Solving nonlinear parabolic problems with result verification. Part I: one-space dimensional case

AbstractWe propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension. It also means that one can obtain a posteriori error bounds for the approximate solutions of the problems. Based upon Schauder's fixed-point theorem, a verification condition is formulated and, by the use of finite-element approximation and its error estimates for a simple parabolic problem, we present a numerical verification algorithm of exact solutions in a computer. Some numerical examples which are verified by the method are illustrated

Sciduction: Combining Induction, Deduction, and Structure for Verification and Synthesis

Even with impressive advances in automated formal methods, certain problems in system verification and synthesis remain challenging. Examples include the verification of quantitative properties of software involving constraints on timing and energy consumption, and the automatic synthesis of systems from specifications. The major challenges include environment modeling, incompleteness in specifications, and the complexity of underlying decision problems. This position paper proposes sciduction, an approach to tackle these challenges by integrating inductive inference, deductive reasoning, and structure hypotheses. Deductive reasoning, which leads from general rules or concepts to conclusions about specific problem instances, includes techniques such as logical inference and constraint solving. Inductive inference, which generalizes from specific instances to yield a concept, includes algorithmic learning from examples. Structure hypotheses are used to define the class of artifacts, such as invariants or program fragments, generated during verification or synthesis. Sciduction constrains inductive and deductive reasoning using structure hypotheses, and actively combines inductive and deductive reasoning: for instance, deductive techniques generate examples for learning, and inductive reasoning is used to guide the deductive engines. We illustrate this approach with three applications: (i) timing analysis of software; (ii) synthesis of loop-free programs, and (iii) controller synthesis for hybrid systems. Some future applications are also discussed

An Algebraic Framework for the Real-Time Solution of Inverse Problems on Embedded Systems

This article presents a new approach to the real-time solution of inverse problems on embedded systems. The class of problems addressed corresponds to ordinary differential equations (ODEs) with generalized linear constraints, whereby the data from an array of sensors forms the forcing function. The solution of the equation is formulated as a least squares (LS) problem with linear constraints. The LS approach makes the method suitable for the explicit solution of inverse problems where the forcing function is perturbed by noise. The algebraic computation is partitioned into a initial preparatory step, which precomputes the matrices required for the run-time computation; and the cyclic run-time computation, which is repeated with each acquisition of sensor data. The cyclic computation consists of a single matrix-vector multiplication, in this manner computation complexity is known a-priori, fulfilling the definition of a real-time computation. Numerical testing of the new method is presented on perturbed as well as unperturbed problems; the results are compared with known analytic solutions and solutions acquired from state-of-the-art implicit solvers. The solution is implemented with model based design and uses only fundamental linear algebra; consequently, this approach supports automatic code generation for deployment on embedded systems. The targeting concept was tested via software- and processor-in-the-loop verification on two systems with different processor architectures. Finally, the method was tested on a laboratory prototype with real measurement data for the monitoring of flexible structures. The problem solved is: the real-time overconstrained reconstruction of a curve from measured gradients. Such systems are commonly encountered in the monitoring of structures and/or ground subsidence.Comment: 24 pages, journal articl

Stochastic Stability Analysis of Discrete Time System Using Lyapunov Measure

In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based stochastic stability theorems. Furthermore, connection between Lyapunov functions, a popular tool for stochastic stability verification, and Lyapunov measures is established. Using the duality property between the linear transfer Perron-Frobenius and Koopman operators, we show the Lyapunov measure and Lyapunov function used for the verification of stochastic stability are dual to each other. Set-oriented numerical methods are proposed for the finite dimensional approximation of the Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability results in finite dimensional approximation space are also presented. Finite dimensional approximation is shown to introduce further weaker notion of stability referred to as coarse stochastic stability. The results in this paper extend our earlier work on the use of Lyapunov measures for almost everywhere stability verification of deterministic dynamical systems ("Lyapunov Measure for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol. 53, No. 1, Feb. 2008).Comment: Proceedings of American Control Conference, Chicago IL, 201

Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179

Numerical Verification of Affine Systems with up to a Billion Dimensions

Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to scalable analysis for more complex systems. In this paper, we improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system. The reachable states of affine systems can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We overcome these challenges by combining several methods that leverage common problem structure. Memory is reduced by exploiting initial states that are not full-dimensional and safety properties (outputs) over a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use Krylov-subspace simulation approaches based on the Arnoldi or Lanczos iterations. Our method produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, can analyze a system with one billion real-valued state variables