We present abstraction techniques that transform a given non-linear dynamical system into a linear system or an algebraic system described by polynomials of bounded degree, such that, invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-basis transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change of basis transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems. The techniques presented here allow us to discover, given a non-linear system, if a change of bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. This approach is further extended to search for a change of bases transformation that abstracts a given non-linear system into a system of linear differential inclusions. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.
Finally, our approach is closely related to that can be used to linearize a given differential equation with polynomial right-hand sides @cite_24 . The standard Carlemann embedding technique creates an infinite dimensional linear system, wherein, each dimension corresponds to a monomial or a basis polynomial. In practice, it is possible to create a linear approximation with known error bounds by truncating the monomial terms beyond a degree cutoff. Our approach for differential equation abstractions can be seen as a search for a finite submatrix'' inside the infinite matrix created by the Carleman linearization. The rows and columns of this submatrix correspond to monomials such that the derivative of each monomial in the submatrix is a linear combination of monomials that belong the submatrix. Note, however, that while Carleman embedding is defined using some basis for polynomials (usually power-products), our approach can derive transformations that may involve polynomials as opposed to just power-products.
@cite_24
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The numerical approximation of parametric partial differential equations D(u,y)=0 is a computational challenge when the dimension d of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown in [1,2] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map y↦u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis in [1,2] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for classes of parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z↦u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying semi-axes lengths of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case d=∞. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDEs that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDEs, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allow us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions. For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model. We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem. *Research of the first author is suppo... In this paper, we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the diagonal quadratic approximation method DQAM of Mulvey and Ruszczynski [A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 1992, pp. 205–215] and the parallel coordinate descent method PCDM of Richtarik and Takac [Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873]. We show that the two methods are equivalent for feasibility problems up to the selection of a step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8L′ Lω−12 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.