Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems

LU and Cholesky factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction and implementation of these factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and vice versa. Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these round off errors altogether and to do so efficiently to ensure practicality. Firstly, this work introduces two round off-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations. Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries' bit-length--- i.e., the number of bits required for expression--- is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property. Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demonstrate the predominance of the featured framework over Q-matrices, which comprise an alternative integer-preserving approach relying on the basis adjunct matrix. Thirdly, this work develops special algorithms for updating the REF factorizations. This is necessary because applying the traditional approach to the REF factorizations is inefficient in terms of entry growth and computational effort. In fact, these inefficiencies virtually wipe out all the computational savings commonly expected of factorization updates. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. The featured REF updates are column/row addition, deletion, and replacement

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

Computational Methods in Science and Engineering : Proceedings of the Workshop SimLabs@KIT, November 29 - 30, 2010, Karlsruhe, Germany

In this proceedings volume we provide a compilation of article contributions equally covering applications from different research fields and ranging from capacity up to capability computing. Besides classical computing aspects such as parallelization, the focus of these proceedings is on multi-scale approaches and methods for tackling algorithm and data complexity. Also practical aspects regarding the usage of the HPC infrastructure and available tools and software at the SCC are presented

Proceedings of the 1994 Monterey Workshop, Increasing the Practical Impact of Formal Methods for Computer-Aided Software Development: Evolution Control for Large Software Systems Techniques for Integrating Software Development Environments

Office of Naval Research, Advanced Research Projects Agency, Air Force Office of Scientific Research, Army Research Office, Naval Postgraduate School, National Science Foundatio

Fundamental Approaches to Software Engineering

computer software maintenance; computer software selection and evaluation; formal logic; formal methods; formal specification; programming languages; semantics; software engineering; specifications; verificatio

A Parallel Adaptive Mesh Refinement Library for Cartesian Meshes

abstract: This dissertation introduces FARCOM (Fortran Adaptive Refiner for Cartesian Orthogonal Meshes), a new general library for adaptive mesh refinement (AMR) based on an unstructured hexahedral mesh framework. As a result of the underlying unstructured formulation, the refinement and coarsening operators of the library operate on a single-cell basis and perform in-situ replacement of old mesh elements. This approach allows for h-refinement without the memory and computational expense of calculating masked coarse grid cells, as is done in traditional patch-based AMR approaches, and enables unstructured flow solvers to have access to the automated domain generation capabilities usually only found in tree AMR formulations. The library is written to let the user determine where to refine and coarsen through custom refinement selector functions for static mesh generation and dynamic mesh refinement, and can handle smooth fields (such as level sets) or localized markers (e.g. density gradients). The library was parallelized with the use of the Zoltan graph-partitioning library, which provides interfaces to both a graph partitioner (PT-Scotch) and a partitioner based on Hilbert space-filling curves. The partitioned adjacency graph, mesh data, and solution variable data is then packed and distributed across all MPI ranks in the simulation, which then regenerate the mesh, generate domain decomposition ghost cells, and create communication caches. Scalability runs were performed using a Leveque wave propagation scheme for solving the Euler equations. The results of simulations on up to 1536 cores indicate that the parallel performance is highly dependent on the graph partitioner being used, and differences between the partitioners were analyzed. FARCOM is found to have better performance if each MPI rank has more than 60,000 cells.Dissertation/ThesisDoctoral Dissertation Aerospace Engineering 201

Die Integration von Verifikation und Test in Übersetzungssysteme

In dieser Arbeit wird die Architektur für einen Compiler vorgestellt, der die Korrektheit der übersetzten Quellen als Teil des Übersetzungsvorgangs überprüfen kann. Dabei soll es möglich sein, verschiedene Methoden, wie etwa formaler Test und formaler Beweis, einzusetzen, um die Korrektheit nachzuweisen. Ein vollautomatischer Nachweis ist sehr aufwendig und häufig auch gar nicht möglich. Es ist also nicht praktikabel, aus Spezifikation und Programm die Korrektheit automatisch abzuleiten. Wir erweitern daher die Sprache um Korrektheitsnachweise (justifications), die der Benutzer in den Quelltext einfügen muß ("literate justification"). Je nach gewählter Methode muß der Benutzer den Korrektheitsnachweis mehr oder weniger genau ausführen. Durch die Einführung der Korrektheitsnachweise muß der Übersetzer Beweise nur noch überprüfen anstatt sie automatisch abzuleiten. Die Überprüfung der Korrektheitsnachweise kann in den Übersetzer integriert werden oder an ein externes Werkzeug delegiert werden. Ein externes Werkzeug erlaubt die Einbindung bereits existierender Werkzeuge, aber auch eine Neuentwicklung eigener Werkzeuge ist möglich. Wir zeigen am Beispiel eines taktischen Theorembeweisers, daß eine Eigenentwicklung nicht unbedingt aufwendiger ist als die Anpassung eines vorhandenen Werkzeugs. Um Tests während der Übersetzung durchführen zu können, muß ein Interpreter zur Verfügung stehen. Die Ausführung ungetesteten Codes birgt allerdings auch Sicherheitsprobleme. Wir diskutieren verschiedene Möglichkeiten, mit diesem Problem umzugehen. Der Korrektheit einer Übersetzungseinheit entspricht in der Semantik die Konsistenz einer algebraischen Spezifikation. Wir betrachten zwei Beweismethoden: zum einen durch Konstruktion eines Modells und zum andern durch Nachweis einer korrektheitserhaltenden Relation. Die Beweisverpflichtungen ergeben sich zunächst aus der Beweismethode, außerdem werden Beweisverpflichtungen eingeführt, um die Korrektheit von zusammengesetzten (modularen) Programmen zuzusichern. Die in dieser Arbeit beschriebene Architektur ist prototypisch implementiert worden. Dazu wurde das Opal-System um Elemente zur Spezifikation und zur Beschreibung von Korektheitsnachweisen erweitert. In der Arbeit werden einige kurze Beispiele vorgeführt. Der Opal/J-Prototyp ist seit Version 2.3e Teil der Opal-Distribution.In this thesis we present a compiler architecture that enables the compiler to check the correctness of the source code as part of the compilation process. It allows to perform these correctness checks with different methods, in particular formal testing and formal proof. A fully automated check is very expensive and often impossible. Hence, it is not feasible to check correctness automatically with the help of specification and implementation. We extend the programming language by (correctness) justifications that the user must insert into the source code ("literate justification"). Depending on the chosen justification method the user must work out the justification in more or less detail. The introduction of justifications changes the compiler's task from deriving a correctness proof by itself to checking a correctness proof provided by the user. The correctness check for justifications can be integrated into the compiler or delegated to an external tool. An external tool allows the integration of existing tools but the development of specialized tools is also possible. The example development of a specialized tactical theorem-prover shows that the development of a specialized tool is not necessarily more expensive than the adaptation of an existing tool. For test execution during the compilation an interpreter must be available. The execution of untested code causes security risks. We discuss different possibilities to deal with this problem. The correctnessof a compilation unit corresponds to the consistency of an algebraic specification. We study two proof methods: either by construction of a model or by establishing a correctness-preserving relation. The proofo bligations result from the proof method, in addition proof obligations arei ntroduced to ensure the correctness of modular programs. The compiler architecture described in this thesis has been prototypically implemented. The Opal system has been extended with language elements to denote specifications and (correctness) justifications. The thesis presents some short examples. The Opal/J prototype is part of the Opal distribution since version 2.3e

Proceedings of the XXIIIrd TELEMAC-MASCARET User Conference 2016, 11 to 13 October 2016, Paris, France

Water Qualit

Investigation, Development, and Evaluation of Performance Proving for Fault-tolerant Computers

A number of methodologies for verifying systems and computer based tools that assist users in verifying their systems were developed. These tools were applied to verify in part the SIFT ultrareliable aircraft computer. Topics covered included: STP theorem prover; design verification of SIFT; high level language code verification; assembly language level verification; numerical algorithm verification; verification of flight control programs; and verification of hardware logic