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Measurement and analysis of high energetic particles for scientific, medical or industrial applications is a complex procedure, requiring the design of sophisticated detector and data processing systems. The development of adaptive and differentiable software pipelines using a combination of conventional and machine learning algorithms is therefore getting ever more important to optimize and operate the system efficiently while maintaining end-to-end (E2E) differentiability. We propose for the application of charged particle tracking an E2E differentiable decision-focused learning scheme using graph neural networks with combinatorial components solving a linear assignment problem for each detector layer. We demonstrate empirically that including differentiable variations of discrete assignment operations allows for efficient network optimization, working better or on par with approaches that lack E2E differentiability. In additional studies, we dive deeper into the optimization process and provide further insights from a loss landscape perspective. We demonstrate that while both methods converge into similar performing, globally well-connected regions, they suffer under substantial predictive instability across initialization and optimization methods, which can have unpredictable consequences on the performance of downstream tasks such as image reconstruction. We also point out a dependency between the interpolation factor of the gradient estimator and the prediction stability of the model, suggesting the choice of sufficiently small values. Given the strong global connectivity of learned solutions and the excellent training performance, we argue that E2E differentiability provides, besides the general availability of gradient information, an important tool for robust particle tracking to mitigate prediction instabilities by favoring solutions that perform well on downstream tasks.

We have applied an operator-overloading forward-mode algorithmic differentiation tool to the Monte-Carlo particle simulation toolkit Geant4. Our differentiated version of Geant4 allows computing mean pathwise derivatives of user-defined outputs of Geant4 applications with respect to user-defined inputs. This constitutes a major step towards enabling gradient-based optimization techniques in high-energy physics, as well as other application domains of Geant4. This is a preliminary report on the technical aspects of applying operator-overloading AD to Geant4, as well as a first analysis of some results obtained by our differentiated Geant4 prototype. We plan to follow up with a more refined analysis.

Among the well-known methods to approximate derivatives of expectancies computed by Monte-Carlo simulations, averages of pathwise derivatives are often the easiest one to apply. Computing them via algorithmic differentiation typically does not require major manual analysis and rewriting of the code, even for very complex programs like simulations of particle-detector interactions in high-energy physics. However, the pathwise derivative estimator can be biased if there are discontinuities in the program, which may diminish its value for applications. This work integrates algorithmic differentiation into the electromagnetic shower simulation code HepEmShow based on G4HepEm, allowing us to study how well pathwise derivatives approximate derivatives of energy depositions in a sampling calorimeter with respect to parameters of the beam and geometry. We found that when multiple scattering is disabled in the simulation, means of pathwise derivatives converge quickly to their expected values, and these are close to the actual derivatives of the energy deposition. Additionally, we demonstrate the applicability of this novel gradient estimator for stochastic gradient-based optimization in a model example., Comment: 12 pages, 11 figures, 2 tables

In shared-memory parallel automatic differentiation, shared inputs among simultaneous thread-local preaccumulations lead to data races if Jacobians are accumulated with a single, shared vector of adjoint variables. In this work, we discuss the benefits and tradeoffs of re-enabling such preaccumulations by a transition to suitable local adjoint variables. In particular, we assess the performance of mapped local adjoints in discrete adjoint computations in the multiphysics simulation suite SU2., Comment: 9 pages, 4 figures, 1 table

The open-source multiphysics suite SU2 features discrete adjoints by means of operator overloading automatic differentiation (AD). While both primal and discrete adjoint solvers support MPI parallelism, hybrid parallelism using both MPI and OpenMP has only been introduced for the primal solvers so far. In this work, we enable hybrid parallel discrete adjoint solvers. Coupling SU2 with OpDiLib, an add-on for operator overloading AD tools that extends AD to OpenMP parallelism, marks a key step in this endeavour. We identify the affected parts of SU2's advanced AD workflow and discuss the required changes and their tradeoffs. Detailed performance studies compare MPI parallel and hybrid parallel discrete adjoints in terms of memory and runtime and unveil key performance characteristics. We showcase the effectiveness of performance optimizations and highlight perspectives for future improvements. At the same time, this study demonstrates the applicability of OpDiLib in a large code base and its scalability on large test cases, providing valuable insights for future applications both within and beyond SU2., Comment: 28 pages, 9 figures, 2 listings; new layout, revised section structure, polishing and small updates

We formulate and solve data-driven aerodynamic shape design problems with distributionally robust optimization (DRO) approaches. Building on the findings of the work \cite{gotoh2018robust}, we study the connections between a class of DRO and the Taguchi method in the context of robust design optimization. Our preliminary computational experiments on aerodynamic shape optimization in transonic turbulent flow show promising design results.

In this document we describe a model of an array of water Cherenkov detectors proposed to study ultra-high-energy gamma rays in the southern hemisphere, and a continuous model of secondary particles produced on the ground from gamma and proton showers. We use the model of the detector and the parametrization of showers for the identification of the most promising configuration of detector elements, using a likelihood ratio test statistic to classify showers and a stochastic gradient descent technique to maximize a utility function describing the measurement precision on the gamma-ray flux., Comment: 47 pages, 15 figures

In this article we examine recent developments in the research area concerning the creation of end-to-end models for the complete optimization of measuring instruments. The models we consider rely on differentiable programming methods and on the specification of a software pipeline including all factors impacting performance -- from the data-generating processes to their reconstruction and the extraction of inference on the parameters of interest of a measuring instrument -- along with the careful specification of a utility function well aligned with the end goals of the experiment. Building on previous studies originated within the MODE Collaboration, we focus specifically on applications involving instruments for particle physics experimentation, as well as industrial and medical applications that share the detection of radiation as their data-generating mechanism., Comment: 70 pages, 17 figures. To be submitted to journal

We present a first-order method for solving constrained optimization problems. The method is derived from our previous work, a modified search direction method inspired by singular value decomposition. In this work, we simplify its computational framework to a ``gradient descent akin'' method (GDAM), i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. We give fundamental theoretical guarantees on the global convergence of the method. This work focuses on the algorithms and applications of GDAM. We present computational algorithms that adapt common strategies for the gradient descent method. We demonstrate the potential of the method using two engineering applications, shape optimization and sensor network localization. When practically implemented, GDAM is robust and very competitive in solving the considered large and challenging optimization problems.

Broadband noise reduction is a significant problem in aerospace and industrial applications. Specifically, the noise generated from the trailing edge of an airfoil poses a challenging problem with various proposed solutions. This study investigates the porous trailing edge treatment. We use surrogate-based gradient-free optimization and an empirical noise model to efficiently explore the design space and find the optimal porosity distribution. As a result, a predicted 8-10 dB reduction in the broadband 300-5000 Hz was achieved. Furthermore, the optimal design emphasizes the design space's complexity and global exploration's difficulty. Further, the optimal design presents a low porous solution while constituting significant noise reduction., Comment: 5 pages, 6 figures, added affiliation, fixed typos, homog. citation style

Tools for algorithmic differentiation (AD) provide accurate derivatives of computer-implemented functions for use in, e. g., optimization and machine learning (ML). However, they often require the source code of the function to be available in a restricted set of programming languages. As a step towards making AD accessible for code bases with cross-language or closed-source components, we recently presented the forward-mode AD tool Derivgrind. It inserts forward-mode AD logic into the machine code of a compiled program using the Valgrind dynamic binary instrumentation framework. This work extends Derivgrind, adding the capability to record the real-arithmetic evaluation tree, and thus enabling operator overloading style reverse-mode AD for compiled programs. We maintain the high level of correctness reported for Derivgrind's forward mode, failing the same few testcases in an extensive test suite for the same well-understood reasons. Runtime-wise, the recording slows down the execution of a compiled 64-bit benchmark program by a factor of about 180., Comment: 17 pages, 5 figures, 1 listing

Algorithmic differentiation (AD) is a set of techniques that provide partial derivatives of computer-implemented functions. Such a function can be supplied to state-of-the-art AD tools via its source code, or via an intermediate representation produced while compiling its source code. We present the novel AD tool Derivgrind, which augments the machine code of compiled programs with forward-mode AD logic. Derivgrind leverages the Valgrind instrumentation framework for a structured access to the machine code, and a shadow memory tool to store dot values. Access to the source code is required at most for the files in which input and output variables are defined. Derivgrind's versatility comes at the price of scaling the run-time by a factor between 30 and 75, measured on a benchmark based on a numerical solver for a partial differential equation. Results of our extensive regression test suite indicate that Derivgrind produces correct results on GCC- and Clang-compiled programs, including a Python interpreter, with a small number of exceptions. While we provide a list of scenarios that Derivgrind does not handle correctly, nearly all of them are academic counterexamples or originate from highly optimized math libraries. As long as differentiating those is avoided, Derivgrind can be applied to an unprecedentedly wide range of cross-language or partially closed-source software with little integration efforts., Comment: 21 pages, 3 figures, 3 tables, 5 listings

Aerosol particles play an important role in the climate system by absorbing and scattering radiation and influencing cloud properties. They are also one of the biggest sources of uncertainty for climate modeling. Many climate models do not include aerosols in sufficient detail due to computational constraints. In order to represent key processes, aerosol microphysical properties and processes have to be accounted for. This is done in the ECHAM-HAM global climate aerosol model using the M7 microphysics, but high computational costs make it very expensive to run with finer resolution or for a longer time. We aim to use machine learning to emulate the microphysics model at sufficient accuracy and reduce the computational cost by being fast at inference time. The original M7 model is used to generate data of input-output pairs to train a neural network on it. We are able to learn the variables' tendencies achieving an average $R^2$ score of $77.1\% $. We further explore methods to inform and constrain the neural network with physical knowledge to reduce mass violation and enforce mass positivity. On a GPU we achieve a speed-up of up to over 64x compared to the original model., Comment: arXiv admin note: text overlap with arXiv:2109.10593

The full optimization of the design and operation of instruments whose functioning relies on the interaction of radiation with matter is a super-human task, given the large dimensionality of the space of possible choices for geometry, detection technology, materials, data-acquisition, and information-extraction techniques, and the interdependence of the related parameters. On the other hand, massive potential gains in performance over standard, "experience-driven" layouts are in principle within our reach if an objective function fully aligned with the final goals of the instrument is maximized by means of a systematic search of the configuration space. The stochastic nature of the involved quantum processes make the modeling of these systems an intractable problem from a classical statistics point of view, yet the construction of a fully differentiable pipeline and the use of deep learning techniques may allow the simultaneous optimization of all design parameters. In this document we lay down our plans for the design of a modular and versatile modeling tool for the end-to-end optimization of complex instruments for particle physics experiments as well as industrial and medical applications that share the detection of radiation as their basic ingredient. We consider a selected set of use cases to highlight the specific needs of different applications., Comment: 109 pages, 32 figures. To be submitted to Reviews in Physics

Objective. Algorithmic differentiation (AD) can be a useful technique to numerically optimize design and algorithmic parameters by, and quantify uncertainties in, computer simulations. However, the effectiveness of AD depends on how "well-linearizable" the software is. In this study, we assess how promising derivative information of a typical proton computed tomography (pCT) scan computer simulation is for the aforementioned applications. Approach. This study is mainly based on numerical experiments, in which we repeatedly evaluate three representative computational steps with perturbed input values. We support our observations with a review of the algorithmic steps and arithmetic operations performed by the software, using debugging techniques. Main results. The model-based iterative reconstruction (MBIR) subprocedure (at the end of the software pipeline) and the Monte Carlo (MC) simulation (at the beginning) were piecewise differentiable. Jumps in the MBIR function arose from the discrete computation of the set of voxels intersected by a proton path. Jumps in the MC function likely arose from changes in the control flow that affect the amount of consumed random numbers. The tracking algorithm solves an inherently non-differentiable problem. Significance. The MC and MBIR codes are ready for the integration of AD, and further research on surrogate models for the tracking subprocedure is necessary., Comment: 27 pages, 11 figures

On the one hand, Sobolev gradient smoothing can considerably improve the performance of aerodynamic shape optimization and prevent issues with regularity. On the other hand, Sobolev smoothing can also be interpreted as an approximation for the shape Hessian. This paper demonstrates, how Sobolev smoothing, interpreted as a shape Hessian approximation, offers considerable benefits, although the parameterization is smooth in itself already. Such an approach is especially beneficial in the context of simultaneous analysis and design, where we deal with inexact flow and adjoint solutions, also called One Shot optimization. Furthermore, the incorporation of the parameterization allows for direct application to engineering test cases, where shapes are always described by a CAD model. The new methodology presented in this paper is used for reference test cases from aerodynamic shape optimization and performance improvements in comparison to a classical Quasi-Newton scheme are shown., Comment: Version 2 of the paper. We included revisions and suggestions from peer review. We corrected spelling and grammar errors throughout the paper

We present the new software OpDiLib, a universal add-on for classical operator overloading AD tools that enables the automatic differentiation (AD) of OpenMP parallelized code. With it, we establish support for OpenMP features in a reverse mode operator overloading AD tool to an extent that was previously only reported on in source transformation tools. We achieve this with an event-based implementation ansatz that is unprecedented in AD. Combined with modern OpenMP features around OMPT, we demonstrate how it can be used to achieve differentiation without any additional modifications of the source code; neither do we impose a priori restrictions on the data access patterns, which makes OpDiLib highly applicable. For further performance optimizations, restrictions like atomic updates on adjoint variables can be lifted in a fine-grained manner. OpDiLib can also be applied in a semi-automatic fashion via a macro interface, which supports compilers that do not implement OMPT. We demonstrate the applicability of OpDiLib for a pure operator overloading approach in a hybrid parallel environment. We quantify the cost of atomic updates on adjoint variables and showcase the speedup and scaling that can be achieved with the different configurations of OpDiLib in both the forward and the reverse pass., Comment: 31 pages, 13 figures, 3 tables, 13 listings; new layout, additional references, refocused Section 3 (former Section 4), extended performance tests, overall polishing and shortening

With more efficient structures, last trends in aeronautics have witnessed an increased flexibility of wings, calling for adequate design and optimization approaches. To correctly model the coupled physics, aerostructural optimization has progressively become more important, being nowadays performed also considering higher-fidelity discipline methods, i.e., CFD for aerodynamics and FEM for structures. In this paper a methodology for high-fidelity gradient-based aerostructural optimization of wings, including aerodynamic and structural nonlinearities, is presented. The main key feature of the method is its modularity: each discipline solver, independently employing algorithmic differentiation for the evaluation of adjoint-based sensitivities, is interfaced at high-level by means of a wrapper to both solve the aerostructural primal problem and evaluate exact discrete gradients of the coupled problem. The implemented capability, ad-hoc created to demonstrate the methodology, and freely available within the open-source SU2 multiphysics suite, is applied to perform aerostructural optimization of aeroelastic test cases based on the ONERA M6 and NASA CRM wings. Single-point optimizations, employing Euler or RANS flow models, are carried out to find wing optimal outer mold line in terms of aerodynamic efficiency. Results remark the importance of taking into account the aerostructural coupling when performing wing shape optimization.

The identification of primal variables and adjoint variables is usually done via indices in operator overloading algorithmic differentiation tools. One approach is a linear management scheme, which is easy to implement and supports memory optimization for copy statements. An alternative approach performs a reuse of indices, which requires more implementation effort but results in much smaller adjoint vectors. Therefore, the vector mode of algorithmic differentiation scales better with the reuse management scheme. In this paper, we present a novel approach that reuses the indices and allows the copy optimization, thus combining the advantages of the two aforementioned schemes. The new approach is compared to the known approaches on a simple synthetic test case and a real-world example using the computational fluid dynamics solver SU2., Comment: 15 pages, 4 figures, 4 tables

We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness and the applicability of AutoMat by integrating it into the FFT-based homogenization scheme of Moulinec and Suquet and discuss the benefits of using AutoMat with respect to runtime and solution accuracy for an elasto-viscoplastic example., Comment: 28 pages, 15 figures, 7 tables; new layout, more detailed proof of Theorem 1

Most machine learning methods require careful selection of hyper-parameters in order to train a high performing model with good generalization abilities. Hence, several automatic selection algorithms have been introduced to overcome tedious manual (try and error) tuning of these parameters. Due to its very high sample efficiency, Bayesian Optimization over a Gaussian Processes modeling of the parameter space has become the method of choice. Unfortunately, this approach suffers from a cubic compute complexity due to underlying Cholesky factorization, which makes it very hard to be scaled beyond a small number of sampling steps. In this paper, we present a novel, highly accurate approximation of the underlying Gaussian Process. Reducing its computational complexity from cubic to quadratic allows an efficient strong scaling of Bayesian Optimization while outperforming the previous approach regarding optimization accuracy. The first experiments show speedups of a factor of 162 in single node and further speed up by a factor of 5 in a parallel environment., Comment: 14 pages; 6 figures; 4 tables; Accepted in proceedings of MLHPC 2019: Fifth International Workshop on Machine Learning in High Performance Computing Environments, Super Computing Conference 2019, Denver, Colorado

Current training methods for deep neural networks boil down to very high dimensional and non-convex optimization problems which are usually solved by a wide range of stochastic gradient descent methods. While these approaches tend to work in practice, there are still many gaps in the theoretical understanding of key aspects like convergence and generalization guarantees, which are induced by the properties of the optimization surface (loss landscape). In order to gain deeper insights, a number of recent publications proposed methods to visualize and analyze the optimization surfaces. However, the computational cost of these methods are very high, making it hardly possible to use them on larger networks. In this paper, we present the GradVis Toolbox, an open source library for efficient and scalable visualization and analysis of deep neural network loss landscapes in Tensorflow and PyTorch. Introducing more efficient mathematical formulations and a novel parallelization scheme, GradVis allows to plot 2d and 3d projections of optimization surfaces and trajectories, as well as high resolution second order gradient information for large networks., Comment: 10 pages, 8 figures

Global aerodynamic design optimization using Euler or Navier-Stokes equations requires very reliable surrogate modeling techniques since the computational effort for the underlying flow simulations is usually high. In general, for such problems, the number of samples that can be generated to train the surrogate models is very limited due to restricted computational resources. On the other hand, recent developments in adjoint methods enable nowadays evaluation of gradient information at a reasonable computational cost for a wide variety of engineering problems. Therefore, a much richer data set can be collected using an adjoint solver in a Design of Experiment framework. In the present work, we present a novel aggregation method, which enables the state of the art surrogate models to incorporate extra gradient information without causing overfitting problems. Therefore, accurate surrogate models with relatively large number of design parameters can be trained from a small set of samples. We also present results two well known benchmark design optimization problems showing efficiency and robustness of the new method., Comment: 10 pages, 2 figures

As more and more multiphysics effects are entering the field of CFD simulations, this raises the question how they can be accurately captured in gradient computations for shape optimization. The latter has been successfully enriched over the last years by the use of (discrete) adjoints. One can think of them as Lagrange multipliers to the flow field problem linked to an objective function that depends on quantities like pressure or momentums, and they will set also the framework for this paper. It is split into two main parts: First, we show how one can compute coupled discrete adjoints using automatic differentiation in an effective way that is still easily extendable for all kinds of other couplings. Second, we suppose that a valuable first application are so-called conjugate heat transfer problems which are gaining more and more interest from the automobile and aeronautics industry. Therefore we present an implementation for this capability within the open-source solver SU2 as well as for the generic adjoint computation algorithm.

An improved understanding of the divergence-free constraint for the incompressible Navier--Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of {\em pressure-robustness} allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order $k$ are comparably accurate than non-pressure-robust methods of formal order $2k$ on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments., Comment: 43 pages, 18 figures, 2 tables

A novel methodology to efficiently approximate the Hessian for numerical shape optimization is considered. The method enhances operator symbol approximations by including body fitted coordinates and spatially changing symbols in a semi automated framework based on local Fourier analysis. Contrary to classical operator symbol methods, the proposed strategy will identify areas in which a non-smooth design is physically meaningful and will automatically turn off smoothing in these regions. A new strategy to also numerically identify the analytic symbol is derived, extending the procedure to a wide variety of problems. The effectiveness is demonstrated by using drag minimization in Stokes and Navier-Stokes flows.

In this paper, we discuss selected adjoint approaches for the turbulent flow control. In particular, we focus on the application of adjoint solvers for the scope of noise reduction, in which flow solutions are obtained by large eddy and direct numerical simulations. Optimization results obtained with round and plane jet configurations are presented. The results indicate that using large control horizons poses a serious problem for the control of turbulent flows due to existence of very large sensitivity values with respect to control parameters. Typically these sensitivities grow in time and lead to arithmetic overflow in the computations. This phenomena is illustrated by a sensitivity study performed with an exact tangent-linear solver obtained by algorithmic differentiation techniques., Comment: 33 pages, 14 figures

Algorithmic Differentiation (AD) can be used to automate the generation of derivatives in arbitrary software projects. This will generate maintainable derivatives, that are always consistent with the computation of the software. If a domain specific language (DSL) is used in a software the state of the art approach is to differentiate the DSL library with the same AD tool. The drawback of this solution is the reduced performance since the compiler is no longer able to optimize the e.g. SIMD operations. The new approach in this paper integrates the types and operations of the DSL into the AD tool. It will be an operator overloading tool that is generated from an abstract definition of a DSL. This approach enables the compiler to optimize again e.g. for SIMD operation since all calculations are still performed with the original data types. This will also reduce the required memory for AD since the statements inside the DLS implementation are no longer seen by the AD tool. The implementation is presented in the paper and first results for the performance of the solution are presented., Comment: 6 pages, 4 figures

This paper presents a non-intrusive framework for integrating existing unsteady partial differential equation (PDE) solvers into a parallel-in-time simultaneous optimization algorithm. The time-parallelization is provided by the non-intrusive software library XBraid, which applies an iterative multigrid reduction technique to the time domain of existing time-marching schemes for solving unsteady PDEs. Its general user-interface has been extended for computing adjoint sensitivities such that gradients of output quantities with respect to design changes can be computed parallel-in-time alongside with the primal PDE solution. In this paper, the primal and adjoint XBraid iterations are embedded into a simultaneous optimization framework, namely the One-shot method. In this method, design updates towards optimality are employed after each state and adjoint update such that optimality and feasibility of the design and the PDE solution are reached simultaneously. The time-parallel optimization method is validated on an advection-dominated flow control problem which shows significant speedup over a classical time-serial optimization algorithm.

There are several AD tools available, which all implement different strategies for the reverse mode of AD. The major strategies are primal value taping (implemented e.g. by ADOL-c) and Jacobi taping (implemented e.g. by adept and dco/c++). Especially for Jacobi taping, recent advances by using expression templates make this approach very attractive for large scale software. The current implementations are either closed source or miss essential features and flexibility. Therefore, we present the new AD tool CoDiPack (Code Differentiation Package) in this paper. It is specifically designed for a minimal memory consumption and optimal runtime, such that it can be used for the differentiation of large scale software. An essential part of the design of CoDiPack is the modular layout and the recursive data structures, which do not only allow the efficient implementation of the Jacobi taping approach, but will also enable other approaches like the primal value taping or new research ideas. We will also present the performance value of CoDiPack on a generic PDE example and on the SU2 code., Comment: 21 pages, 11 figures, 6 tables, CoDiPack: https://github.com/SciCompKL/CoDiPack

In this paper, an adjoint solver for the multigrid in time software library XBraid is presented. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain. It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. Techniques from Automatic Differentiation are used to develop a consistent discrete adjoint solver which provides sensitivity information of output quantities with respect to design parameter changes. The adjoint code runs backwards through the primal XBraid actions and accumulates gradient information parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. The adjoint code is validated on advection-dominated flow with periodic upstream boundary condition. It provides similar strong scaling results as the primal XBraid solver and offers great potential for speeding up the overall computational costs for sensitivity analysis using multiple processors.

In the following, an illustrative example concerning difficulties in differentiating stiff ODEs is presented. In the given example, the solution of a completely deterministic system becomes chaotic due to computational noise introduced by the numerical algorithm.

The single-step one-shot method has proven to be very efficient for PDE-constrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is non-trivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The One-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented.