Bit Complexity of Computing Solutions for Symmetric Hyperbolic Systems of PDEs (Extended Abstract) (original) (raw)

Computing Solutions of Symmetric Hyperbolic Systems of PDE's

Electronic Notes in Theoretical Computer Science, 2008

We study the computability properties of symmetric hyperbolic systems of PDE' . . . , 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 operator that sends (for any fixed computable matrices A, B 1 , . . . , Bm satisfying some natural conditions) any initial function ϕ ∈ C k+1 (Q, R n ), k ≥ 1, to the unique solution u ∈ C k (H, R n ), where Q = [0, 1] m and H is the nonempty domain of correctness of the system.

Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs

2013

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 conditions, and of the Cauchy problem for the same system (we also prove computable dependence on the coefficients) in a cube QsubseteqmathbbRmQ\subseteq\mathbb R^mQsubseteqmathbbRm. Such systems describe a wide variety of physical processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many boundary-value problems for the wave equation also can be reduced to this case, thus we partially answer a question raised in Weihrauch and Zhong (2002). Compared with most of other existing methods of proving computability for PDEs, this method does not require existence of explicit solution formulas and is thus applicable to a broader class of (systems of) equations.

Computing the Solution Operators of Symmetric Hyperbolic Systems of PDE

Journal of Universal Computer Science, 2009

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

What Is the Complexity of Solution-Restricted Operator Equations?

Journal of Complexity, 1995

We s t u d y t h e w orst case complexity of operator equations Lu = f , where L : G ! X is a bounded linear injection, G is a Hilbert space, and X is a normed linear space. Past work on the complexity o f s u c h problems has generally assumed that the class F of problem elements f to be the unit ball of X. H o wever, there are many problems for which t h i s c hoice of F yields unsatisfactory results. Mixed elliptic-hyperbolic problems are one example, the di culty being that our technical tools are not strong enoguh to give good complexity bounds. Ill-posed problems are another example, because we know that the complexity of computing nite-error approximations is in nite if F is a ball in X. In this paper, we pursue another idea. Rather than directly restrict the class F of problem elements f , w e will consider problems that are solution-restricted, i.e., we restrict the class U of solution elements u. In particular, we assume that U is the unit ball of a Hilbert space W continuously embedded in G. The main idea is that our problem can be reduced to the standard approximation problem of approximating the embedding of W into G. T h i s a l l o ws us to characterize optimal information and algorithms for our problem. Then, we consider speci c applications. The rst application we consider is any problem for which G and W are standard Sobolev Hilbert spaces we call this the \standard problem" since it includes many problems of practical interest. We show that nite element information and generalized Galerkin methods are nearly optimal for standard problems. We then look at elliptic boundary-value problems, Fredholm integral equations of the second kind, the Tricomi problem (a mixed hyperbolic-elliptic problem arising in the study of transonic ow), the inverse nite Laplace transform, and the backwards heat equation. (Note that with the exception of the backwards heat equation, all of these are standard problems. Moreover, the inverse nite Laplace transform and the backwards heat equation are ill-posed problems.) We determine the problem complexity and derive nearly optimal algorithms for all these problems.

Computability, noncomputability, and hyperbolic systems

Applied Mathematics and Computation, 2012

In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot (though we prove they are semi-computable). We also show that Smale's horseshoe, the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit, is computable. arXiv:1201.0164v1 [math.LO]

Complexity of parabolic systems

Publications mathématiques de l'IHÉS, 2020

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces in arbitrary codimension. We also show sharp bounds for codimension in arguably some of the most important situations of general ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at −∞ must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension. The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow. We show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. [CM11].