Bernard Haasdonk

Olena Burkovska, Bernard Haasdonk, Julien Salomon et al.

In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.

Gabriele Santin, Bernard Haasdonk

Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent \textit{$P$-greedy} algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be near-optimal in the case of kernels generating Sobolev spaces. As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced.

Babak Maboudi Afkham, Ashish Bhatt, Bernard Haasdonk et al.

In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper, we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm-bounded reduced basis. The accuracy and the stability of this model reduction technique are illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation.

Patrick Buchfink, Ashish Bhatt, Bernard Haasdonk

Parametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

Dominik Wittwar, Gabriele Santin, Bernard Haasdonk

In this paper we consider the problem of approximating vector-valued functions over a domain $Ω$. For this purpose, we use matrix-valued reproducing kernels, which can be related to Reproducing kernel Hilbert spaces of vectorial functions and which can be viewed as an extension to the scalar-valued case. These spaces seem promising, when modelling correlations between the target function components, as the components are not learned independently of one another. We focus on the interpolation with such matrix-valued kernels. We derive error bounds for the interpolation error in terms of a generalized power-function and we introduce a subclass of matrix-valued kernels whose power-functions can be traced back to the power-function of scalar-valued reproducing kernels. Finally, we apply these kind of kernels to some artificial data to illustrate the benefit of interpolation with matrix-valued kernels in comparison to a componentwise approach.

Bernard Haasdonk, Boumediene Hamzi, Gabriele Santin et al.

For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.

Dominik Garmatter, Bernard Haasdonk, Bastian Harrach

We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and want to couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.

Steffen Waldherr, Bernard Haasdonk

Background: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. Results: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. Conclusions: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.

Gabriele Santin, Dominik Wittwar, Bernard Haasdonk

Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some advantage over exact interpolation: one can obtain the same approximation order while solving a better conditioned linear system. This method is well suited also for noisy data values, where exact interpolation is not meaningful. Moreover, it allows more flexibility in the kernel choice, since approximation problems can be solved also for non strictly positive definite kernels. We discuss in this paper a greedy algorithm to compute a sparse approximation of the kernel regularized interpolant. This sparsity is a desirable property when the approximant is used as a surrogate of an expensive function, since the resulting model is fast to evaluate. Moreover, we derive convergence results for the approximation scheme, and we prove that a certain greedy selection rule produces asymptotically quasi-optimal error rates.

Johannes Rettberg, Jonas Kneifl, Julius Herb et al.

Conventional physics-based modeling techniques involve high effort, e.g., time and expert knowledge, while data-driven methods often lack interpretability, structure, and sometimes reliability. To mitigate this, we present a data-driven system identification framework that derives models in the port-Hamiltonian (pH) formulation. This formulation is suitable for multi-physical systems while guaranteeing the useful system theoretical properties of passivity and stability. Our framework combines linear and nonlinear reduction with structured, physics-motivated system identification. In this process, high-dimensional state data obtained from possibly nonlinear systems serves as input for an autoencoder, which then performs two tasks: (i) nonlinearly transforming and (ii) reducing this data onto a low-dimensional latent space. In this space, a linear pH system, that satisfies the pH properties per construction, is parameterized by the weights of a neural network. The mathematical requirements are met by defining the pH matrices through Cholesky factorizations. The neural networks that define the coordinate transformation and the pH system are identified in a joint optimization process to match the dynamics observed in the data while defining a linear pH system in the latent space. The learned, low-dimensional pH system can describe even nonlinear systems and is rapidly computable due to its small size. The method is exemplified by a parametric mass-spring-damper and a nonlinear pendulum example, as well as the high-dimensional model of a disc brake with linear thermoelastic behavior.

Raphael Leiteritz, Patrick Buchfink, Bernard Haasdonk et al.

Neural networks can be used as surrogates for PDE models. They can be made physics-aware by penalizing underlying equations or the conservation of physical properties in the loss function during training. Current approaches allow to additionally respect data from numerical simulations or experiments in the training process. However, this data is frequently expensive to obtain and thus only scarcely available for complex models. In this work, we investigate how physics-aware models can be enriched with computationally cheaper, but inexact, data from other surrogate models like Reduced-Order Models (ROMs). In order to avoid trusting too-low-fidelity surrogate solutions, we develop an approach that is sensitive to the error in inexact data. As a proof of concept, we consider the one-dimensional wave equation and show that the training accuracy is increased by two orders of magnitude when inexact data from ROMs is incorporated.

Tizian Wenzel, Gabriele Santin, Bernard Haasdonk

In this paper, we leverage a recent deep kernel representer theorem to connect kernel based learning and (deep) neural networks in order to understand their interplay. In particular, we show that the use of special types of kernels yields models reminiscent of neural networks that are founded in the same theoretical framework of classical kernel methods, while benefiting from the computational advantages of deep neural networks. Especially the introduced Structured Deep Kernel Networks (SDKNs) can be viewed as neural networks (NNs) with optimizable activation functions obeying a representer theorem. This link allows us to analyze also NNs within the framework of kernel networks. We prove analytic properties of the SDKNs which show their universal approximation properties in three different asymptotic regimes of unbounded number of centers, width and depth. Especially in the case of unbounded depth, more accurate constructions can be achieved using fewer layers compared to corresponding constructions for ReLU neural networks. This is made possible by leveraging properties of kernel approximation.

Tizian Wenzel, Marius Kurz, Andrea Beck et al.

Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that is capable of dealing with high-dimensional and huge datasets - and enjoys typical standard machine learning approximation properties. We extend the SDKN to combine it with standard machine learning modules and compare it with Neural Networks on the scientific challenge of data-driven prediction of closure terms of turbulent flows. We show experimentally that the SDKNs are capable of dealing with large datasets and achieve near-perfect accuracy on the given application.

Bernard Haasdonk, Boumediene Hamzi, Gabriele Santin et al.

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.

Bernard Haasdonk, Tizian Wenzel, Gabriele Santin et al.

Greedy kernel approximation algorithms are successful techniques for sparse and accurate data-based modelling and function approximation. Based on a recent idea of stabilization of such algorithms in the scalar output case, we here consider the vectorial extension built on VKOGA. We introduce the so called $γ$-restricted VKOGA, comment on analytical properties and present numerical evaluation on data from a clinically relevant application, the modelling of the human spine. The experiments show that the new stabilized algorithms result in improved accuracy and stability over the non-stabilized algorithms.

Roman Föll, Bernard Haasdonk, Markus Hanselmann et al.

Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems, so called recurrent models are frequently used. In this paper we introduce several new Deep recurrent Gaussian process (DRGP) models based on the Sparse Spectrum Gaussian process (SSGP) and the improved version, called variational Sparse Spectrum Gaussian process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific variational sparse Nyström approximation, the recurrent Gaussian process (RGP). Similar to previous work, we also variationally integrate out the input-space and hence can propagate uncertainty through the Gaussian process (GP) layers. Our approach can deal with a larger class of covariance functions than the RGP, because its spectral nature allows variational integration in all stationary cases. Furthermore, we combine the (variational) Sparse Spectrum ((V)SS) approximations with a well known inducing-input regularization framework. We improve over current state of the art methods in prediction accuracy for experimental data-sets used for their evaluation and introduce a new data-set for engine control, named Emission.

Alessandro Alla, Bernard Haasdonk, Andreas Schmidt

In this paper we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct nonuniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.

Roman Föll, Bernard Haasdonk, Markus Hanselmann et al.

Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems so called recurrent models are used. In this article we introduce two new Deep Recurrent Gaussian Process (DRGP) models based on the Sparse Spectrum Gaussian Process (SSGP) and the improved variational version called Variational Sparse Spectrum Gaussian Process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific sparse Nyström approximation. Therefore, we also variationally integrate out the input-space and hence can propagate uncertainty through the layers. We can show that for the resulting lower bound an optimal variational distribution exists. Training is realized through optimizing the variational lower bound. Using Distributed Variational Inference (DVI), we can reduce the computational complexity. We improve over current state of the art methods in prediction accuracy for experimental data-sets used for their evaluation and introduce a new data-set for engine control, named Emission. Furthermore, our method can easily be adapted for unsupervised learning, e.g. the latent variable model and its deep version.

Kevin Carlberg, Lukas Brencher, Bernard Haasdonk et al.

This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the state are available. The method adopts the parareal framework for time parallelism, which is defined by an initialization method, a coarse propagator, and a fine propagator. Rather than employing usual approaches for initialization and coarse propagation, we propose novel data-driven techniques that leverage the available time-evolution bases. The coarse propagator is defined by a forecast (proposed in Ref. [12]) applied locally within each coarse time interval, which comprises the following steps: (1) apply the fine propagator for a small number of time steps, (2) approximate the state over the entire coarse time interval using gappy POD with the local time-evolution bases, and (3) select the approximation at the end of the time interval as the propagated state. We also propose both local-forecast and global-forecast initialization. The method is particularly well suited for POD-based reduced-order models (ROMs). In this case, spatial parallelism quickly saturates, as the ROM dynamical system is low dimensional; thus, time parallelism is needed to enable lower wall times. Further, the time-evolution bases can be extracted from the (readily available) right singular vectors arising during POD computation. In addition to performing analyses related to the method's accuracy, speedup, stability, and convergence, we also numerically demonstrate the method's performance. Here, numerical experiments on ROMs for a nonlinear convection-reaction problem demonstrate the method's ability to realize near-ideal speedups; global-forecast initialization with a local-forecast coarse propagator leads to the best performance.