# Svetlana SelivanovaKorea Advanced Institute of Science and Technology | KAIST · School of Computing

Svetlana Selivanova

PhD

## About

27

Publications

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129

Citations

Introduction

Currently I am working on an interdisciplinary project devoted to rigorous computational analysis of differential equations. The main focus is on PDEs and (i) developing numerical methods based on the Exact Real Computation approach; (ii) relating well known PDE problems to ``discrete'' complexity classes such as P, NP, #P, PSPACE etc by establishing their optimal computational complexity bounds in a rigorous sense.
My research interests extend to geometry and applications of DEs.

## Publications

Publications (27)

This survey paper summarizes the main results of Professor Victor Selivanov’s research, which together highlight the important advances he achieved in mathematical logic and theoretical computer science.

We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison’s theorem about real closu...

This paper provides a brief survey of recent achievements in characterizing computational complexity of partial differential equations (PDEs), as well as computing solutions with guaranteed precision within the exact real computation approach. The emphasis is on classical solutions and linear PDE systems, since these are the cases where most of the...

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals and apply them to several problems of algebra and analysis. In particular, we find a primitive recursive analogue of Ershov-Madison’s theorem about the computable real closure, relate primitive recursive fields of reals to the...

We devise and analyze the bit-cost of solvers for linear evolutionary systems of Partial Differential Equations (PDEs) with given analytic initial conditions. Our algorithms are rigorous in that they produce approximations to the solution up to guaranteed absolute error \(1/2^n\) for any desired number n of output bits. Previous work has shown that...

Finite Elements are a common method for solving differential equations via discretization. Under suitable hypotheses, the solution \(\mathbf {u}=\mathbf {u}(t,\vec x)\) of a well-posed initial/boundary-value problem for a linear evolutionary system of PDEs is approximated up to absolute error \(1/2^n\) by repeatedly (exponentially often in n) multi...

We establish upper bounds on bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs, combining symbolic and approximate algorithms to obtain the solutions with guaranteed prescribed precision. Restricting to algebraic real inputs allows us to use the classical (“discrete”) bit complexity concept.

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals, and then apply them to proving primitive recursiveness of some natural problems in linear algebra and analysis. In particular, we find a partial primitive recursive analogue of Ershov-Madison's theorem about real closures of...

We re-consider the problem of solving systems of differential equations approximately up to guaranteed absolute error \(1/2^n\) from the rigorous perspective of sequential and parallel time (i.e. Boolean circuit depth, equivalently: Turing machine space) complexity. While solutions to general smooth ODEs are known “PSPACE-complete” [Kawamura’10], w...

Choosing an encoding over binary strings is usually straightforward or inessential for computations over countable universes (like of graphs), but crucially affects already the computability of problems involving continuous data (like real numbers), and even more their complexity. We introduce a condition for complexity-theoretically reasonable enc...

We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in our papers of 2009 and 2017, where computability, in the rigorous sense of computable analysis, has been established for solution operators of Cauchy and dissipative boundary-value problems f...

We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in in our revious publications where computability, in the rigorous sense of computable analysis, has been established for solution operators of Cauchy and dissipative boundary-value problems fo...

In this paper we find a connection between constructive number fields and computable reals. This connection is applied to prove the computability in the rigorous sense of computable analysis) of solutions of some important systems of partial differential equations, by means of algorithms which are really used in numerical analysis.

We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary condition...

We provide an algorithm of computing Clebsch-Gordan coefficients for
irreducible representations, with integer weights, of the rotation group SO(3)
and demonstrate the convenience of this algorithm for constructing new (to our
knowledge) models in anisotropic elasticity theory.

We investigate local and metric geometry of weighted Carnot-Carath\'eodory
spaces which are a wide generalization of sub-Riemannian manifolds and arise in
nonlinear control theory, subelliptic equations etc. For such spaces the
intrinsic Carnot-Carath\'eodory metric might not exist, and some other new
effects take place. We describe the local algeb...

In this paper we study the local geometry of Carnot manifolds in a neighborhood of a singular point in the case when horizontal
vector fields are 2M-smooth. Here M is the depth of a Carnot manifold.
Keywords and phrasesCarnot manifold–singular point–tangent cone

We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is introduced in the general framework of quasimetric spaces. Considering quasimetrics allows us to cover a gene...

We propound some convergence theory for quasimetric spaces that includes as a particular case the Gromov-Hausdorff theory
for metric spaces. We prove the existence of the tangent cone (with respect to the introduced convergence) to a quasimetric
space with dilations and, as a corollary, to a regular quasimetric Carnot-Carathéodory space. This resul...

S study was conducted to investigate algebraic and analytic properties of topological spaces on which dilations were defined. The properties of the tangent cone to a (quasi)metric space endowed with a dilation structure were also investigated. The system of axioms introduced ensured the existence of a tangent cone at any fixed point of a quasimetri...

The existence of a tangent cone to a quasimetric Carnot-Caratéodory space at a regular point is studied. The structure of the space is specified by smooth vector fields and its local Lie group is a Carnot group. The distance between quasimetric spaces is defined as the infimum and is finite for bounded quasimetric spaces. A sequence of compact quas...

We study the computability properties of symmetric hyperbolic systems of PDE A ∂u ∂t + m � i=1 B i ∂u ∂xi =0 ,A = A ∗ > 0, Bi = B∗ i , with the initial condition u|t=0 = ϕ(x1 ,...,x m). Such systems first considered by K.O. Friedrichs can be used to describe a wide variety of physical processes. Using the difference equations approach, we prove com...

We study the computability properties of symmetric hyperbolic systems of PDE's A∂u∂t+∑i=1mBi∂u∂xi=0, A=A∗>0, Bi=Bi∗, with the initial condition u|t=0=φ(x1,…,xm). Such systems first considered by K.O. Friedrichs can be used to describe a wide variety of physical processes. Using the difference equations approach, we prove computability of the operat...

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed.
An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic
approach is used, which does not require any geometric constructions related to the analysis of s...