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Introduction

## Publications

Publications (154)

We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our nonsmooth nonconvex problem setting is motivated by machine learning, since the broad class of abs-smooth functions includes, for instance, the squared $\ell_2$-error of a neural network with ReLU or hinge...

This paper considers an indirect measurement approach to reconstruct a defect in a two-dimensional waveguide model for a non-destructive ultrasonic inspection via derivative-based optimization. The propagation of the mechanical waves is simulated by the scaled boundary finite element method that builds on a semi-analytical approach. The simulated d...

Semiconductor quantum dots embedded in optical cavities are promising on-demand sources of single photons. Here, we theoretically study single photon emission from an optically driven two-photon Raman transition between the biexciton and the ground state of a quantum dot. The advantage of this process is that it allows all-optical control of the pr...

This chapter presents two optimization algorithms to solve non-smooth optimization problems subject to PDE constraints. Throughout, all non-differentiabilities are assumed to be caused by the Lipschitz-continuous operator abs() as well as the related min() and max() operators. The two approaches are based on a special treatment of the absolute valu...

This paper considers the reconstruction of a defect in a two-dimensional waveguide using a derivative-based optimization approach. The propagation of the waves is simulated by the Scaled Boundary Finite Element Method (SBFEM) that builds on a semi-analytical approach. The simulated data is then fitted to a given set of data describing the reflectio...

Semiconductor quantum dots embedded in optical cavities are promising on-demand sources of single photons. Here, we theoretically study single photon emission from an optically driven two-photon Raman transition between the biexciton and the ground state of a quantum dot. The advantage of this process is that it allows all-optical control of the pr...

The progress in numerical methods and simulation tools promotes the use of inverse problems in material characterisation problems. A newly developed procedure can be used to identify the behaviour of piezoce-ramic discs over a wide frequency range using a single specimen via fitting simulated and measured impedances by optimising the underlying mat...

For piecewise linear functions f:Rn↦R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex fˇ and a concave part fˆ , including a pair of generalized gradients gˇ∈Rn∋gˆ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a...

The increasingly simulation-driven design process of ultrasonic transducers requires several reliable parameters for the description of the material behaviour. Exact results can only be achieved when a single specimen is used in the identification process, which typically is prone to the problem of low sensitivities to certain material parameters a...

Computer Aided Design (CAD) systems and tools are considered essential for industrial design. They construct and manipulate the geometry of a certain component with an arbitrary set of design parameters. However, it is a challenging task to incorporate the parametric definition in a gradient-based shape optimization loop, since the CAD libraries us...

For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f and a concave part f ^ , including a pair of generalized gradients g ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentia...

We present and analyze the solution of nonsmooth optimization problems by a quadratic overestimation method in a function space setting. Under certain assumptions on a suitable local model, we show convergence to first-order minimal points. Subsequently, we discuss an approach to generate such a local model using the so-called abs-linearization. Fi...

For more than 30 years much of the research and development in nonsmooth optimization has been predicated on the assumption that the user provides an oracle that evaluates at any given \({\boldsymbol x} \in \mathbb {R}^n\) the objective function value φ(x) and a generalized gradient g ∈ ∂φ(x) in the sense of Clarke. We will argue here that, if ther...

Algorithmic differentiation (AD), also known as automatic differentiation, is a technology for accurate and efficient evaluation of derivatives of a function given as a computer model. The evaluations of such models are essential building blocks in numerous scientific computing and data analysis applications, including optimization, parameter ident...

In the paper [Optim. Methods Softw., 31 (2016), pp. 904–930] we derived first order (KKT) and second order (second order sufficiency condition (SOSC)) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. In that analysis, a key assumption on the local piecewise linearization was the lin...

We previously derived first-order (KKT) and second-order (SOSC) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. For this class of problems we showed that the algorithm of successive abs-linear minimization with a proximal term (SALMIN) achieves a linear or even quadratic rate of co...

For its usage in simulation-based design processes a precise knowledge of the employed material properties is inevitable. In the case of piezoelectric ceramics, the provided material parameters often suffer from large uncertainties and even inconsistencies since the standardised measurement procedure needs several specimens to determine a single se...

The one-shot approach is often applied for design optimization tasks involving a slowly converging Newton-like solver for the underlying partial differential equations. The state solver is augmented with an adjoint solver to obtain reduced derivatives for an optimization step. The idea of the one-shot approach is to simultaneously pursue state and...

Numerical optimization is becoming an essential industrial method in engineering design for shapes immersed in fluids. High-fidelity optimization requires fine design spaces with many design variables, which can only be tackled efficiently with gradient-based optimization methods. CAD packages that are open-source or commercially available do not p...

We present an optimization method for Lipschitz continuous, piecewise smooth (PS) objective functions based on successive piecewise linearization. Since, in many realistic cases, nondifferentiabilities are caused by the occurrence of abs(), max(), and min(), we concentrate on these nonsmooth elemental functions. The method’s idea is to locate an op...

Data of material properties given by manufacturers of piezoelectric ceramics is often flawed due to, for example, slightly different manufacturing conditions for each production batch. Hence, the need for more reliable data arises. Recently published material parameter estimation methods are based on the solution of an inverse problem fitting imped...

This paper concerns the implementation and application of the extended one-shot approach including additional equality constraints to achieve a direct transition from simulation to optimization. The approach can be applied for different areas of scientific computing where partial differential equations are treated by using a fixed-point solver. The...

Computer-aided design (CAD) tools are extensively used to design industrial components, however, contrary to e.g. computational fluid dynamics (CFD) solvers, shape sensitivities for gradient-based optimization of CAD-parametrized geometries have only been available with inaccurate and non-robust finite differences. Here, algorithmic differentiation...

Inversion and PDE-constrained optimization problems often rely on solving the adjoint problem to calculate the gradient of the objec- tive function. This requires storing large amounts of intermediate data, setting a limit to the largest problem that might be solved with a given amount of memory available. Checkpointing is an approach that can redu...

We propose a novel approach using shape derivatives to solve sharp interface geometric inverse optimization problems governed by Maxwell's equations. A tracking-type target functional determines the distance between the solution of a 3D time-dependent Maxwell problem and given measured data in an L2-norm. Minimization is conducted using H¹-gradient...

A complete characterisation of piezo ceramic materials typically requires several differently processed specimen. Since the results of all these measurements are combined to form one single set of material parameters, inconsistencies occur naturally. The problem when considering only one specimen is the lack of sensitivity for several material para...

This paper presents a minimization method for Lipschitz continuous, piecewise smooth objective functions based on algorithmic differentiation (AD). We assume that all nondifferentiabilities are caused by abs(), min(), and max(). The optimization method generates successively piecewise linearizations in abs-normal form and solves these local subprob...

Any piecewise smooth function that is specified by an evaluation procedure involving smooth elemental functions and piecewise linear functions like and can be represented in the so-called abs-normal form. By an extension of algorithmic, or automatic, differentiation, one can then compute certain first- and second-order derivative vectors and matric...

A class of trust-region algorithms is developed and analyzed for the solution of optimization problems with nonlinear equality and inequality constraints. These algorithms are developed for problem classes where the constraints are not available in an open, equation-based form, and constraint Jacobians are of high dimension and are expensive to cal...

For design optimization tasks, quite often a so-called one-shot approach is used. It augments the solution of the state equation with a suitable adjoint solver yielding approximate reduced derivatives that can be used in an optimization iteration to change the design. The coordination of these three iterative processes is well established when only...

Advanced process optimization technologies play a critical role in improving economics of carbon capture technologies. Pressure swing adsorption (PSA) has received recent attention as a potential process for economically removing CO2 from flue and/or shifted syngas for carbon capture and storage. In this work, we present state-of-the-art methods fo...

Zusammenfassung
Große zylindrische Stahlprüflinge werden mittels der Methode der finiten Differenzen im Zeitbereich (engl.

We address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function. They possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term. This overestimating local model is continuous but general...

The proceedings represent the state of knowledge in the area of algorithmic differentiation (AD). The 31 contributed papers presented at the AD2012 conference cover the application of AD to many areas in science and engineering as well as aspects of AD theory and its implementation in tools. For all papers the referees, selected from the program co...

In light of an increasing awareness of environmental challenges, extensive research is underway to develop new light-weight materials. A problem arising with these materials is their increased response to vibration. This can be solved using a new composite material that contains embedded hollow spheres that are partially filled with particles. Prog...

The use of short glass fiber-reinforced thermoplastics for the production of highly stressed parts in the plastics processing industry has experienced an enormous boom in the last few years. The reasons for this are primarily the improvements to the stiffness and strength properties brought about by fiber reinforcement.
These positive characteristi...

Efficient calculation of the solutions of nonlinear optimal control problems (NOCPs) is becoming more and more important for today’s control engineers. The systems to be controlled are typically described using differential-algebraic equations (DAEs), which can be conveniently formulated in Modelica. In addition, the corresponding optimization prob...

For the modeling and simulation of wave propagation in geometrically simple waveguides such as plates or rods, one may employ the analytical global matrix method. That is, a certain (global) matrix depending on the two parameters wavenumber and frequency is built. Subsequently, one must calculate all parameter pairs within the domain of interest wh...

Computer aided simulation of guided acoustic waves in single- or multilayered waveguides is an essential tool for several applications of acoustics and ultrasonics (i.e. pipe inspection, noise reduction). To simulate wave propagation in geometrically simple waveguides (plates or rods), one may employ the analytical Global Matrix Method [3]. This re...

Nonsmoothness is a typical characteristic of numerous objective functions in optimisation that arises from applications. The standard approach in algorithmic differentiation (AD) is to consider only differentiable functions that are defined by an evaluation program. We extend this functionality by allowing also the functions abs(), min() and max()...

In this paper, we consider an optimization problem for the complete design chain of an airfoil. Starting with a parameter vector, one has to perform a three step procedure to evaluate the desired objective: Generate a grid around the airfoil, compute the flow around the airfoil, and compute the objective. Applying a gradient-based optimization meth...

This paper presents some details for the development, analysis, and implementation of efficient numerical optimization algorithms using algorithmic differentiation (AD) in the context of partial differential equation (PDE) constrained optimization. This includes an error analysis for the discrete adjoints computed with AD and a systematic structure...

In this work we demonstrate the advantage of algorithmic differentiation [2] over finite difference estimates by approximating the set of optimal compromises of several conflicting objectives. We use a multiobjective optimization method related to the method developed in [8] to compute the approximated set of optimal compromises. The method combine...

In contrast to integration, the differentiation of a function is an ill-conditioned process, if only an oracle is available for its pointwise evaluation. That is, unrelated small variations in the value of the composite function are allowed at nearly identical arguments. In contrast, we show here that, if the function is defined by an evaluation pr...

The generation of specific high harmonics for an optical two-level system is elucidated. The desired emitted radiation can be induced by a carefully designed excitation pulse, which is found by a multiparameter optimization procedure. The presented mechanism can also be applied to semiconductor structures for which the calculations result in much h...

High harmonic generation is investigated for a two-band model of a
semiconductor nanostructure. Similar to an atomic two-level system, the
semiconductor emits high harmonic radiation. We show how one can
specifically enhance the emission for a given frequency by applying a
non-trivially shaped laser pulse. Therefore, the semiconductor Bloch
equatio...

We discuss the design, implementation and performance of algorithms suitable for the efficient computation of sparse Jacobian and Hessian matrices using automatic differentiation via operator overloading on multicore architectures. The procedure for exploiting sparsity (for runtime and memory efficiency) in serial computation involves a number of s...

The exploitation of sparsity forms an important ingredient for the efficient solution of large-scale problems. For this purpose, this paper discusses two algorithms to detect the sparsity pattern of Hessians: An approach for the computation of exact sparsity patterns and a second one for the overestimation of sparsity patterns. For both algorithms,...

We are simulating a new lightweight material that could potentially be used in several technical applications, such as machine casings to rapidly damp vibrations, reducing wear and tear. This is achieved by employing embedded hollow spheres that are filled with a granular material, such as a ceramic powder. Energy is dissipated via friction caused...

Optimal power flow problems arise in the context of the optimization and secure exploitation of electrical power in alternating current (AC) networks. This optimization problem evaluates the flow on each line and to ensure that they are under line thermal limits. To improve the reliability of the power supply, a secure network is necessary, i.e., i...

For numerous applications, the computation and provision of exact derivative information plays an important role for optimizing the considered system. This paper introduces the technique of algorithmic differentiation, a method to compute derivatives of arbitrary order within working precision. This derivative information will be combined with a ca...

Viele Aufgabenstellungen aus dem täglichen Leben, wie etwa die Bestimmung eines kürzesten Weges mithilfe eines Navigationsgerätes,
lassen sich sehr gut durch lineare Zusammenhänge beschreiben. Ein solcher linearer Zusammenhang liegt vor, wenn die Veränderung
einer Größe sich proportional auf eine andere Größe auswirkt. Ein lineares Verhalten findet...

We analyze the sensitivity of dielectric waveguides with respect to design
parameters such as permittivity values or simple geometric dependencies.
Based on a discretization using the Finite Integration Technique the
eigenvalue problem for the wave number is shown to be non-Hermitian with
possibly complex solutions even in the lossless case. Nevert...

We present the adaptation and implementation of a composite-step trust region algorithm, developed in (Walther, SIAM J. Optim.
19(1):307–325, 2008), that incorporates the approximation of the Jacobian of the equality constraints with a specialized quasi-Newton method.
The forming and/or factoring of the exact Jacobian in each optimization step is a...

In this work we combine a recently developed method, Discrete Mechanics and Optimal Control (DMOC), with the well established Automatic Differentiation package ADOL-C. DMOC is based on the discretization of the variational structure of the mechanical system which leads to structure (symplectic-momentum) preserving time-stepping equations. The discr...

A class of trust-region algorithms is developed and analyzed for the solution of minimization problems with nonlinear inequality constraints. Based on composite-step trust region methods with barrier functions, the resulting algorithm also does not require the computation of exact Ja-cobians; only Jacobian vector products are used along with approx...

For numerous applications, the computation and provision of exact derivative information plays an important role for optimizing the considered system but quite often also for its simulation. This presentation introduces the technique of Algorithmic Differentiation (AD), a method to compute derivatives of arbitrary order within working precision. Qu...

Periodic adsorption processes have gained increasing commercial importance as an energy-efficient separation technique over the past two decades. Based on fluid−solid interactions, these systems never reach steady state. Instead they operate at cyclic steady state, where the bed conditions at the beginning of the cycle match with those at the end o...

COmet Nucleus Sounding Experiment by Radio Wave Transmission (CONSERT) is one of 20 experiments onboard the ESA mission Rosetta and aimed at the reconstruction of the unknown internal material parameter distribution of a comet nucleus. The details on the experiment setup can be found in [1], [2]. CONSERT consists of a lander module which attaches t...

The numerical solution of nonlinear equation systems is often achieved by so-called quasi-Newton methods. They preserve the
rapid local convergence of Newton’s method at a significantly reduced cost per step by successively approximating the system
Jacobian though low-rank updates. We analyze two variants of the recently proposed adjoint Broyden up...

Frequently, the computation of derivatives for optimizing time-dependent problems is based on the integration of the adjoint differential equation. For this purpose, the knowledge of the complete forward solution may be required. Similar information is needed in the context of a posteriori error estimation with respect to a given functional. In the...

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order derivatives. A technique based on algorithmic differentiation is presented which allows for a precise calculation o...

Periodic Adsorption Processes (PAPs) have gained increasing commercial acceptance as an energy-efficient separation technique over the past two decades. These systems never reach steady state, instead they operate at cyclic steady state (CSS), where the bed conditions at the beginning of the cycle match with those at the end of the cycle. This CSS...

The computation of a sparse Hessian matrix H using automatic differentiation (AD) can be made efficient using the following four-step procedure: (1) Determine the sparsity structure of H, (2) obtain a seed matrix S that defines a column partition of H using a specialized coloring on the adjacency graph of H, (3) compute the compressed Hessian matri...

In this paper we introduce a quasi-Newton method for the solution of systems of non-linear equations based on the nested application of adjoint Broyden updates. In combination with a suitable line search this method yields convergence of the iteration under the same requirements on F as Newton's method. The successive use of adjoint Broyden updates...

Partikelgefüllte metallische Hohlkugeln stellen die Grundbausteine dar, aus denen durch Kleben, Löten, Gießen und Sintern eine neue Klasse von Leichtbauwerkstoffen zur Körperschalldämpfung aufgebaut wird. Durch die verschiedenen Verbindungstechniken werden die Verbundwerkstoffe unterschiedlichen Anforderungen bezüglich Einsatzbedingungen, Herstellu...

The computation of derivatives for optimizing time-dependent flow problems is often based on the integration of the adjoint differential equation. For this purpose, the knowledge of the complete forward solution is required. In the area of flow control, especially for three-dimensional problems, it may be impossible to keep track of the full forwar...

We present a sequential quadratic programming (SQP) type algorithm, based on quasi-Newton approximations of Hessian and Jacobian matrices, which is suitable for the solution of general nonlinear programming problems involving equality and inequality constraints. In contrast to most existing SQP methods, no evaluation of the exact constraint Jacobia...

The C++ package ADOL-C described in this paper facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The numerical values of derivative vectors are obtained free of truncation errors at mostly a small multiple of the run time and a fix small multiple random access m...

Periodic Adsorption Processes (PAPs) have gained increasing commercial acceptance as an efficient separation technique for a wide range of applications. These processes consist of vessels or beds packed with solid adsorbent. The adsorbent is brought in contact with a multi-component feed and the separation of the components is based on the differen...

According to the directions of European Union for nearly all machines noise control arrangements are needed (EU environmental noise directive 2002/49/EC). Additional e.g. automotive industry requires lightweight materials to reduce the total weight of their cars which led to significant reduction of the CO 2 emission. The combination of both requir...

A common way to solve PDE constrained optimal control problems by automatic differentiation (AD) is the full black box approach.
This technique may fail because of the large memory requirement. In this paper we present two alternative approaches. First,
we exploit the structure in time yielding a reduced memory requirement. Second, we additionally...

The reverse mode of automatic differentiation allows the computation of gradients at a temporal complexity that is only a small multiple of the temporal complexity to evaluate the function itself. However, the memory requirement of the reverse mode in its basic form is proportional to the operation count of the function to be differentiated. For it...

Quasi-Newton methods based on least change secant updating formulas that solve linear equations Ax=b in n=dim(x)=dim(b) steps can be expected to solve the smooth nonlinear systems n-step quadratically, i.e. with an r-order of ρ=21/n =1+1/n+O(1/n ²). The best rate one can generally expect is ρn−k for some fixed k, where ρn is the positive root of ρn...

A new approach for computing a sparsity pattern for a Hessian is presented: nonlinearity information is propagated through the function evaluation yielding the nonzero structure. A complexity analysis of the proposed algorithm is given. Once the sparsity pattern is available, coloring algorithms can be applied to compute a seed matrix. To evaluate...

A class of trust-region sequential quadratic programming algorithms for the solution of minimization problems with nonlinear equality constraints is analyzed. The considered class of optimization methods does not require the exact evaluation of the constraint Jacobian in each optimization step but uses only an approximation of this first-order deri...