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We present and discuss seven different open problems in applied combinatorics. The application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems., Comment: 43 pages, 5 figures

The realtime follow-up of neutrino events is a promising approach to search for astrophysical neutrino sources. It has so far provided compelling evidence for a neutrino point source: the flaring gamma-ray blazar TXS 0506+056 observed in coincidence with the high-energy neutrino IceCube-170922A detected by IceCube. The detection of very-high-energy gamma rays (VHE, $\mathrm{E} > 100\,\mathrm{GeV}$) from this source helped establish the coincidence and constrained the modeling of the blazar emission at the time of the IceCube event. The four major imaging atmospheric Cherenkov telescope arrays (IACTs) - FACT, H.E.S.S., MAGIC, and VERITAS - operate an active follow-up program of target-of-opportunity observations of neutrino alerts sent by IceCube. This program has two main components. One are the observations of known gamma-ray sources around which a cluster of candidate neutrino events has been identified by IceCube (Gamma-ray Follow-Up, GFU). Second one is the follow-up of single high-energy neutrino candidate events of potential astrophysical origin such as IceCube-170922A. GFU has been recently upgraded by IceCube in collaboration with the IACT groups. We present here recent results from the IACT follow-up programs of IceCube neutrino alerts and a description of the upgraded IceCube GFU system., Presented at the 37th International Cosmic Ray Conference (ICRC 2021). See arXiv:2107.06966 for all IceCube contributions. See arXiv:2108.05257 for all H.E.S.S. contributions

We propose a generalization of separability in the context of global optimization. Our results apply to objective functions implemented as differentiable computer programs. They are presented in the context of a simple branch and bound method. The often significant search space reduction can be expected to yield an acceleration of any global optimization method. We show how to utilize interval derivatives calculated by adjoint algorithmic differentiation to examine the monotonicity of the objective with respect to so called structural separators and how to verify the latter automatically., 15 pages, 1 figure

We demonstrate the flexibility and ease of use of C++ algorithmic differentiation (AD) tools based on overloading to numerical patterns (kernels) arising in computational finance. While adjoint methods and AD have been known in the finance literature for some time, there are few tools capable of handling and integrating with the C++ codes found in production. Adjoint methods are also known to be very powerful but to potentially have infeasible memory requirements. We present several techniques for dealing with this problem and demonstrate them on numerical kernels which occur frequently in finance. We build the discussion around our own AD tool dco/c++ which is designed to handle arbitrary C++ codes and to be highly flexible, however the sketched concepts can certainly be transferred to other AD solutions including in-house tools. An archive of the source code for the numerical kernels as well as all the AD solutions discussed can be downloaded from an accompanying website. This includes documentation for the code and dco/c++. Trial licences for dco/c++ are available from NAG.

Simulations in cardiac electrophysiology use the bidomain equations to describe the electrical potential in the heart. If only the electrical activation sequence in the heart is needed, then the full bidomain equations can be substituted by the Eikonal equation which allows much faster responses w.r.t. the changed material parameters in the equation. We use our Eikonal solver optimized for memory usage and parallelization. Patient-specific simulations in cardiac electrophysiology require patient-specific conductivity parameters which are not accurately available in vivo. One chance to improve the given conductivity parameters consists in comparing the computed activation sequence on the heart surface with the measured ECG on the torso mapped onto this surface. By minimizing the squared distance between the measured solution and the Eikonal computed solution we are able to determine the material parameters more accurately. To reduce the number of optimization parameters in this process, we group the material parameters and introduce a specific scaling parameter \(\gamma _k\) for each group. The minimization takes place w.r.t. the scaling \({\gamma }\). We solve the minimization problem by the BFGS method and adaptive step size control. The required gradient \(\nabla _\gamma f({\gamma })\) is computed either via finite differences or algorithmic differentiation using Open image in new window in tangent as well as in adjoint mode. We present convergence behavior as well as runtime and scaling results.

Many problems in science and engineering require uncertainty quantification that accounts for observed data. For example, in computational neuroscience, Neural Population Models (NPMs) are mechanistic models that describe brain physiology in a range of different states. Within computational neuroscience there is growing interest in the inverse problem of inferring NPM parameters from recordings such as the EEG (Electroencephalogram). Uncertainty quantification is essential in this application area in order to infer the mechanistic effect of interventions such as anaesthesia. This paper presents C++ software for Bayesian uncertainty quantification in the parameters of NPMs from approximately stationary data using Markov Chain Monte Carlo (MCMC). Modern MCMC methods require first order (and in some cases higher order) derivatives of the posterior density. The software presented offers two distinct methods of evaluating derivatives: finite differences and exact derivatives obtained through Algorithmic Differentiation (AD). For AD, two different implementations are used: the open source Stan Math Library and the commercially licenced dco/c++ tool distributed by NAG (Numerical Algorithms Group). The use of derivative information in MCMC sampling is demonstrated through a simple example, the noise-driven harmonic oscillator. And different methods for computing derivatives are compared. The software is written in a modular object-oriented way such that it can be extended to derivative based MCMC for other scientific domains.

In this work we present useful techniques and possible enhancements when applying an Algorithmic Differentiation (AD) tool to the linear algebra library Eigen using our in-house AD by overloading (AD-O) tool Open image in new window as a case study. After outlining performance and feasibility issues when calculating derivatives for the official Eigen release, we propose Eigen-AD, which enables different optimization options for an AD-O tool by providing add-on modules for Eigen. The range of features includes a better handling of expression templates for general performance improvements as well as implementations of symbolically derived expressions for calculating derivatives of certain core operations. The software design allows an AD-O tool to provide specializations to automatically include symbolic operations and thereby keep the look and feel of plain AD by overloading. As a showcase, Open image in new window is provided with such a module and its significant performance improvements are validated by benchmarks.

Many problems in science and engineering require uncertainty quantification that accounts for observed data. For example, in computational neuroscience, Neural Population Models (NPMs) are mechanistic models that describe brain physiology in a range of different states. Within computational neuroscience there is growing interest in the inverse problem of inferring NPM parameters from recordings such as the EEG (Electroencephalogram). Uncertainty quantification is essential in this application area in order to infer the mechanistic effect of interventions such as anaesthesia. This paper presents C++ software for Bayesian uncertainty quantification in the parameters of NPMs from approximately stationary data using Markov Chain Monte Carlo (MCMC). Modern MCMC methods require first order (and in some cases higher order) derivatives of the posterior density. The software presented offers two distinct methods of evaluating derivatives: finite differences and exact derivatives obtained through Algorithmic Differentiation (AD). For AD, two different implementations are used: the open source Stan Math Library and the commercially licenced dco/c++ tool distributed by NAG (Numerical Algorithms Group). The use of derivative information in MCMC sampling is demonstrated through a simple example, the noise-driven harmonic oscillator. And different methods for computing derivatives are compared. The software is written in a modular object-oriented way such that it can be extended to derivative based MCMC for other scientific domains.

Many studies apply sensitivity analysis to explore the impact of reaction kinetic parameters on model predictions. The importance of thermochemical and transport data is often assumed to be relatively low. While this is true for specific combustion properties of hydrocarbons, the role of thermochemical and transport data in combustion processes of nitrogen-containing molecules remains to be investigated. Thus, this work applies adjoint sensitivity analysis to the complete set of parameters in combustion models, i.e., kinetics, thermodynamics, and transport data. This integral approach increases the number of parameters considered in the sensitivity analysis drastically. Compared to forward sensitivity analysis, the adjoint approach is very efficient for a large number of parameters, and analysis with several thousand parameters can be performed in seconds. Nitrogen oxide formation in methane/air flames and laminar burning velocities of ammonia/air flames are considered as prediction targets. Sensitivity analysis results for kinetic, thermochemical, and transport data are compared by jointly considering all appearing parameter uncertainties. The comparison reveals that, due to their importance for the equilibrium constants of elementary reactions, the optimization potential of thermodynamic properties is often similarly high as that of the kinetics parameters. Transport parameters are found to be of the lowest priority for the model development due to their low uncertainties, even though high sensitivities are determined for several of them. More specifically, the analysis for the laminar burning velocities of ammonia/air flames reveals a high optimization potential for parameters in the N2-amine chemistry, including the molar heat capacities of N2H2, N2H3, and NH. Interestingly, analyses with different mechanisms reveal strongly diverging results, especially regarding the importance of reactions with OH, which is uncommon when considering the combustion of hydrocarbons.

Energies 13(21), 5592 (2020). doi:10.3390/en13215592, Published by MDPI, Basel

Optimal neural network architecture is a very important factor for computational complexity and memory footprints of neural networks. In this regard, a robust pruning method based on interval adjoints significance analysis is presented in this paper to prune irrelevant and redundant nodes from a neural network. The significance of a node is defined as a product of a node’s interval width and an absolute maximum of first-order derivative of that node’s interval. Based on the significance of nodes, one can decide how much to prune from each layer. We show that the proposed method works effectively on hidden and input layers by experimenting on famous and complex datasets of machine learning. In the proposed method, a node is removed based on its significance and bias is updated for remaining nodes.

The determination of the neutrino mass is one of the major challenges in astroparticle physics today. Direct neutrino mass experiments, based solely on the kinematics of $$\upbeta $$β-decay, provide a largely model-independent probe to the neutrino mass scale. The Karlsruhe Tritium Neutrino (KATRIN) experiment is designed to directly measure the effective electron antineutrino mass with a sensitivity of $$0.2\hbox { eV}$$0.2eV ($$90\%$$90% CL). In this work we report on the first operation of KATRIN with tritium which took place in 2018. During this commissioning phase of the tritium circulation system, excellent agreement of the theoretical prediction with the recorded spectra was found and stable conditions over a time period of 13 days could be established. These results are an essential prerequisite for the subsequent neutrino mass measurements with KATRIN in 2019.

We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented for the one-dimensional Burgers equation and the Euler equations. Using the essential routines of a state-of-the-art code for computational fluid dynamics (CFD) as a starting point, three modifications are required to apply the introduced calculus. First, the CFD code is modified to solve an additional equation for the shock location. Second, we customize the computation of the corresponding tangent to the shock location. Finally, the modified method is enhanced by algorithmic differentiation. Applying the introduced calculus to problems of the Burgers equation and the Euler equations, it is found that correct sensitivities can be computed, whereas the application of black-box algorithmic differentiation fails.

Biotechnology & bioengineering (2019). doi:10.1002/bit.27099, Published by Wiley, New York, NY [u.a.]

Computational science at the gates of nature : International Conference on Computational Science (ICCS 2015) ; Reykjavik, Iceland, 1 - 3 June 2015 / [organised by: Háskólinn í Reykjavík ...]. Ed.: Slawomir Koziel ... - Pt. 1 International Conference On Computational Science, ICCS 2015, Reykjavík, Iceland, 1 Jun 2015 - 3 Jun 2015; Red Hook, NY : Curran, Procedia computer science, 51, 231-238 (2015). doi:10.1016/j.procs.2015.05.232, Published by Curran, Red Hook, NY

Computational science at the gates of nature : International Conference on Computational Science (ICCS 2015) ; Reykjavik, Iceland, 1 - 3 June 2015 / [organised by: Háskólinn í Reykjavík ...]. Ed.: Slawomir Koziel ... - Pt. 1 International Conference On Computational Science, ICCS 2015, Reykjavík, Iceland, 1 Jun 2015 - 3 Jun 2015; Red Hook, NY : Curran, Procedia computer science, 51, 19-28 (2015). doi:10.1016/j.procs.2015.05.181, Published by Curran, Red Hook, NY

An extension of approximate computing, significance-based computing exploits applications' inherent error resiliency and offers a new structural paradigm that strategically relaxes full computational precision to provide significant energy savings with minimal performance degradation.

Four-dimensional variational data assimilation (4D-Var) produces unavoidable inaccuracies in the models initial state vector. In this paper the authors investigate a novel variational error estimation method to calculate these inaccuracies. The impacts of model, background, and observational errors on the state estimate produced by 4D-Var are analyzed by applying the variational error estimation method. The structure of the method is similar to the conventional 4D-Var, with the differences in that (i) instead of observations it assimilates observational errors, and (ii) the original model equations (used in 4D-Var as constraints) are first linearized with respect to a small perturbation in the initial state vector and then used as the constraints. The authors then carry out a proof-of-concept study and validate the reliability of this method through multiple twin experiments on the basis of a 2D shallow-water model. All required differentiated models were generated by means of algorithmic differentiation directly from the nonlinear model source code. The experiments reveal that the suggested method works well in a wide range of assimilation windows and types of observational and model errors and can be recommended for error estimation and prediction in data assimilation.

The Enceladus Explorer project is a preparatory study for a future space mission to Saturn’s moon, Enceladus. Its ultimate goal is to probe liquid-water pockets below the ice surface of Enceladus for signatures of life. A probe could be based on the IceMole concept, which melts curved trajectories through the ice. In the Enceladus Explorer project, a specialized IceMole probe for a terrestrial test scenario is in development. The goal of this exploratory study is to probe water from a liquid crevasse close to Blood Falls at Taylor Glacier, Antarctica. To navigate such a probe it is essential to be able to determine its position and monitor its trajectory. Part of the navigation system is the in-ice acoustic positioning system. For this, the head of the IceMole is equipped with acoustic sensors, which receive signals from synchronized acoustic emitters situated at the ice surface. Based on the measured propagation times, the speed of sound in ice and the positions of the emitters at the surface, the position of the IceMole can be determined by trilateration techniques. Here we present the developed acoustic positioning system, which is designed to track the in-ice melting probe up to distances of 100 m in glacier ice. Results from full-system tests in water and a first field test on Morteratschgletscher, Switzerland, are discussed.

Computer physics communications 200, 300-311 (2016). doi:10.1016/j.cpc.2015.10.027, Published by North Holland Publ. Co., Amsterdam

Infrared limb sounding from aircraft can provide 2-D curtains of multiple trace gas species. However, conventional limb sounders view perpendicular to the aircraft axis and are unable to resolve the observed airmass along their line-of-sight. GLORIA (Gimballed Limb Observer for Radiance Imaging of the Atmosphere) is a new remote sensing instrument that is able to adjust its horizontal view angle with respect to the aircraft flight direction from 45° to 135°. This will allow for tomographic measurements of mesoscale structures for a wide variety of atmospheric constituents. Many flights of the GLORIA instrument will not follow closed curves that allow measuring an airmass from all directions. Consequently, it is examined by means of simulations, what spatial resolution can be expected under ideal conditions from tomographic evaluation of measurements made during a straight flight. It is demonstrated that the achievable horizontal resolution in the line-of-sight direction could be reduced from over 200 km to around 70 km compared to conventional retrievals and that the tomographic retrieval is also more robust against horizontal gradients in retrieved quantities in this direction. In a second step, it is shown that the incorporation of channels exhibiting different optical depth can further enhance the spatial resolution of 3-D retrievals enabling the exploitation of spectral samples usually not used for limb sounding due to their opacity. A second problem for tomographic retrievals is that advection, which can be neglected for conventional retrievals, plays an important role for the time-scales involved in a tomographic measurement flight. This paper presents a method to diagnose the effect of a time-varying atmosphere on a 3-D retrieval and demonstrates an effective way to compensate for effects of advection by incorporating wind-fields from meteorological datasets as a priori information.

Goal oriented dual weight error estimation has been used in context of computational fluid dynamics for several years. The adaptation of this method to geophysical models is the subject of this paper. A differentiation-enabled prototype of the NAG Fortran compiler is used to generate a discrete adjoint version of such a geophysical model and allows to compute the required goal sensitivities. Numerical results are presented for a shallow water configuration of the Icosahedral Non-hydrostatic General Circulation Model (ICON). A special treatment of the underlying linear solver is discussed yielding improved scalability of this approach and a significant reduction in memory consumption and runtime.

We describe the use of first- and second-order tangent-linear and adjoint models of the residual of linear-implicit autonomous differential algebraic systems in the context of an extrapolated Euler scheme. The derivative code compiler dcc is applied to a C-implementation of the residual to get first derivative code. Second-(and higher-)order derivative models are obtained by reapplication of dcc to its own output. The resulting solver serves as a first proof of concept of a new platform for source-level manipulation of mathematical models that is currently under development at RWTH Aachen University.

Runs of numerical computer programs can be visualized as directed acyclic graphs (DAGs). We consider the problem of restoring the intermediate values computed by such a program (the vertices in the DAG) in reverse order for a given upper bound on the available memory. The minimization of the associated computational cost in terms of the number of performed arithmetic operations is shown to be NP-complete. The reversal of the data-flow finds application, for example, in the efficient evaluation of adjoint numerical programs. We derive special cases of numerical programs that require the intermediate values exactly in reverse order, thus establishing the NP-completeness of the optimal adjoint computation problem. Last but not least we review some state-of-the-art approaches to efficient data-flow reversal taken by existing software tools for automatic differentiation.

The source transformation tool for automatic differentiation of Fortran programs ADIFOR uses a preaccumulation technique to speed up tangent-linear codes significantly compared to the standard forward mode. Reverse mode automatic differentiation is applied to all scalar assignments to generate efficient code for the computation of local gradients. It has been well known for some time that reverse mode is not necessarily the optimal choice for the computation of these statement-level gradients as it does not minimize the number of operations required. This article presents an efficient algorithm for the solution of this combinatorial optimization problem. The corresponding software is freely available for downloading on our website. Developers of software for automatic differentiation are invited to integrate the algorithm into their tools. Gradients of scalar multivariate functions can be computed by elimination methods on the linearized computational graph. The combinatorial optimization problem that aims to minimize the number of arithmetic operations performed by the elimination algorithm is known to be NP-complete. In this article we present a polynomial algorithm for solving a relevant subclass of this problem's instances. The proposed method relies on the ability to compute vertex covers in bipartite graphs in polynomial time. A simplified version of this graph algorithm is used in a research prototype of the differentiation-enabled NAGWare Fortran compiler for the preaccumulation of local gradients of scalar assignments in the context of automatic generation of efficient tangent-linear code for numerical programs.

The reverse mode of automatic differentiation is widely used in science and engineering. A severe bottleneck for the performance of the reverse mode, however, is the necessity to recover certain intermediate values of the program in reverse order. Among these values are computed addresses, which traditionally are recovered through forward recomputation and storage in memory. We propose an alternative approach for recovery that uses inverse computation based on dependency information. Address storage constitutes a significant portion of the overall storage requirements. An example illustrates substantial gains that the proposed approach yields, and we show use cases in practical applications.

Adjoint based calculation of sensitivities pertaining to a Computational Fluid Dynamics (CFD) Solver has proven to be a vital tool in tackling large scale problems often found in industrial applications. There are two basic approaches for applying Algorithmic Differentiation (AD) to a CFD solver, namely, Operator Overloading and Source Transformation. Unfortunately, in both cases, it is still necessary to invest a significant amount of manual coding in order to get to an application that performs acceptably in terms of memory consumption and runtime. In this paper reverse mode of AD has been applied to an unstructured pressure based steady Navier-Stokes solver. We explore the feasibility of combining both kinds of AD approaches to show where and how the advantages of each method can be exploited in order to reach a suitable compromise between performance, simplicity and efficiency. Additionally,we propose a methodology to automate the implementation of an adjoint software in order to minimize work the developer must carry out to produce the desired derivative. We investigate the effectiveness of this approach for relevant flow test cases.

International Conference on Computational Science 2016, ICCS 2016, 6-8 June 2016, San Diego, California, USA / Edited by Ilkay Altintas, Michael Norman, Jack Dongarra, Valeria V. Krzhizhanovskaya, Michael Lees and Peter M.A. Sloot International Conference on Computational Science 2016, ICCS 2016, San Diego, CA, USA, 6 Jun 2017 - 8 Jun 2017; Amsterdam [u.a.] : Elsevier, Procedia computer science, 80, 2231-2235 (2016). doi:10.1016/j.procs.2016.05.388, Published by Elsevier, Amsterdam [u.a.]

We present here an efficient first-order second moment method using Algorithmic Differentiation (FOSM/AD) which can be applied to quantify uncertainty/sensitivities in morphodynamic models. Changes with respect to variable flow and sediment input parameters are estimated with machine accuracy using the technique of Algorithmic Differentiation (AD). This method is particularly attractive for process-based morphodynamic models like the Telemac-2D/Sisyphe model considering the large number of input parameters and CPU time associated to each simulation.The FOSM/AD method is applied to identify the relevant processes in a trench migration experiment (van Rijn, 1987). A Tangent Linear Model (TLM) of the Telemac-2D/Sisyphe morphodynamic model (release 6.2) was generated using the AD-enabled NAG Fortran compiler. One single run of the TLM is required per variable input parameter and results are then combined to calculate the total uncertainty.The limits of the FOSM/AD method have been assessed by comparison with Monte Carlo (MC) simulations. Similar results were obtained assuming small standard deviation of the variable input parameters. Both settling velocity and grain size have been identified as the most sensitive input parameters and the uncertainty as measured by the standard deviation of the calculated bed evolution increases with time. A first-order second moment method (FOSM) is applied to quantify uncertainty.This method uses Algorithmic Differentiation (AD) and a Tangent Linear Model (TLM).The method is compared with Monte Carlo analysis in a trench migration test case.A TLM of the Telemac-2d/Sisyphe morphodynamic model has been applied.The FOSM/AD method is an efficient alternative to Monte Carlo simulations.

International Conference on Computational Science 2016, ICCS 2016, 6-8 June 2016, San Diego, California, USA / Edited by Ilkay Altintas, Michael Norman, Jack Dongarra, ValeriaV. Krzhizhanovskaya, Michael Lees and Peter M.A. Sloot International Conference on Computational Science 2016, ICCS 2016, San Diego, California, USA, 6 Jun 2016 - 8 Jun 2016; Amsterdam [u.a.] : Elsevier, Procedia computer science, 80, 201-211 (2016). doi:10.1016/j.procs.2016.05.310, Published by Elsevier, Amsterdam [u.a.]

Gradients of high-dimensional functions can be computed efficiently and with machine accuracy by so-called adjoint codes. We present an L-attributed grammar for the single-pass generation of intraprocedural adjoint code for a subset of Fortran. Our aim is to integrate the syntax directed approach into the front-end of the NAGWare Fortran compiler. Research prototypes of this compiler that build adjoint code based on an abstract intermediate representation have been developed for several years. We consider the syntax-directed generation of adjoint code as a low development cost alternative to more sophisticated algorithms. The price to pay comes in form of a very limited set of code optimizations that can be performed in a single-pass setting.

Numerical simulations of physical, chemical, and economical processes play an increasingly important role in modern science and engineering. The implementation of mathematical models of real-world applications on a computer facilitates both the speed and the depth of our understanding of the behavior of the respective systems. Derivative models of the computer programs are required to make the transition from pure simulation to the highly desirable optimization of the numerical models with respect to a potentially very large number of input parameters. Such models can be generated automatically from the given numerical program by a source transformation technique known as automatic differentiation. Reversal of the control flow is especially important for the generation of adjoint derivative models. We describe an approach to the control flow reversal of structured programs motivated by the significant weakness of approaches that do not exploit the result of control-flow analysis. Our approach is used to automatically generate adjoint code for numerical programs by semantic source transformation. After a short introduction to applications and the implementation tool set, we motivate the proposed approach with a simple example. We present a novel preaccumulation algorithm for local Jacobian matrices at the level of basic blocks. The main part of the paper covers the reversal of structured control flow graphs. First we show the algorithmic steps for simple branches and loops. We give a detailed algorithm for the reversal of arbitrary combinations of loops and branches based only on the structural information in a general control flow graph. Dependencies between computations and their enclosing control flow constructs can lead to inefficient adjoint code. We formulate a set of conditions that allows for considerable efficiency gains in the adjoint code while permitting a reasonably simple modification of the reversal algorithms for specially designated control flow subgraphs. We present a sensitivity computation of an oceanographic application that illustrates the benefits of this modified approach.

The automatic generation of adjoints of mathematical models that are implemented as computer programs is receiving increased attention in the scientific and engineering communities. Reverse-mode automatic differentiation is of particular interest for large-scale optimization problems. It allows the computation of gradients at a small constant multiple of the cost for evaluating the objective function itself, independent of the number of input parameters. Source-to-source transformation tools apply simple differentiation rules to generate adjoint codes based on the adjoint version of every statement. In order to guarantee correctness, certain values that are computed and overwritten in the original program must be made available in the adjoint program. For their determination we introduce a static data-flow analysis called "to be recorded" analysis. Possible overestimation of this set must be kept minimal to get efficient adjoint codes. This efficiency is essential for the applicability of source-to-source transformation tools to real-world applications.

Theranostics 4(10), 960-971 (2014). doi:10.7150/thno.9293, Published by Ivyspring, Wyoming, NSW

The Julich-Aachen Dynamic optimization Environment (JADE) is employed for computing first- and second-order parameter sensitivities of a metabolically and isotopically non-stationary biochemical network model. Based on a Modelica representation of the model, code generation, algorithmic differentiation and first- and second-order adjoint sensitivity analysis are employed for computing the gradient and the Hessian of a parameter estimation objective function. In particular, we use composite adjoints, an extension of the classical adjoint sensitivity analysis, and a numerical integrator based a modification of secondorder discrete adjoints of the extrapolated linearlyimplicit Euler method. Therewith, the 116 116Hessian of the objective function with respect to 116 model parameters can be computed at the cost equivalent to only 18 objective function evaluations, while computing the same Hessian with the cheapest finitedifference formula would require 6845 evaluations of