes the optimization problem on the global domain into subproblems on overlapping subdomains, each subdomain is further decomposed to form an additive Schwarz preconditioner for solving these smaller subproblems simultaneously with a preconditioned Krylov subspace method. For each subproblem, the overlapping part of the solution is discarded and the remaining non-overlapping part of the solution is put together to obtain an approximated global solution to the inverse problem. Since all the subproblems are solved independently, the multilevel domain decomposition method has the advantage of higher degree of parallelism. Numerical experiments show that the algorithm is accurate in terms of the reconstruction error and has reasonably good speedup in terms of the computing time. The efficiency and robustness of the proposed approach on a parallel computer with more than 1, 000 processors are reported.

Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

We consider the problem of designing a cloak for waves described by the Helmholtz equation from an integer programming point of view. The problem can be modeled as a PDE-constrained optimization problem with integer-valued control inputs that are distributed in the computational domain. A first-discretize-then-optimize approach results in a large-scale mixed-integer nonlinear program that is in general intractable because of the large number of integer variables that arise from the discretization of the domain. Instead, we propose an efficient algorithm that is able to approximate the local infima of the underlying nonconvex infinite-dimensional problem arbitrarily close without the need to solve the discretized finite-dimensional integer programs to optimality. We optimize only the continuous relaxations of the approximations for local minima and then apply the sum-up rounding methodology to obtain integer-valued controls. If the solutions of the discretized continuous relaxations converge to a local minimizer of the continuous relaxation, then the resulting discrete-valued control sequence converges weakly $$^*$$ in $$L^\infty$$ to the same local minimizer. These approximation properties follow under suitable refinements of the involved discretization grids. Our results use familiar concepts arising from the analytical properties of the underlying PDE and complement previous results, derived from a topology optimization point of view.

Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes

Abstract