arXiv subjects

Michael Griebel

Publications and source records attributed to Michael Griebel.

At least 19 recordsLinked to original sources

arXiv2026

Near-Optimal Learning of Gaussian Sobolev Operators

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

math.NA↗

arXiv2026

Sparse Configuration Interaction for the Electronic Schr\"odinger Equation Revisited: Complete Basis Set Limit Complexity and Quantum-Encoding Impact

In this article we revisit regularity results for eigenfunctions in the discrete spectrum of the electronic Schr\"odinger equation and study their consequences for approximation complexity. In particular, for the convergence to the complete basis set limit, it can be shown that the curse of dimensionality in the leading algebraic exponent can be mitigated. That is, for general sparse grid constructions, the main term of the convergence rate with respect to the number of degrees of freedom is independent of the number of electrons. These insights indicate potential benefits for classical numerical solvers of the electronic Schr\"odinger equation and also for quantum-computing approaches through new qubit-efficient wavefunction encodings.

quant-ph↗

arXiv2025

A Parallel-in-Time Combination Method for Parabolic Problems

In this article, we present a parallel discretization and solution method for parabolic problems with a higher number of space dimensions. It consists of a parallel-in-time approach using the multigrid reduction-in-time algorithm MGRIT with its implementation in the library XBraid, the sparse grid combination method for discretizing the resulting elliptic problems in space, and a domain decomposition method for each of the subproblems in the combination method based on the space-filling curve approach. As a result, we obtain an extremely fast and embarrassingly parallel solver with excellent speedup and scale-up qualities, which is perfectly suited for parabolic problems with up to six space dimensions. We describe our new parallel approach and show its superior parallelization properties for the heat equation, the chemical master equation and some exemplary stochastic differential equations.

math.NA↗

arXiv2025

Kernel interpolation on generalized sparse grids

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.

math.NA↗

arXiv2025

Small distance behavior of one-particle Green's functions in electronic structure theory

Within the framework of many-particle perturbation theory, we develop an analytical approach that allows us to determine the small distance behavior of Green's functions and related quantities in electronic structure theory. As a case study, we consider the one-particle Green's function up to 2nd order in the perturbation approach. We derive explicit expressions for the leading order terms of the asymptotic small distance behavior. In particular, we demonstrate the appearance of a logarithmic term in the corresponding 2nd order Feynman diagrams. Our asymptotic analysis leads to an improved classification scheme for the diagrams, which takes into account not only the perturbation order, but also the asymptotic smoothness properties near their diagonals. Such a classification may be useful in the design of numerical algorithms and helps to improve their efficiency.

math-ph↗

arXiv2024

On Multilevel Energy-Based Fragmentation Methods

Energy-based fragmentation methods approximate the potential energy of a molecular system as a sum of contribution terms built from the energies of particular subsystems. Some such methods reduce to truncations of the many-body expansion (MBE); others combine subsystem energies in a manner inspired by the principle of inclusion/exclusion (PIE). The combinatorial technique of M\"obius inversion of sums over partially ordered sets, which generalizes the PIE, is known to provide a non-recursive expression for the MBE contribution terms, and has also been connected to related cluster expansion methods. We build from these ideas a very general framework for decomposing potential functions into energetic contribution terms associated with elements of particular partially ordered sets (posets) and direct products thereof. Specific choices immediately reproduce not only the MBE, but also a number of other existing decomposition forms, including, e.g., the multilevel ML-BOSSANOVA schema. Furthermore, a different choice of poset product leads to a setup familiar from the combination technique for high-dimensional approximation, which has a known connection to quantum-chemical composite methods. We present the ML-SUPANOVA decomposition form, which allows the further refinement of the terms of an MBE-like expansion of the Born-Oppenheimer potential according to systematic hierarchies of ab initio methods and of basis sets. We outline an adaptive algorithm for the a posteori construction of quasi-optimal truncations of this decomposition. Some initial experiments are reported and discussed.

math.NA↗

arXiv2024

On the Numerical Approximation of the Karhunen-Lo\`{e}ve Expansion for Random Fields with Random Discrete Data

In many applications, random fields reflect uncertain parameters, and often their moments are part of the modeling process and thus well known. However, there are practical situations where this is simply not the case. Therefore, we do not assume that we know moments or expansion terms of the random fields, but only have discretized samples of them. The main contribution of this paper concerns the approximation of the true covariance operator from these finite measurements. We derive explicit error estimates that include the finite-rank approximation error of the covariance operator, the Monte Carlo-type error for sampling in the stochastic domain, and the numerical discretization error in the physical domain. For this purpose, we use modern tapering covariance estimators adapted to high-dimensional applications, where the dimension is introduced by the resolution of the measurement process. This allows us to give sufficient conditions on the three discretization parameters to guarantee that the error is kept below a prescribed accuracy $\varepsilon$.

math.NA↗

arXiv2024

The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures

Operator learning, the approximation of mappings between infinite-dimensional function spaces using machine learning, has gained increasing research attention in recent years. Operator approximations can serve as efficient surrogate models for problems in computational science and engineering, complementing traditional methods. However, despite their empirical success, our understanding of the underlying mathematical theory is in large part still incomplete. In this paper, we study the approximation of Lipschitz operators with respect to Gaussian measures. We prove higher Gaussian Sobolev regularity of Lipschitz operators and establish lower and upper bounds on the Hermite polynomial approximation error. We then study general reconstruction strategies of Lipschitz operators from $m$ arbitrary (potentially adaptive) linear samples. As a key finding, we tightly characterize the corresponding sample complexity, that is, the smallest achievable worst-case error among all possible choices of (adaptive) sampling and reconstruction strategies, in terms of $m$. As a consequence, we identify an inherent curse of sample complexity: No method to approximate Lipschitz operators based on $m$ linear samples can achieve algebraic convergence rates in $m$. On the positive side, we prove that a sufficiently fast spectral decay of the covariance operator of the underlying Gaussian measure guarantees convergence rates which are arbitrarily close to any algebraic rate. Overall, by tightly characterizing the sample complexity, our work confirms the intrinsic difficulty of learning Lipschitz operators, regardless of the data or learning technique.

cs.LG↗

arXiv2024

Efficient solution of ill-posed integral equations through averaging

This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavorably with the number of evaluation points, which can result in computational inefficiency. To address this issue, we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple, we focus only on one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to more complicated two- and three-dimensional problems with appropriate modifications.

math.NA↗

arXiv2023

Convergence analysis of online algorithms for vector-valued kernel regression

We consider the problem of approximating the regression function $f_\mu:\, \Omega \to Y$ from noisy $\mu$-distributed vector-valued data $(\omega_m,y_m)\in\Omega\times Y$ by an online learning algorithm using a reproducing kernel Hilbert space $H$ (RKHS) as prior. In an online algorithm, i.i.d. samples become available one by one via a random process and are successively processed to build approximations to the regression function. Assuming that the regression function essentially belongs to $H$ (soft learning scenario), we provide estimates for the expected squared error in the RKHS norm of the approximations $f^{(m)}\in H$ obtained by a standard regularized online approximation algorithm. In particular, we show an order-optimal estimate $$ \mathbb{E}(\|\epsilon^{(m)}\|_H^2)\le C (m+1)^{-s/(2+s)},\qquad m=1,2,\ldots, $$ where $\epsilon^{(m)}$ denotes the error term after $m$ processed data, the parameter $0<s\leq 1$ expresses an additional smoothness assumption on the regression function, and the constant $C$ depends on the variance of the input noise, the smoothness of the regression function, and other parameters of the algorithm. The proof, which is inspired by results on Schwarz iterative methods in the noiseless case, uses only elementary Hilbert space techniques and minimal assumptions on the noise, the feature map that defines $H$ and the associated covariance operator.

stat.ML↗

arXiv2023

On the expected uniform error of Brownian motion approximated by the Lévy-Ciesielski construction

It is known that the Brownian bridge or Lévy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the uniform error. In particular, we show constructively that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal{O}(\sqrt{\ln d/d})$, matching the known orders. We apply the results to an option pricing example.

math.NA↗

arXiv2023

A Dimension-adaptive Combination Technique for Uncertainty Quantification

We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Loève expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Loève coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.

math.NA↗

arXiv2022

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Let $Ω_i\subset\mathbb{R}^{n_i}$, $i=1,\ldots,m$, be given domains. In this article, we study the low-rank approximation with respect to $L^2(Ω_1\times\dots\timesΩ_m)$ of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare \cite{GH14,GH19}, we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.

math.NA↗

arXiv2021

On the Numerical Approximation of the Karhunen-Loève Expansion for Random Fields with Random Discrete Data

Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro--differential equations with uncertainties in the coefficient functions described by random fields. They also play a dominant role in problems in machine learning. In this article, we do not assume to have knowledge of the moments or expansion terms of the random fields but we instead have only given discretized samples for them. We thus model some measurement process for this discrete information and then approximate the covariance operator of the original random field. Of course, the true covariance operator is of infinite rank and hence we can not assume to get an accurate approximation from a finite number of spatially discretized observations. On the other hand, smoothness of the true (unknown) covariance function results in effective low rank approximations to the true covariance operator. We derive explicit error estimates that involve the finite rank approximation error of the covariance operator, the Monte-Carlo-type errors for sampling in the stochastic domain and the numerical discretization error in the physical domain. This permits to give sufficient conditions on the three discretization parameters to guarantee that an error below a prescribed accuracy $\varepsilon$ is achieved.

math.NA↗

arXiv2021

Deep Neural Networks and PIDE discretizations

In this paper, we propose neural networks that tackle the problems of stability and field-of-view of a Convolutional Neural Network (CNN). As an alternative to increasing the network's depth or width to improve performance, we propose integral-based spatially nonlocal operators which are related to global weighted Laplacian, fractional Laplacian and inverse fractional Laplacian operators that arise in several problems in the physical sciences. The forward propagation of such networks is inspired by partial integro-differential equations (PIDEs). We test the effectiveness of the proposed neural architectures on benchmark image classification datasets and semantic segmentation tasks in autonomous driving. Moreover, we investigate the extra computational costs of these dense operators and the stability of forward propagation of the proposed neural networks.

cs.LG↗

arXiv2021

A dimension-oblivious domain decomposition method based on space-filling curves

In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning approach that is applicable to any discretization, i.e. it directly operates on the assembled matrix equations. Moreover, it allows for the effective use of arbitrary processor numbers independent of the dimension of the underlying partial differential equation while maintaining optimal convergence behavior. This is the core property required to attain a sparse grid based combination method with extreme scalability which can utilize exascale parallel systems efficiently. Moreover, this approach provides a basis for the development of a fault-tolerant solver for the numerical treatment of high-dimensional problems. To achieve the required data redundancy we are therefore concerned with large overlaps of our domain decomposition which we construct via space-filling curves. In this paper, we propose our space-filling curve based domain decomposition solver and present its convergence properties and scaling behavior. The results of numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in arbitrary dimension utilizing arbitrary processor numbers.

math.NA↗

arXiv2021

Sparse tensor product approximation for a class of generalized method of moments estimators

Generalized Method of Moments (GMM) estimators in their various forms, including the popular Maximum Likelihood (ML) estimator, are frequently applied for the evaluation of complex econometric models with not analytically computable moment or likelihood functions. As the objective functions of GMM- and ML-estimators themselves constitute the approximation of an integral, more precisely of the expected value over the real world data space, the question arises whether the approximation of the moment function and the simulation of the entire objective function can be combined. Motivated by the popular Probit and Mixed Logit models, we consider double integrals with a linking function which stems from the considered estimator, e.g. the logarithm for Maximum Likelihood, and apply a sparse tensor product quadrature to reduce the computational effort for the approximation of the combined integral. Given Hölder continuity of the linking function, we prove that this approach can improve the order of the convergence rate of the classical GMM- and ML-estimator by a factor of two, even for integrands of low regularity or high dimensionality. This result is illustrated by numerical simulations of Mixed Logit and Multinomial Probit integrals which are estimated by ML- and GMM-estimators, respectively.

stat.ME↗

arXiv2021

Analysis of tensor approximation schemes for continuous functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate weights.

math.NA↗