Adjoint State Research Articles

Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems

We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.

Non-coercive Neumann Boundary Control Problems

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.

Acoustic shape optimization using energy stable curvilinear finite differences

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inverting for the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance.

Multimaterial filtering applied to the topology optimization of a permanent magnet synchronous machine

PurposeThis study aims to propose a general methodology to handle multimaterial filtering for density-based topology optimization containing periodic or antiperiodic boundary conditions, which are expected to reduce the simulation time of electrical machines. The optimization of the material distribution in a permanent magnet synchronous machine rotor illustrates the relevance of this approach.Design/methodology/approachThe optimization algorithm relies on an augmented Lagrangian with a projected gradient descent. The 2D finite element method computes the physical and adjoint states to evaluate the objective function and its sensitivities. Concerning regularization, a mathematical development leads to a multimaterial convolution filtering methodology that is consistent with the boundary conditions and helps eliminate artifacts.FindingsThe method behaves as expected and shows the superiority of multimaterial topology optimization over bimaterial topology optimization for the chosen test case. Unlike the standard approach that uses a cropped convolution kernel, the proposed methodology does not artificially reflect the limits of the simulation domain in the optimized material distribution.Originality/valueAlthough filtering is a standard tool in topology optimization, no attention has previously been paid to the influence of periodic or antiperiodic boundary conditions when dealing with different natures of materials. The comparison between the bimaterial and multimaterial topology optimization of a permanent magnet machine rotor without symmetry constraints constitutes another originality of this work.

Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation

The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided.

Turbulence driven goal-oriented anisotropic mesh adaptation for RANS simulations in aerodynamics

Anisotropic mesh adaptation is starting to play a role of great importance within the CFD community. In fact, such a tool allows to accurately capture the physics of the problem in an automatic way, by reducing at the same time the computational resources, in contrast to the best-practice meshing guidelines, which require an a priori knowledge of the flow solution and, for this reason, are approximate and error-prone. In the present work, we consider anisotropic mesh adaptation for steady flows to get mesh-independent solutions with a small number of vertices, that is, we achieve the so-called early capturing property. Two classes of methods exist: feature-based mesh adaptation where the minimization of the interpolation error associated with sensor field is targeted and goal-oriented mesh adaptation where the minimization of the discretization error associated with a goal functional is carried out, such as the lift or the drag coefficients. The goal-oriented mesh adaptation requires the adjoint state computation, thus a crucial part of the treatment deals with this topic. The novelty of the present approach is the taking into account of the turbulent primal and adjoint variables, and the corresponding fluxes (including turbulence sources) inside the mesh adaptation process and the adjoint computation, as up to now only the inviscid and the viscous contributions have been considered in this kind of processes. We show how such additions are extremely beneficial for goal-oriented mesh adaptation, as the resulting meshes are able to capture the physics of the problem with a higher accuracy, in contrast to the exclusively inviscid- or viscous-driven adaptation. Eventually, as the mesh adaptation process tends to produce highly unstructured anisotropic meshes, a relevant part of the present work is devoted to the description of a robust solver capable of solving the RANS equations on such meshes.

Adjoint Synthesis for Trans‐Oceanic Tsunami Waveforms and Simultaneous Inversion of Fault Geometry and Slip Distribution

AbstractTsunamis propagate over long distances and can cause widespread damage even after crossing ocean basins. Prediction of tsunamis in distant areas based on observations near their sources is critical to mitigating damage. In recent years, the accuracy of numerical models of trans‐oceanic tsunami propagation has improved significantly due to the incorporation of effects such as the solid earth response to tsunami loading and wave dispersion. However, these models are computationally expensive and have not been fully utilized for real‐time prediction. Here, we derive the adjoint operator for the linear set of equations describing deep‐ocean tsunami propagation and show how a pre‐computed database of adjoint states can achieve rapid synthesis of tsunami waveforms at target sites from nonpoint arbitrary tsunami sources. The adjoint synthesis method allows for an exhaustive parameter search for tsunami source estimation. A method for simultaneous inversion of fault geometry and slip distribution using adjoint synthesis with Sequential Monte Carlo method was proposed and applied to the 2012 Haida Gwaii earthquake tsunami. The influence of model accuracy and the amount of observed data on the estimation of tsunami sources and waveforms was examined. It was found that with a highly accurate propagation model, using only a limited amount of observed data produced source and waveform estimates very similar to the final models obtained with much larger data sets. The final inferred fault model involved megathrust slip distributed between the Haida Gwaii trench and the Queen Charlotte fault. The proposed method can also quantify the uncertainty of the waveform forecasts.

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.

An optimal control problem for diffusion–precipitation model

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

Automatic design of interpretable control laws through parametrized Genetic Programming with adjoint state method gradient evaluation

This work investigates the application of a Local Search (LS) enhanced Genetic Programming (GP) algorithm to the control scheme’s design task. The combination of LS and GP aims to produce an interpretable control law as similar as possible to the optimal control scheme reference. Inclusive Genetic Programming (IGP), a GP heuristic capable of promoting and maintaining the population diversity, is chosen as the GP algorithm since it proved successful on the considered task. IGP is enhanced with the Operators Gradient Descent (OPGD) approach, which consists of embedding learnable parameters into the GP individuals. These parameters are optimized during and after the evolutionary process. Moreover, the OPGD approach is combined with the adjoint state method to evaluate the gradient of the objective function. The original OPGD was formulated by relying on the backpropagation technique for the gradient’s evaluation, which is impractical in an optimization problem involving a dynamical system because of scalability and numerical errors. On the other hand, the adjoint method allows for overcoming this issue. Two experiments are formulated to test the proposed approach, named Operator Gradient Descent - Inclusive Genetic Programming (OPGD-IGP): the design of a Proportional–Derivative (PD) control law for a harmonic oscillator and the design of a Linear Quadratic Regulator (LQR) control law for an inverted pendulum on a cart. OPGD-IGP proved successful in both experiments, being capable of autonomously designing an interpretable control law similar to the optimal ones, both in terms of shape and control gains.

Existence of solution for an optimal control problem in a heterogeneous porous medium

Abstract The paper explores an optimal control problem concerning transport processes in a porous medium. The transport phenomena is governed by diffusion-reaction equations, which is basically a semi-linear parabolic system. An $L^{2}$-cost (energy) functional is introduced, and control (distributed) is applied in the porous part of the medium. The primary objective is to characterize a given control to be an optimal control and analyse the relationship between optimal control and the adjoint state. The analysis commences with the microscopic description of the controlled system followed by upscaling the system via periodic homogenization.

Model order reduction based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods.

Error estimates of characteristic finite elements for bilinear convection–diffusion optimal control problems

This paper investigates a fully discrete characteristic finite element approximation of bilinear unsteady convection–diffusion optimal control problems. The characteristic line method is used to treat the convection term and the finite element method is adopted to treat the diffusion term. The state and adjoint state are discretized by piecewise linear functions, the control is approximated by piecewise constant functions. A priori error estimates are derived for the state, adjoint state and control variables. Some numerical examples are provided to confirm our theoretical findings.

Characterization of train kinematics and source wavelets from near-field seismic data

SUMMARY Train traffic is a powerful source of seismic waves, with many applications for passive seismic imaging. Seismic signals were recorded a few metres away from the railway track. These records display harmonious waveforms below 15 Hz for trains driving at speeds of around 100 km hr−1. The sensors record an apparent wavelet emitted by the interaction of the axle on a few of the closest sleepers. From this, we build a simple modelling tool using shifted wavelets to simulate a train signal. The relationship involves the varying train speed, the distances between each axle, and the wavelet emitted by each axle. We propose a nonlinear deconvolution method to invert this relationship. We use a local minimization algorithm with gradients derived by the adjoint state method, and use a frequency continuation technique. A linearized picking-based inversion initializes the nonlinear inversion. On real data, we apply this automatic workflow to 300 train passages, with an excellent match between the best simulation and the data. We identify the trains decelerating as they enter a train station. We also identify the train type based on inverted wheel spacing with centimetric accuracy. The inverted wavelets are consistent with the assumption that trains emit seismic waves by bending the rail above sleepers, although the theory does not explain why the inverted wavelet is not zero phase. This automated kinematic inversion algorithm may allow for contactless railway monitoring, and be used for source characterization for subsurface monitoring below railway tracks.

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Interior point methods in optimal control

This paper deals with Interior Point Methods (IPMs) for Optimal Control Problems (OCPs) with pure state and mixed constraints. This paper establishes a complete proof of convergence of IPMs for a general class of OCPs. Convergence results are proved for primal variables, namely state and control variables, and for dual variables, namely, the adjoint state, and the constraints multipliers. In addition, the presented convergence result does not rely on a strong convexity assumption. Finally, this paper compares the performances of a primal and a primal-dual implementation of IPMs in optimal control in three examples.

Modified [formula omitted] interior penalty analysis for fourth order Dirichlet boundary control problem and a posteriori error estimates

We revisit the L2 norm error estimate for the C0-interior penalty analysis of fourth order Dirichlet boundary control problem. The L2 norm error estimate for the optimal control is derived under reduced regularity assumption and this analysis can be carried out on any convex polygonal domain. Moreover, residual based a posteriori error bounds are derived for the optimal control, state and adjoint state variables under minimal regularity assumptions. The error estimator is shown to be reliable and locally efficient. The theoretical findings are illustrated by numerical experiments.

Eikonal-equation-based characteristic reflection traveltime tomography

Automatically picking reflection traveltimes from the prestack shot gathers and positioning and updating the reflector depths during inversion are challenging in conventional eikonal-equation-based adjoint-state reflection traveltime tomography methods. To solve these problems, we develop an eikonal-equation-based adjoint-state characteristic reflection traveltime tomography (ASCRT) method. This method automatically extracts the reflection traveltimes by sequentially applying characteristic reflector picking in the depth migration section, followed by eikonal-equation-based traveltime calculation and an event-tracking algorithm. We also develop an efficient eikonal-equation-based kinematic imaging method to rapidly position and update the reflector depths. With our ASCRT method, the characteristic reflector positions and the velocity above the reflector are alternately updated. Our ASCRT method is performed with a layer-stripping strategy that sequentially uses selected characteristic reflectors to update the velocity model from shallow to deep. Compared with the wave equation-based reflection traveltime inversion methods, the efficiency of the ASCRT method is improved by several orders of magnitude, and the memory consumption is remarkably reduced. Synthetic and field data examples are presented to demonstrate the effectiveness and efficiency of our method.

Converted-wave reflection traveltime inversion with free-surface multiples for ocean-bottom-node data

Advancements in ocean-bottom-node (OBN) technologies have enabled the recording of high-quality multicomponent seismic data, which can provide crucial information about subsurface properties through elastic seismic imaging. However, imaging multicomponent data is challenging due to the requirement for P- and S-wave velocity models. Conventional processing relies on estimating the S-wave velocity model using a primary converted wave (C wave) because the pure S-wave primaries are not available if using standard air-gun sources. When a free surface is present, the C waves recorded with geophones can be contaminated by source-side P-wave multiples, especially water-layer reverberations, which are difficult to remove even using state-of-the-art wavefield decomposition methods. These C-wave multiples can cause significant mismatches between synthetic and observed data during reflection traveltime or waveform inversion if only primary P-to-S (PS) conversions are simulated. To address these issues and reliably build an S-wave velocity model for PS imaging, we develop a C-wave reflection traveltime inversion method that takes source-side free-surface multiples into account. The traveltime misfit of C-wave data is extracted using dynamic image warping to ensure a better match between the synthetic and observed data. The functional gradient of the objective function is derived using the adjoint state method. Based on sensitivity kernel analyses, a convenient gradient preconditioning strategy is developed to robustly separate the contribution of primary and multiple C waves during backpropagation. This strategy can effectively mitigate the high-wavenumber crosstalk associated with nonphysical wavepaths in the gradient and thus allow more accurate updating for the S-wave velocity model. The synthetic data and shallow-water OBN data from the East China Sea indicate that this approach can reliably recover the S-wave macrovelocity structures for PS imaging.

Damage identification based on XFEM: inverse analysis and experimental validation

AbstractModel‐based inverse analyses emerge as an alternative means to reconstruct buried objects in structures. In this paper, we introduced an inverse analysis using XFEM and adjoint method to identify the cracks, which is based on laboratory experimental data. XFEM model allows precisely reproducing the crack forms in the structures, whilst adjoint state with level set for inverse problem allows calculating the gradient of complex geometrical structures and avoiding the local minima traps. Besides proposing a modification XFEM to model reinforced concrete behaviors, we also establish the adjoint formulas for inverse analysis based on the measured displacement and strain in the experiments. Finally, the preliminary results are verified with experimental data.