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Markus Lange-Hegermann

Publications and source records attributed to Markus Lange-Hegermann.

At least 37 records · Page 2Linked to original sources
arXiv2023

PGNAA Spectral Classification of Metal with Density Estimations

For environmental, sustainable economic and political reasons, recycling processes are becoming increasingly important, aiming at a much higher use of secondary raw materials. Currently, for the copper and aluminium industries, no method for the non-destructive online analysis of heterogeneous materials are available. The Prompt Gamma Neutron Activation Analysis (PGNAA) has the potential to overcome this challenge. A difficulty when using PGNAA for online classification arises from the small amount of noisy data, due to short-term measurements. In this case, classical evaluation methods using detailed peak by peak analysis fail. Therefore, we propose to view spectral data as probability distributions. Then, we can classify material using maximum log-likelihood with respect to kernel density estimation and use discrete sampling to optimize hyperparameters. For measurements of pure aluminium alloys we achieve near perfect classification of aluminium alloys under 0.25 second.

cs.LG↗
arXiv2022

"Prompt-Gamma Neutron Activation Analysis (PGNAA)" Metal Spectral Classification using Deep Learning Method

There is a pressing market demand to minimize the test time of Prompt Gamma Neutron Activation Analysis (PGNAA) spectra measurement machine, so that it could function as an instant material analyzer, e.g. to classify waste samples instantaneously and determine the best recycling method based on the detected compositions of the testing sample. This article introduces a new development of the deep learning classification and contrive to reduce the test time for PGNAA machine. We propose both Random Sampling Methods and Class Activation Map (CAM) to generate "downsized" samples and train the CNN model continuously. Random Sampling Methods (RSM) aims to reduce the measuring time within a sample, and Class Activation Map (CAM) is for filtering out the less important energy range of the downsized samples. We shorten the overall PGNAA measuring time down to 2.5 seconds while ensuring the accuracy is around 96.88 % for our dataset with 12 different species of substances. Compared with classifying different species of materials, it requires more test time (sample count rate) for substances having the same elements to archive good accuracy. For example, the classification of copper alloys requires nearly 24 seconds test time to reach 98 % accuracy.

cs.LG↗
arXiv2022

Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations

Data in many applications follows systems of Ordinary Differential Equations (ODEs). This paper presents a novel algorithmic and symbolic construction for covariance functions of Gaussian Processes (GPs) with realizations strictly following a system of linear homogeneous ODEs with constant coefficients, which we call LODE-GPs. Introducing this strong inductive bias into a GP improves modelling of such data. Using smith normal form algorithms, a symbolic technique, we overcome two current restrictions in the state of the art: (1) the need for certain uniqueness conditions in the set of solutions, typically assumed in classical ODE solvers and their probabilistic counterparts, and (2) the restriction to controllable systems, typically assumed when encoding differential equations in covariance functions. We show the effectiveness of LODE-GPs in a number of experiments, for example learning physically interpretable parameters by maximizing the likelihood.

cs.LG↗
arXiv2022

On boundary conditions parametrized by analytic functions

Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gröbner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations.

cs.LG↗
arXiv2022

Designing Robust Biotechnological Processes Regarding Variabilities using Multi-Objective Optimization Applied to a Biopharmaceutical Seed Train Design

Development and optimization of biopharmaceutical production processes with cell cultures is cost- and time-consuming and often performed rather empirically. Efficient optimization of multiple-objectives like process time, viable cell density, number of operating steps & cultivation scales, required medium, amount of product as well as product quality depicts a promising approach. This contribution presents a workflow which couples uncertainty-based upstream simulation and Bayes optimization using Gaussian processes. Its application is demonstrated in a simulation case study for a relevant industrial task in process development, the design of a robust cell culture expansion process (seed train), meaning that despite uncertainties and variabilities concerning cell growth, low variations of viable cell density during the seed train are obtained. Compared to a non-optimized reference seed train, the optimized process showed much lower deviation rates regarding viable cell densities (<~10% instead of 41.7%) using 5 or 4 shake flask scales and seed train duration could be reduced by 56 h from 576 h to 520 h. Overall, it is shown that applying Bayes optimization allows for optimization of a multi-objective optimization function with several optimizable input variables and under a considerable amount of constraints with a low computational effort. This approach provides the potential to be used in form of a decision tool, e.g. for the choice of an optimal and robust seed train design or for further optimization tasks within process development.

cs.LG↗
arXiv2021

Including Sparse Production Knowledge into Variational Autoencoders to Increase Anomaly Detection Reliability

Digitalization leads to data transparency for production systems that we can benefit from with data-driven analysis methods like neural networks. For example, automated anomaly detection enables saving resources and optimizing the production. We study using rarely occurring information about labeled anomalies into Variational Autoencoder neural network structures to overcome information deficits of supervised and unsupervised approaches. This method outperforms all other models in terms of accuracy, precision, and recall. We evaluate the following methods: Principal Component Analysis, Isolation Forest, Classifying Neural Networks, and Variational Autoencoders on seven time series datasets to find the best performing detection methods. We extend this idea to include more infrequently occurring meta information about production processes. This use of sparse labels, both of anomalies or production data, allows to harness any additional information available for increasing anomaly detection performance.

cs.LG↗
arXiv2021

Singularities of Algebraic Differential Equations

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notoriously difficult notion ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

math.AC↗
arXiv2021

Linearly Constrained Gaussian Processes with Boundary Conditions

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Gröbner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.

cs.LG↗
arXiv2021

An algorithmic approach to Chevalley's Theorem on images of rational morphisms between affine varieties

The goal of this paper is to introduce a new constructive geometric proof of the affine version of Chevalley's Theorem. This proof is algorithmic and a verbatim implementation resulted in an efficient code for computing the constructible image of rational maps between affine varieties. Our approach extends the known descriptions of uniform matrix product states to $\operatorname{uMPS}(2,2,5)$

math.AG↗
arXiv2020

Elimination via saturation

This short paper presents saturation-based algorithms for homogenization and elimination. This algorithm can compute elimination ideals by using syzygies and ideal membership test, hence it works with any} monomial order, in particular without the use of block-elimination orders. The used saturation is a translation of the geometric fact that the projective closure of an affine scheme has no components in the hyperplane at infinity.

math.AC↗
arXiv2020

Thomas Decomposition and Nonlinear Control Systems

This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas' algorithm is a symbolic method which splits a given system of nonlinear partial differential equations into a finite family of so-called simple systems which are formally integrable and define a partition of the solution set of the original differential system. Different simple systems of a Thomas decomposition describe different structural behavior of the control system in general. The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied. A Maple implementation of Thomas' algorithm is used to illustrate the techniques on explicit examples.

math.OC↗
arXiv2019

Algorithmic Linearly Constrained Gaussian Processes

We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gröbner bases. If successful, a push forward Gaussian process along the paramerization is the desired prior. We consider several examples from physics, geomathematics and control, among them the full inhomogeneous system of Maxwell's equations. By bringing together stochastic learning and computer algebra in a novel way, we combine noisy observations with precise algebraic computations.

stat.ML↗
arXiv2018

The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.

physics.comp-ph↗
arXiv2017

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring $R$, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over $R$. For a finitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

math.AC↗
arXiv2017

A constructive approach to the module of twisted global sections on relative projective spaces

The ideal transform of a graded module $M$ is known to compute the module of twisted global sections of the sheafification of $M$ over a relative projective space. We introduce a second description motivated by the relative BGG-correspondence. However, our approach avoids the full BGG-correspondence by replacing the Tate resolution with the computationally more efficient purely linear saturation and the Castelnuovo-Mumford regularity with the often enough much smaller linear regularity. This paper provides elementary, constructive, and unified proofs that these two descriptions compute the (truncated) modules of twisted global sections. The main argument relies on an established characterization of Gabriel monads.

math.AG↗
arXiv2015

The Differential Counting Polynomial

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional asumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions.

math.AP↗
arXiv2014

On monads of exact reflective localizations of Abelian categories

In this paper we define Gabriel monads as the idempotent monads associated to exact reflective localizations in Abelian categories and characterize them by a simple set of properties. The coimage of a Gabriel monad is a Serre quotient category. The Gabriel monad induces an equivalence between its coimage and its image, the localizing subcategory of local objects.

math.CT↗