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Complexity and information
Part I. Fundamentals: 1. Introduction 2. Information-based complexity 3. Breaking the curse of dimensionality Part II. Some Interesting Topics: 4. Very high-dimensional integration and mathematicalExpand
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Computational complexity of one-step methods for systems of differential equations
Abstract : The problem is to calculate an approximate solution of an initial value problem for an autonomous system of N ordinary differential equations. Using fast power series techniques, weExpand
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Computational complexity of differential and integral equations - an information-based approach
  • A. Werschulz
  • Mathematics, Computer Science
  • Oxford mathematical monographs
  • 7 November 1991
Introduction EXAMPLE: A TWO-POINT BOUNDARY VALUE PROBLEM: Introduction Error, cost, and complexity A minimal error algorithm Complexity bounds Comparison with the finite element method StandardExpand
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What is the complexity of ill-posed problems?
This paper deals with the optimal solution of ill-posed linear problems, i.e..linear problems for which the solution operator is unbounded. We consider worst-case ar,and averagecase settings. OurExpand
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The Complexity of Definite Elliptic Problems with Noisy Data
  • A. Werschulz
  • Computer Science, Mathematics
  • J. Complex.
  • 1 December 1996
TLDR
We study the complexity of 2mth order definite elliptic problemsLu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain ?, error being measured in theHm(?)-norm. We find that a noisy finite element method with quadrature is a minimal error algorithm. Expand
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Finite element methods are not always optimal
Consider a regularly elliptic 2mth order boundary value problem Lu = f with f @e H^r(@W), where r $@?m. In previous work, we showed that the finite element method (FEM) using piecewise polynomials ofExpand
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What Is the Complexity of Surface Integration?
We study the worst case complexity of computing ?-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface.Expand
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Where does smoothness count the most for Fredholm equations of the second kind with noisy information?
  • A. Werschulz
  • Computer Science, Mathematics
  • J. Complex.
  • 1 December 2003
TLDR
We study the complexity of Fredholm problems (I - Tk)u = f of the second kind on Id = [0,1]d, where Tk is an integral operator with kernel k. Expand
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Tractability of multivariate approximation over weighted standard Sobolev spaces
TLDR
We study multivariate approximation for complex-valued functions defined over the d -dimensional torus, these functions belonging to weighted standard Sobolev spaces of smoothness  r . Expand
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Optimal error properties of finite element methods for second order elliptic Dirichlet problems
We use the informational approach of Traub and Wozniakowski [9] to study the variational form of the second order elliptic Dirichlet problem Lu = f on Q C RN. For f G HW(Q), where r -1, aExpand
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