Uniform Weak Tractability of Multivariate Problems

In this dissertation we introduce a new notion of tractability which is called uniform weak tractability. We give necessary and sufficient conditions on uniform weak tractability of homogeneous linear tensor product problems in the worst case, average case and randomized settings. We then turn to the study of approximation problems defined over spaces of functions with varying regularity with respect to successive variables. In the worst case setting we study approximation problems defined over suitable Korobov and Sobolev spaces. In the average case setting we study approximation problems defined over the space of continuous functions C([0, 1]^d ) equipped with a zero-mean Gaussian measure whose covariance operator is given by Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters

Contributions to Information-Based Complexity and to Quantum Computing

Multivariate continuous problems are widely encountered in physics, chemistry, finance and in computational sciences. Unfortunately, interesting real world multivariate continuous problems can almost never be solved analytically. As a result, they are typically solved numerically and therefore approximately. In this thesis we deal with the approximate solution of multivariate problems. The complexity of such problems in the classical setting has been extensively studied in the literature. On the other hand the quantum computational model presents a promising alternative for dealing with multivariate problems. The idea of using quantum mechanics to simulate quantum physics was initially proposed by Feynman in 1982. Its potential was demonstrated by Shor's integer factorization algorithm, which exponentially improves the cost of the best classical algorithm known. In the first part of this thesis we study the tractability of multivariate problems in the worst and average case settings using the real number model with oracles. We derive necessary and sufficient conditions for weak tractability for linear multivariate tensor product problems in those settings. More specifically, we initially study necessary and sufficient conditions for weak tractability on linear multivariate tensor product problems in the worst case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the squares of the singular values of the solution operator for the univariate problem. We show a condition on the singular values that is sufficient for weak tractability. The same condition is known to be necessary for weak tractability. Then, we study linear multivariate tensor product problems in the average case setting under the absolute error criterion. The complexity of such problems depends on the rate of decay of the eigenvalues of the covariance operator of the induced measure of the one dimensional problem. We derive a necessary and sufficient condition on the eigenvalues for such problems to be weakly tractable but not polynomially tractable. In the second part of this thesis we study quantum algorithms for certain eigenvalue problems and the implementation and design of quantum circuits for a modification of the quantum NAND evaluation algorithm on k-ary trees, where k is a constant. First, we study quantum algorithms for the estimation of the ground state energy of the multivariate time-independent Schrodinger equation corresponding to a multiparticle system in a box. The dimension d of the problem depends linearly to the number of particles of the system. We design a quantum algorithm that approximates the lowest eigenvalue with relative error ε for a non-negative potential V, where V as well as its first order partial derivatives are continuous and uniformly bounded by one. The algorithm requires a number of quantum operations that depends polynomially on the inverse of the accuracy and linearly on the number of the particles of the system. We note that the cost of any classical deterministic algorithm grows exponentially in the number of particles. Thus we have an exponential speedup with respect to the dimension of the problem d, when compared to the classical deterministic case. We extend our results to convex non-negative potentials V, where V as well as its first order partial derivatives are continuous and uniformly bounded by constants C and C' respectively. The algorithm solves the eigenvalue problem for a sequence of convex potentials in order to obtain its final result. More specifically, the quantum algorithm estimates the ground state energy with relative error ε a number of quantum operations that depends polynomially on the inverse of the accuracy, the uniform bound C on the potential and the dimension d of the problem. In addition, we present a modification of the algorithm that produces a quantum state which approximates the ground state eigenvector of the discretized Hamiltonian within Δ. This algorithm requires a number of quantum operations that depends pollynomially on the inverse of ε, the inverse of Δ, the uniform bound C on the potential and the dimension d of the problem. Finally, we consider the algorithm by Ambainis et.al. that evaluates balanced binary NAND formulas. We design a quantum circuit that implements a modification of the algorithm for k-ary trees, where k is a constant. Furthermore, we design another quantum circuit that consists exclusively of Clifford and T gates. This circuit approximates the previous one with error ε using the Solovay-Kitaev algorithm

The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces

We study linear problems defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the singular values of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems we give characterizations for polynomial tractability and strong polynomial tractability in terms of the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schr\"odinger equation.Comment: Extended version (53 pages); corrected typos, added journal referenc

Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces

Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k >1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k

Some Results on the Complexity of Numerical Integration

This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected

On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.Comment: 18 page