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Frank Werner

Publications and source records attributed to Frank Werner.

At least 19 recordsLinked to original sources
arXiv2026

Minimizing the Makespan Approximately on Two Identical Parallel Machines with a Loading--Unloading Server

We study makespan minimisation on two identical parallel machines that share a single server for both loading and unloading. Each job must be loaded, processed without interruption on its assigned machine, and unloaded immediately after processing, with a common positive integer duration for all loading and unloading operations. We prove that the decision problem is NP-complete for every fixed server-operation duration and strongly NP-complete when this duration is part of the input. We then analyse ordinary list scheduling and the longest-processing-time rule in the non-unit setting. List scheduling has a tight supremum ratio of two. For the longest-processing-time rule, we obtain the exact worst-case ratio when all processing times are at least the server-operation duration, and derive new parameter-dependent lower and upper bounds for unrestricted instances. The results show that both processing-time granularity and blocking generated by short jobs shape the approximation behaviour of the common-server problem.

cs.DS↗
arXiv2026

Parallel Machine Scheduling with a Singler Server and Loading-Unloading Operations

This paper investigates a parallel machine scheduling problem featuring a single common server responsible for both loading and unloading operations. Each job consists of a unit-time loading operation, non-preemptive processing on one of \(m\) identical machines, and a unit-time unloading operation executed by the same server. The objective is to minimize the makespan. Unlike classical loading-only common-server models, our setting requires the server to handle an unloading operation immediately after a job's processing phase concludes. We prove that the corresponding decision problem is strongly NP-complete when the number of machines is given as part of the input. Furthermore, we analyse the worst-case performance of the List Scheduling (LS) and Longest Processing Time (LPT) heuristics. For three machines, we establish that Algorithm LS achieves an approximation ratio of at most \(5/2\). For an arbitrary fixed \(m \ge 3\), we show that the general LS bound approaches \(4-3/m\) as the number of jobs grows, whereas for Algorithm LPT, we prove the finite-instance ratio $3-\frac{2}{m}+\frac{(m-1)(m-2)}{mn}.$ Thus, for each fixed \(m\), the LPT bound approaches \(3-2/m\) as the number of jobs grows.

math.OC↗
arXiv2026

Expanding Flow Shop Tasks Based on Recursive Functions

The paper discusses several extensions of the recursive representation of the flow shop scheduling problem. It is shown that recursive functions make it possible to describe multiple extensions in a single problem. The paper considers altogether six extensions. The examples consider three types of recursive functions: functions associated with the machine, functions that adjust the procession time based on constraints, and functions that control the feasibility of the schedule. The structure of the superpositions of these functions is presented, and also descriptions of several objective functions by recursive functions are presented. Then the general requirements for a recursive function are formulated and its properties are described. Finally, a demonstration of the formulation of new problems is provided using examples of simple flow shop extensions and branch and bound optimization.

math.OC↗
arXiv2026

Stochstic Sampling for Generative Diffusion Models: From Euler-Maruyama to Higher-Order Schemes

We develop a convergence analysis for generative diffusion models that simultaneously accounts for the three principal sources of error in stochastic sampling: initialization error, score-matching error, and discretization of the reverse-time SDE. Our central tool is the notion of a general strong scheme, a broad class of discretization methods for the reverse dynamics defined via explicit, index-wise tolerances on their It\^o-Taylor coefficients. This notion extends the classical strong-scheme framework of Kloeden and Platen to an iterate-wise formulation, which is strictly stronger and recovers their bound as a corollary. We prove a convergence theorem in the 2-Wasserstein distance that applies to this entire class of schemes at once, reducing the analysis of any concrete sampler to a finite verification checklist, and covers general forward processes with time-dependent, spatially linear drift and spatially independent diffusion coefficient, rather than a fixed variance-preserving, variance-exploding, or Ornstein--Uhlenbeck schedule. We instantiate this theorem for the Euler--Maruyama scheme, the exponential integrator, and, as our main application, a derivative-free stochastic Runge-Kutta scheme of strong order 1.5, yielding the first stochastic sampler for generative diffusion models with a provably higher convergence order than Euler--Maruyama. We further derive the resulting iteration complexity and an accompanying parameter-selection rule for the terminal time, score accuracy, and step size, and discuss the dissipative setting, in which the discretization and score-matching errors decouple from the terminal time. Numerical experiments on Gaussian toy models and the CIFAR-10 benchmark confirm the predicted convergence orders. Code available at: https://github.com/emanuelpfarr/SSGDM.

math.NA↗
arXiv2025

L^1 data fitting for Inverse Problems yields optimal rates of convergence in case of discretized white Gaussian noise

It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In this paper, we therefore analyze generalized Tikhonov regularization with L^1 data fidelity for Inverse Problems F(u) = g in a general setting, including general measurement errors and errors in the forward operator. The derived results are then applied to the situation of discretized Gaussian white noise, and we show that the resulting error bounds allow for order-optimal rates of convergence. These findings are also investigated in numerical simulations.

math.NA↗
arXiv2025

A Simple Combination of Diffusion Models for Better Quality Trade-Offs in Image Denoising

Diffusion models have garnered considerable interest in computer vision, owing both to their capacity to synthesize photorealistic images and to their proven effectiveness in image reconstruction tasks. However, existing approaches fail to efficiently balance the high visual quality of diffusion models with the low distortion achieved by previous image reconstruction methods. Specifically, for the fundamental task of additive Gaussian noise removal, we first illustrate an intuitive method for leveraging pretrained diffusion models. Further, we introduce our proposed Linear Combination Diffusion Denoiser (LCDD), which unifies two complementary inference procedures - one that leverages the model's generative potential and another that ensures faithful signal recovery. By exploiting the inherent structure of the denoising samples, LCDD achieves state-of-the-art performance and offers controlled, well-behaved trade-offs through a simple scalar hyperparameter adjustment.

cs.CV↗
arXiv2024

A unified concept of the degree of ill-posedness for compact and non-compact linear operator equations in Hilbert spaces under the auspices of the spectral theorem

Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some new facets with respect to this story under the auspices of the spectral theorem. The latter states that any self-adjoint and bounded operator is unitarily equivalent to a multiplication operator on some (semi-finite) measure space. We will exploit this fact and derive a distribution function from the corresponding multiplier, the growth behavior of which at zero allows us to characterize the degree of ill-posedness. We prove that this new concept coincides with the well-known one for compact operators (by means of their singular values), and illustrate the implications along examples including the Hausdorff moment operator and convolutions.

math.NA↗
arXiv2024

An overview of some single machine scheduling problems: polynomial algorithms, complexity and approximability

Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often particular subjects like special objective functions are discussed, or more general scheduling problems were surveyed, where a substantial part is devoted to single machine problems. In this paper we present some results on polynomial algorithms, complexity and approximation issues, where the main focus is on results, which have been published during the last decades in papers, where at least one of the first two authors of this paper was involved. We hope that the reviewed results will stimulate further investigation in related research fields.

cs.DS↗
arXiv2024

Maximum a posteriori testing in statistical inverse problems

This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $\Pi \left(\cdot |Y = y\right)$, we want to infer whether a feature $\langle\varphi, u^\dagger\rangle$ of the unknown quantity of interest $u^\dagger$ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.

math.ST↗
arXiv2023

Adaptive minimax optimality in statistical inverse problems via SOLIT -- Sharp Optimal Lepskii-Inspired Tuning

We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form $\hat f_α= q_α\left(T^*T\right)T^*Y$, where $Y$ is the available data, $T$ the forward operator, $\left(q_α\right)_{α\in \mathcal A}$ an ordered filter, and $α> 0$ a regularization parameter. Whenever such a method is used in practice, $α$ has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible $α$ in the sense that mean squared error (MSE) $\mathbb E [\Vert \hat f_α- f^\dagger\Vert^2]$ w.r.t.~the true solution $f^\dagger$ is minimized. In this paper, we introduce the Sharp Optimal Lepski\uı-Inspired Tuning (SOLIT) method, which yields an a posteriori parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on $Y$ and the noise level $σ$ as well as the operator $T$ and the filter $\left(q_α\right)_{α\in \mathcal A}$ and does not require any problem-dependent tuning of further parameters. We prove an oracle inequality for the corresponding MSE in a general setting and derive the rates of convergence in different scenarios. By a careful analysis we show that no other a posteriori parameter choice rule can yield a better performance in terms of the order of the convergence rate of the MSE. In particular, our results reveal that the typical understanding of Lepski\uı-type methods in inverse problems leading to a loss of a log factor is wrong. In addition, the empirical performance of SOLIT is examined in simulations.

math.ST↗
arXiv2023

Optimal regularized hypothesis testing in statistical inverse problems

Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of Inverse Problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in Inverse Problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions we derive a family of tests with prescribed level $α$ and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.

math.ST↗
arXiv2023

Ensemble Laplacian Biogeography-Based Sine Cosine Algorithm for Structural Engineering Design Optimization Problems

In this paper, an ensemble metaheuristic algorithm (denoted as LX-BBSCA) is introduced. It combines the strengths of Laplacian Biogeography-Based Optimization (LX-BBO) and the Sine Cosine Algorithm (SCA) to address structural engineering design optimization problems. Our primary objective is to mitigate the risk of getting stuck in local minima and accelerate the algorithm's convergence rate. We evaluate the proposed LX-BBSCA algorithm on a set of 23 benchmark functions, including both unimodal and multimodal problems of varying complexity and dimensions. Additionally, we apply LX-BBSCA to tackle five real-world structural engineering design problems, comparing the results with those obtained using other metaheuristics in terms of objective function values and convergence behavior. To ensure the statistical validity of our findings, we employ rigorous tests such as the t-test and the Wilcoxon rank test. The experimental outcomes consistently demonstrate that the ensemble LX-BBSCA algorithm outperforms not only the basic versions of BBO, SCA, and LX-BBO but also other state-of-the-art metaheuristic algorithms.

math.OC↗
arXiv2023

Towards quantitative super-resolution microscopy: Molecular maps with statistical guarantees

Quantifying the number of molecules from fluorescence microscopy measurements is an important topic in cell biology and medical research. In this work, we present a consecutive algorithm for super-resolution (STED) scanning microscopy that provides molecule counts in automatically generated image segments and offers statistical guarantees in form of asymptotic confidence intervals. To this end, we first apply a multiscale scanning procedure on STED microscopy measurements of the sample to obtain a system of significant regions, each of which contains at least one molecule with prescribed uniform probability. This system of regions will typically be highly redundant and consists of rectangular building blocks. To choose an informative but non-redundant subset of more naturally shaped regions, we hybridize our system with the result of a generic segmentation algorithm. The diameter of the segments can be of the order of the resolution of the microscope. Using multiple photon coincidence measurements of the same sample in confocal mode, we are then able to estimate the brightness and number of the molecules and give uniform confidence intervals on the molecule counts for each previously constructed segment. In other words, we establish a so-called molecular map with uniform error control. The performance of the algorithm is investigated on simulated and real data.

stat.AP↗
arXiv2023

Multiscale scanning with nuisance parameters

We develop a multiscale scanning method to find anomalies in a $d$-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic (AMS), which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate.

stat.AP↗
arXiv2023

Scheduling on parallel machines with a common server in charge of loading and unloading operations

This paper addresses the scheduling problem on two identical parallel machines with a single server in charge of loading and unloading operations of jobs. Each job has to be loaded by the server before being processed on one of the two machines and unloaded by the same server after its processing. No delay is allowed between loading and processing, and between processing and unloading. The objective function involves the minimization of the makespan. This problem referred to as P2, S1|sj , tj |Cmax generalizes the classical parallel machine scheduling problem with a single server which performs only the loading (i.e., setup) operation of each job. For this NP-hard problem, no solution algorithm was proposed in the literature. Therefore, we present two mixedinteger linear programming (MILP) formulations, one with completion-time variables along with two valid inequalities and one with time-indexed variables. In addition, we propose some polynomial-time solvable cases and a tight theoretical lower bound. In addition, we show that the minimization of the makespan is equivalent to the minimization of the total idle times on the machines. To solve large-sized instances of the problem, an efficient General Variable Neighborhood Search (GVNS) metaheuristic with two mechanisms for finding an initial solution is designed. The GVNS is evaluated by comparing its performance with the results provided by the MILPs and another metaheuristic. The results show that the average percentage deviation from the theoretical lower-bound of GVNS is within 0.642%. Some managerial insights are presented and our results are compared with the related literature.

math.OC↗
arXiv2023

Scheduling a single machine with compressible jobs to minimize maximum lateness

The problem of scheduling non-simultaneously released jobs with due dates on a single machine with the objective to minimize the maximum job lateness is known to be strongly NP-hard. Here we consider an extended model in which the compression of the job processing times is allowed. The compression is accomplished at the cost of involving additional emerging resources, whose use, however, yields some cost. With a given upper limit $U$ on the total allowable cost, one wishes to minimize the maximum job lateness. It is clear that, by using the available resources, some jobs may complete earlier and the objective function value may respectively be decreased. As we show here, for minimizing the maximum job lateness, by shortening the processing time of some specially determined jobs, the objective value can be decreased. Although the generalized problem is harder than the generic non-compressible version, given a ``sufficient amount'' of additional resources, we can solve the problem optimally. We determine the compression rate for some specific jobs and develop an algorithm that obtains an optimal solution. Such an approach can be beneficial in practice since the manufacturer can be provided with an information about the required amount of additional resources in order to solve the problem optimally. In case the amount of the available additional resources is less than used in the above solution, i.e., it is not feasible, it is transformed to a tight minimal feasible solution.

math.OC↗
arXiv2023

Variational Poisson Denoising via Augmented Lagrangian Methods

In this paper, we denoise a given noisy image by minimizing a smoothness promoting function over a set of local similarity measures which compare the mean of the given image and some candidate image on a large collection of subboxes. The associated convex optimization problem possesses a huge number of constraints which are induced by extended real-valued functions stemming from the Kullback--Leibler divergence. Alternatively, these nonlinear constraints can be reformulated as affine ones, which makes the model seemingly more tractable. For the numerical treatment of both formulations of the model (i.e., the original one as well as the one with affine constraints), we propose a rather general augmented Lagrangian method which is capable of handling the huge amount of constraints. A self-contained, derivative-free, global convergence theory is provided, allowing an extension to other problem classes. For the solution of the resulting subproblems in the setting of our suggested image denoising models, we make use of a suitable stochastic gradient method. Results of several numerical experiments are presented in order to compare both formulations and the associated augmented Lagrangian methods.

math.OC↗