arXiv subjects

Hermann G. Matthies

Publications and source records attributed to Hermann G. Matthies.

At least 19 recordsLinked to original sources

arXiv2026

A Rigorous and Self--Contained Proof of the Grover--Rudolph State Preparation Algorithm

We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum_k\sqrt{p_k}\ket{k}$ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $η$ changes the output distribution by at most $\min(1,nη)$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge\log_2(2nπ/\varepsilon)$ bits and $S\ge 2^{n+1}\log(2/δ)/\varepsilon^2$ shots suffice to achieve accuracy $\varepsilon$ with confidence $1-δ$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.

quant-ph↗

arXiv2026

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

quant-ph↗

arXiv2026

Vistas of Algebraic Probability: Quantum Computation and Information

Kolmogorov's foundation of probability takes measure spaces, $σ$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.

quant-ph↗

arXiv2025

A mathematical model for a universal digital quantum computer with an application to the Grover-Rudolph algorithm

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in $\mathrm{U}(N)$ can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware.

quant-ph↗

arXiv2025

Stochastic Modelling of Elasticity Tensor Fields

We present a novel framework for the probabilistic modelling of random fourth order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memoryless transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors, and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric.

cs.CE↗

arXiv2025

Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations

Systems may depend on parameters which one may control, or which serve to optimise the system, or are imposed externally, or they could be uncertain. This last case is taken as the ``Leitmotiv'' for the following. A reduced order model is produced from the full order model by some kind of projection onto a relatively low-dimensional manifold or subspace. The parameter dependent reduction process produces a function of the parameters into the manifold. One now wants to examine the relation between the full and the reduced state for all possible parameter values of interest. Similarly, in the field of machine learning, also a function of the parameter set into the image space of the machine learning model is learned on a training set of samples, typically minimising the mean-square error. This set may be seen as a sample from some probability distribution, and thus the training is an approximate computation of the expectation, giving an approximation to the conditional expectation, a special case of an Bayesian updating where the Bayesian loss function is the mean-square error. This offers the possibility of having a combined look at these methods, and also of introducing more general loss functions.

cs.LG↗

arXiv2024

Stochastic Modelling of Symmetric Positive Definite Material Tensors

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly 'higher' spatial invariance class. We formulate a modelling framework which not only respects these two requirements$-$positive definiteness and invariance$-$but also allows a fine control over orientation on one hand, and strength/size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest.

math.NA↗

arXiv2022

Computing f-Divergences and Distances of High-Dimensional Probability Density Functions -- Low-Rank Tensor Approximations

Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (pdf) and/or by the corresponding probability characteristic functions (pcf), or by a polynomial chaos (PCE) or similar expansion. Here the interest is mainly to compute characterisations like the entropy, or relations between two distributions, like their Kullback-Leibler divergence. These are all computed from the pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. In this regard, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format. We show how to go from the pcf or functional representation to the pdf. This allows us to reduce the computational complexity and storage cost from an exponential to a linear. The characterisations such as entropy or the $f$-divergences need the possibility to compute point-wise functions of the pdf. This normally rather trivial task becomes more difficult when the pdf is approximated in a low-rank tensor format, as the point values are not directly accessible any more. The data is considered as an element of a high order tensor space. The considered algorithms are independent of the representation of the data as a tensor. All that we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks.

math.NA↗

arXiv2022

Machine learning-based conditional mean filter: a generalization of the ensemble Kalman filter for nonlinear data assimilation

This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) -- a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that the CMF's updated covariance coincides with the expected conditional covariance. Implementing the EnCMF requires computing the conditional mean (CM). A likelihood-based estimator is prone to significant errors for small ensemble sizes, causing the filter divergence. We develop a systematical methodology for integrating machine learning into the EnCMF based on the CM's orthogonal projection property. First, we use a combination of an artificial neural network (ANN) and a linear function, obtained based on the ensemble Kalman filter (EnKF), to approximate the CM, enabling the ML-EnCMF to inherit EnKF's advantages. Secondly, we apply a suitable variance reduction technique to reduce statistical errors when estimating loss function. Lastly, we propose a model selection procedure for element-wisely selecting the applied filter, i.e., either the EnKF or ML-EnCMF, at each updating step. We demonstrate the ML-EnCMF performance using the Lorenz-63 and Lorenz-96 systems and show that the ML-EnCMF outperforms the EnKF and the likelihood-based EnCMF.

cs.LG↗

arXiv2020

Stochastic phase-field modeling of brittle fracture: computing multiple crack patterns and their probabilities

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept of stochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns.

cs.CE↗

arXiv2020

FFT-based homogenisation accelerated by low-rank tensor approximations

Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.

math.NA↗

arXiv2019

Parametric Models Analysed with Linear Maps

Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational burden. Assuming that the parametric entity takes values in a linear space, we show how is is associated to a linear map or operator. This provides a general point of view on how to consider and analyse different representations of such entities. Analysis of the associated linear map in turn connects such representations with reproducing kernel Hilbert spaces and affine- / linear-representations in terms of tensor products. A generalised correlation operator is defined through the associated linear map, and its spectral analysis helps to shed light on the approximation properties of ROMs. This point of view thus unifies many such representations under a functional analytic roof, leading to a deeper understanding and making them available for appropriate analysis.

math.NA↗

arXiv2019

Post-Processing of High-Dimensional Data

Scientific computations or measurements may result in huge volumes of data. Often these can be thought of representing a real-valued function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high degree in some truncated or lossy compressed format. We look at some common post-processing tasks which are not obvious in the compressed format, as such huge data sets can not be stored in their entirety, and the value of an element is not readily accessible through simple look-up. The tasks we consider are finding the location of maximum or minimum, or minimum and maximum of a function of the data, or finding the indices of all elements in some interval --- i.e. level sets, the number of elements with a value in such a level set, the probability of an element being in a particular level set, and the mean and variance of the total collection. The algorithms to be described are fixed point iterations of particular functions of the tensor, which will then exhibit the desired result. For this, the data is considered as an element of a high degree tensor space, although in an abstract sense, the algorithms are independent of the representation of the data as a tensor. All that we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks. We allow the actual computational representation to be a lossy compression, and we allow the algebra operations to be performed in an approximate fashion, so as to maintain a high compression level. One such example which we address explicitly is the representation of data as a tensor with compression in the form of a low-rank representation.

math.NA↗

arXiv2019

Parameter Identification in Viscoplasticity using Transitional Markov Chain Monte Carlo Method

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the difference between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article, the Choboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behavior as experimental data. Then the model parameters are identified by applying an estimation using Bayes's theorem. The Transitional Markov Chain Monte Carlo method (TMCMC) is introduced and employed to estimate the model parameters in the Bayesian setting. The uniform distributions of the parameters representing their priors are considered which literally means no knowledge of the parameters is available. Identified parameters are compared with the true parameters in the simulation, and the efficiency of the identification method is discussed. In fact, the main purpose of this study is to observe the possibility of identifying the model and hardening parameters of a viscoplastic model as a very high non-linear model with only a surface displacement measurement vector in the Bayesian setting using TMCMC and evaluate the number of measurements needed for a very acceptable estimation of the uncertain parameters of the model.

cs.CE↗

arXiv2019

Effect of Load Path on Parameter Identification for Plasticity Models using Bayesian Methods

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article, the Choboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behavior as experimental data. Then the model parameters are identified by applying an estimation using Bayes's theorem. The Gauss-Markov-Kalman filter using functional approximation is introduced and employed to estimate the model parameters in the Bayesian setting. Identified parameters are compared with the true parameters in the simulation, and the efficiency of the identification method is discussed. In the end, the effect of the load path on parameter identification is investigated.

cs.CE↗

arXiv2019

Pivoted Cholesky decomposition by Cross Approximation for efficient solution of kernel systems

Large kernel systems are prone to be ill-conditioned. Pivoted Cholesky decomposition (PCD) render a stable and efficient solution to the systems without a perturbation of regularization. This paper proposes a new PCD algorithm by tuning Cross Approximation (CA) algorithm to kernel matrices which merges the merits of PCD and CA, and proves as well as numerically exemplifies that it solves large kernel systems two-order more efficiently than those resorts to regularization. As a by-product, a diagonal-pivoted CA technique is also shown efficient in eigen-decomposition of large covariance matrices in an uncertainty quantification problem.

math.NA↗

arXiv2019

Accurate computation of conditional expectation for highly non-linear problems

This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes's theorem because it leads to a unique solution of the updated distribution of parameters. Many approaches build on Bayesian updating in terms of probability measures or their densities. However, the uncertainty propagation problems and their discretisation within the stochastic Galerkin or collocation method are naturally formulated for random vectors which calls for updating of random variables, i.e. a filter. Such filters typically build on some approximation to conditional expectation (CE). Specifically, the approximation of the CE with affine functions leads to the familiar Kalman filter which works best on linear or close to linear problems only. Our approach builds on a reformulation, which allows to localise the operator of the CE to the point of measured value. The resulting conditioned expectation (CdE) predicts correctly the quantities of interest, e.g. conditioned mean and covariance, even for general highly non-linear problems. The novel CdE allows straight-forward numerical integration; particularly, the approximated covariance matrix is always positive definite for integration rules with positive weights. The theoretical results are confirmed by numerical examples.

math.NA↗

arXiv2018

Non-deterministic inference using random set models: theory, approximation, and sampling method

A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory of evidence, possibility theory, and set of probability distributions. The aim of this work is to develop a non-deterministic inference framework, including theory, approximation and sampling method, that deals with the inverse problems in which uncertainty is represented using random sets. The proposed inference method yields the posterior random set based on the intersection of the prior and the measurement induced random sets. That inference method is an extension of Dempster's rule of combination, and a generalisation of Bayesian inference as well. A direct evaluation of the posterior random set might be impractical. We approximate the posterior random set by a random discrete set whose domain is the set of samples generated using a proposed probability distribution. We use the capacity transform density function of the posterior random set for this proposed distribution. This function has a special property: it is the posterior density function yielded by Bayesian inference of the capacity transform density function of the prior random set. The samples of such proposed probability distribution can be directly obtained using the methods developed in the Bayesian inference framework. With this approximation method, the evaluation of the posterior random set becomes tractable.

math.NA↗