Complexity and tractability of the heat equation

Abstract

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the heat equation ut = 1u − qu in I d × (0, T ), where I is the unit interval and T is a maximum time value. This problem is to be solved subject to homogeneous Dirichlet boundary conditions, along with the initial conditions u(·, 0) = f over I d . The solution u depends linearly on f , but nonlinearly on q. Here, both f and q are d-variate functions from a reproducing kernel Hilbert space with finite-order weights of order ω. This means that, although d can be arbitrary large, f and q can be decomposed as sums of functions of at most ω variables, with ω independent of d. In this paper, we apply our previous general results to the heat equation. We study both the absolute and normalized error criteria. For either error criterion, we show that the problem is tractable. That is, the number of evaluations of f and q needed to obtain an e-approximation is polynomial in e−1 and d , with the degree of the polynomial depending linearly on ω. In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in e−1, independently of d . We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in d and the integral of the univariate kernel is positive, then the heat equation is strongly tractable.