The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change.

We study COMP-AMS, a distributed optimization framework based on gradient averaging and adaptive AMSGrad algorithm. Gradient compression with error feedback is applied to reduce the communication cost in the gradient transmission process. Our convergence analysis of COMP-AMS shows that such compressed gradient averaging strategy yields same convergence rate as standard AMSGrad, and also exhibits the linear speedup effect w.r.t. the number of local workers. Compared with recently proposed protocols on distributed adaptive methods, COMP-AMS is simple and convenient. Numerical experiments are conducted to justify the theoretical findings, and demonstrate that the proposed method can achieve same test accuracy as the full-gradient AMSGrad with substantial communication savings. With its simplicity and efficiency, COMP-AMS can serve as a useful distributed training framework for adaptive gradient methods.

The Distributed Constraint Optimization Problem (DCOP) formulation is a powerful tool for modeling multi-agent coordination problems. To solve DCOPs in a dynamic environment, Dynamic DCOPs (D-DCOPs) have been proposed to model the inherent dynamism present in many coordination problems. D-DCOPs solve a sequence of static problems by reacting to changes in the environment as the agents observe them. Such reactive approaches ignore knowledge about future changes of the problem. To overcome this limitation, we introduce Proactive Dynamic DCOPs (PD-DCOPs), a novel formalism to model D-DCOPs in the presence of exogenous uncertainty. In contrast to reactive approaches, PD-DCOPs are able to explicitly model possible changes of the problem and take such information into account when solving the dynamically changing problem in a proactive manner. The additional expressivity of this formalism allows it to model a wider variety of distributed optimization problems. Our work presents both theoretical and practical contributions that advance current dynamic DCOP models: (i) We introduce Proactive Dynamic DCOPs (PD-DCOPs), which explicitly model how the DCOP will change over time; (ii) We develop exact and heuristic algorithms to solve PD-DCOPs in a proactive manner; (iii) We provide theoretical results about the complexity of this new class of DCOPs; and (iv) We empirically evaluate both proactive and reactive algorithms to determine the trade-offs between the two classes. The final contribution is important as our results are the first that identify the characteristics of the problems that the two classes of algorithms excel in.

Optimization is a problem associated with the best decision that is effective and efficient decisions whether it is worth maximum or minimum by way of determining a satisfactory solution. Optimization is not a new science. It has grown even since Newton in the 17th century discovered how to count roots. Currently the science of optimization is still evolving in terms of techniques and applications. Many cases or problems in everyday life that involve optimization to solve them.

Tractography has become an indispensable part of brain connectivity studies. However, it is currently facing problems with reliability. In particular, a substantial amount of nerve fiber reconstructions (streamlines) in tractograms produced by state-of-the-art tractography methods are anatomically implausible. To address this problem, tractogram filtering methods have been developed to remove faulty connections in a postprocessing step. This study takes a closer look at one such method, \textit{Spherical-deconvolution Informed Filtering of Tractograms} (SIFT), which uses a global optimization approach to improve the agreement between the remaining streamlines after filtering and the underlying diffusion magnetic resonance imaging data. SIFT is not suitable to judge the plausibility of individual streamlines since its results depend on the size and composition of the surrounding tractogram. To tackle this problem, we propose applying SIFT to randomly selected tractogram subsets in order to retrieve multiple assessments for each streamline. This approach makes it possible to identify streamlines with very consistent filtering results, which were used as pseudo ground truths for training classifiers. The trained classifier is able to distinguish the obtained groups of plausible and implausible streamlines with accuracy above 80%. The software code used in the paper and pretrained weights of the classifier are distributed freely via the Github repository https://github.com/djoerch/randomised_filtering.

Nonnegative matrix factorization (NMF) often relies on the separability condition for tractable algorithm design. Separability-based NMF is mainly handled by two types of approaches, namely, greedy pursuit and convex programming. A notable convex NMF formulation is the so-called self-dictionary multiple measurement vectors (SD-MMV), which can work without knowing the matrix rank a priori, and is arguably more resilient to error propagation relative to greedy pursuit. However, convex SD-MMV renders a large memory cost that scales quadratically with the problem size. This memory challenge has been around for a decade, and a major obstacle for applying convex SD-MMV to big data analytics. This work proposes a memory-efficient algorithm for convex SD-MMV. Our algorithm capitalizes on the special update rules of a classic algorithm from the 1950s, namely, the Frank-Wolfe (FW) algorithm. It is shown that, under reasonable conditions, the FW algorithm solves the noisy SD-MMV problem with a memory cost that grows linearly with the amount of data. To handle noisier scenarios, a smoothed group sparsity regularizer is proposed to improve robustness while maintaining the low memory footprint with guarantees. The proposed approach presents the first linear memory complexity algorithmic framework for convex SD-MMV based NMF. The method is tested over a couple of unsupervised learning tasks, i.e., text mining and community detection, to showcase its effectiveness and memory efficiency.

Unmanned aerial vehicles (UAVs) are widely used platforms to carry data capturing sensors for various applications. The reason for this success can be found in many aspects: the high maneuverability of the UAVs, the capability of performing autonomous data acquisition, flying at different heights, and the possibility to reach almost any vantage point. The selection of appropriate viewpoints and planning the optimum trajectories of UAVs is an emerging topic that aims at increasing the automation, efficiency and reliability of the data capturing process to achieve a dataset with desired quality. On the other hand, 3D reconstruction using the data captured by UAVs is also attracting attention in research and industry. This review paper investigates a wide range of model-free and model-based algorithms for viewpoint and path planning for 3D reconstruction of large-scale objects. The analyzed approaches are limited to those that employ a single-UAV as a data capturing platform for outdoor 3D reconstruction purposes. In addition to discussing the evaluation strategies, this paper also highlights the innovations and limitations of the investigated approaches. It concludes with a critical analysis of the existing challenges and future research perspectives.

We present SymForce, a library for fast symbolic computation, code generation, and nonlinear optimization for robotics applications like computer vision, motion planning, and controls. SymForce combines the development speed and flexibility of symbolic math with the performance of autogenerated, highly optimized code in C++ or any target runtime language. SymForce provides geometry and camera types, Lie group operations, and branchless singularity handling for creating and analyzing complex symbolic expressions in Python, built on top of SymPy. Generated functions can be integrated as factors into our tangent-space nonlinear optimizer, which is highly optimized for real-time production use. We introduce novel methods to automatically compute tangent-space Jacobians, eliminating the need for bug-prone handwritten derivatives. This workflow enables faster runtime code, faster development time, and fewer lines of handwritten code versus the state-of-the-art. Our experiments demonstrate that our approach can yield order of magnitude speedups on computational tasks core to robotics. Code is available at https://github.com/symforce-org/symforce.

Nonconvex regularization has been popularly used in low-rank matrix learning. However, extending it for low-rank tensor learning is still computationally expensive. To address this problem, we develop an efficient solver for use with a nonconvex extension of the overlapped nuclear norm regularizer. Based on the proximal average algorithm, the proposed algorithm can avoid expensive tensor folding/unfolding operations. A special "sparse plus low-rank" structure is maintained throughout the iterations, and allows fast computation of the individual proximal steps. Empirical convergence is further improved with the use of adaptive momentum. We provide convergence guarantees to critical points on smooth losses and also on objectives satisfying the Kurdyka-{\L}ojasiewicz condition. While the optimization problem is nonconvex and nonsmooth, we show that its critical points still have good statistical performance on the tensor completion problem. Experiments on various synthetic and real-world data sets show that the proposed algorithm is efficient in both time and space and more accurate than the existing state-of-the-art.

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.