Anargyros Papageorgiou

Faster evaluation of multidimensional integrals

In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasi-Monte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occurng in mathematical finance. In this paper we report test results of this method on Keisters examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100. For the 25 dimensional integral we achieved accuracy of 0.01 with less than 500 points while the two methods tested by Keister used more than 220,000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n^{-1} rather than the n^{-1/2} of Monte Carlo.

On the average complexity of multivariate problems

Abstract We study the average complexity of linear problems, on a separable Banach space equipped with an orthogonally invariant measure μ. The error and the cost of the algorithms are defined on the average. We exhibit an information operator which is optimal among any linear information operators. We apply the general results to the approximation problem of real functions of d variables. The space is now equipped with a Wiener measure placed on partial derivatives. We show that the average complexity of this problem is almost independent of the dimension d if arbitrary linear functionals are permitted in the information. We conjecture that the same result holds if the information is restricted to function and/or partial derivative evaluations only.

The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration

The Brownian bridge has been suggested as an effective method for reducing the quasi-Monte Carlo error for problems in finance. We give an example of a digital option where the Brownian bridge performs worse than the standard discretization. Hence, the Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. We consider integrals of functions of d variables with Gaussian weights such as the ones encountered in the valuation of financial derivatives and in risk management. Under weak assumptions on the class of functions, we study quasi-Monte Carlo methods that are based on different covariance matrix decompositions. We show that different covariance matrix decompositions lead to the same worst case quasi-Monte Carlo error and are, therefore, equivalent.

Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

We consider the approximation ofd-dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where d is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals isO( p logn=n). Since this is a worst case result, compared to the expected convergence rate O(n 1=2 ) of Monte Carlo, it shows the superiority of quasi-Monte Carlo for this type of integral. This is much faster than the worst case convergence, O(log d n=n), of quasi-Monte Carlo.

Deterministic Simulation for Risk Management

Monte Carlo simulation is widely used in pricing and risk management of complex financial instruments. Deterministic simulation methods (quasi-Monte Carlo methods) are superior to Monte Carlo in terms of accuracy and speed. The authors show how deterministic simulation can be applied to calculate value at risk. They use in their tests a portfolio of collaterized mortgage obligation tranches. One of the deterministic methods consistently outperforms Monte Carlo simulation.

Sufficient conditions for fast quasi-Monte Carlo convergence

We study the approximation of d-dimensional integrals. We present sufficient conditions for fast quasi-Monte Carlo convergence. They apply to isotropic and non-isotropic problems and, in particular, to a number of problems in computational finance. We show that the convergence rate of quasi-Monte Carlo is of order n-1+p{log n}-1/2 with p≥0. This is a worst case result. Compared to the expected rate n-1/2 of Monte Carlo it shows the superiority of quasi-Monte Carlo.

New Results on Deterministic Pricing of Financial Derivatives

Monte Carlo simulation is widely used to price complex financial instruments. Recent theoretical results and extensive computer testing indicate that deterministic methods may be far superior in speed and confidence. In this paper we test the generalized Faure method due to Tezuka on a Collateralized Mortgage Obligation (CMO). This requires integration in 360 dimensions. We conclude that deterministic methods beat Monte Carlo by a wide margin. Among the deterministic methods we have tested, the generalized Faure method is the method of choice. For example, for the hardest CMO tranche, generalized Faure achieves accuracy 10-2 with just 170 points while the Monte Carlo method requires 2700 points for the same accuracy. We introduce a new and more rigorous definition of speed-up. For high accuracy, generalized Faure is 1000 times faster than Monte Carlo.

Quantum algorithm and circuit design solving the Poisson equation

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error e. We assume we are given a superposition of function evaluations of the right-hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in e−1. We present quantum circuit modules together with performance guarantees which can also be used for other problems.

Classical and Quantum Complexity of the Sturm--Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm–Liouville problem in the classical and quantum settings. We consider a univariate Sturm–Liouville eigenvalue problem with a nonnegative function q from the class C2 ([0,1]) and study the minimal number n(ɛ) of function evaluations or queries that are necessary to compute an ɛ-approximation of the smallest eigenvalue. We prove that n(ɛ)=Θ(ɛ−1/2) in the (deterministic) worst case setting, and n(ɛ)=Θ(ɛ−2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grover’s algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp((1/2) iM), where M is an n× n matrix obtained from the standard discretization of the Sturm–Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in n is an open issue. In particular, we show how to compute an ɛ-approximation with probability (3/4) using n(ɛ)=Θ(ɛ−1/3) bit queries. For power queries, we use the phase estimation algorithm as a basic tool and present the algorithm that solves the problem using n(ɛ)=Θ(log ɛ−1) power queries, log 2ɛ−1 quantum operations, and (3/2) log ɛ−1 quantum bits. We also prove that the minimal number of qubits needed for this problem (regardless of the kind of queries used) is at least roughly (1/2) log ɛ−1. The lower bound on the number of quantum queries is proven in Bessen (in preparation). We derive a formula that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem. Many computational problems may be recast as this weighted integration problem, which allows us to solve them with a polylog number of power queries. Examples include Grover’s search, the approximation of the Boolean mean, NP-complete problems, and many multivariate integration problems. In this paper we only provide the relationship formula. The implications are covered in a forthcoming paper (in preparation).

Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.