arXiv subjects

Boleslaw Kacewicz

Publications and source records attributed to Boleslaw Kacewicz.

7 recordsLinked to original sources

arXiv2018

Asymptotically tight worst case complexity bounds for initial-value problems with nonadaptive information

It is known that, for systems of initial-value problems, algorithms using adaptive information perform much better in the worst case setting than the algorithms using nonadaptive information. In the latter case, lower and upper complexity bounds significantly depend on the number of equations. However, in contrast with adaptive information, existing lower and upper complexity bounds for nonadaptive information are not asymptotically tight. In this paper, we close the gap in the complexity exponents, showing asymptotically matching bounds for nonadaptive standard information, as well as for a more general class of nonadaptive linear information.

math.NA↗

arXiv2018

Efficient finite-dimensional solution of initial value problems in infinite-dimensional Banach spaces

We deal with the approximate solution of initial value problems in infinite-dimensional Banach spaces with a Schauder basis. We only allow finite-dimensional algorithms acting in the spaces $\rr^N$, with varying $N$. The error of such algorithms depends on two parameters: the truncation parameters $N$ and a discretization parameter $n$. For a class of $C^r$ right-hand side functions, we define an algorithm with varying $N$, based on possibly non-uniform mesh, and we analyse its error and cost. For constant $N$, we show a matching (up to a constant) lower bound on the error of any algorithm in terms of $N$ and $n$, as $N,n\to \infty$. We stress that in the standard error analysis the dimension $N$ is fixed, and the dependence on $N$ is usually hidden in error coefficient. For a certain model of cost, for many cases of interest, we show tight (up to a constant) upper and lower bounds on the minimal cost of computing an $\e$-approximation to the solution (the $\e$-complexity of the problem). The results are illustrated by an example of the initial value problem in the weighted $\ell_p$ space ($1\leq p<\infty$).

math.NA↗

arXiv2017

Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems

We study adaptive mesh selection for the solution of systems of initial value problems. The goal is a rigorous theoretical analysis of potential advantages of adaption. For an optimal method in the sense of the speed of convergence, we propose an algorithm for successive selection of the mesh points. The selection is based on an upper bound on the local error, and it (asymptotically) guarantees the local errors not exceeding a prescribed level. The mesh selection algorithm can be applied to a general class of methods, not only to the chosen one. We rigorously discuss the cost of the proposed algorithm, comparing it to other algorithms equipped with different mesh selection procedures. We specify a quantitative advantage of the adaptive mesh over the uniform one. Adjustment of the mesh points to a local behavior of the solution yields improved efficiency of the algorithm. Some numerical results illustrating theoretical findings are reported.

math.NA↗

arXiv2017

Adaptive mesh point selection for the efficient solution of scalar IVPs

We discuss adaptive mesh point selection for the solution of scalar IVPs. We consider a method that is optimal in the sense of the speed of convergence, and aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.

math.NA↗

arXiv2006

Almost Optimal Solution of Initial-Value Problems by Randomized and Quantum Algorithms

We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.

quant-ph↗

arXiv2005

Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems

We deal with the problem, initiated in [8], of finding randomized and quantum complexity of initial-value problems. We showed in [8] that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the Hölder class of right-hand side functions with r continuous bounded partial derivatives, with r-th derivative being a Hölder function with exponent ρ, the ε-complexity is shown to be O((1/ε)^{1/(r+ρ+1/3)}) in the randomized setting, and O((1/ε)^{1/(r+ρ+1/2)}) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from [8]. The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is of order (1/ε)^{1/(r+ρ+1/2)} in the randomized setting, and (1/ε)^{1/(r+ρ+1)} in the quantum setting (again up to logarithmic factors).

quant-ph↗

arXiv2003

Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem.

quant-ph↗