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We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in R3 and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calder´on-type preconditioners based on (i) new boundary integral operators, which provide the exact inverses of the weakly singular and hypersingular operators on flat disks, and (ii) stable duality pairings relying on dual meshes. On screens obtained as images of the unit disk under bi-Lipschitz transformations, this approach achieves condition numbers uniformly bounded in the meshwidth even on locally refined meshes. Comprehensive numerical tests also confirm its excellent pre-asymptotic performance.

Quantum Computing commenced in 1980’s with the pioneering work of Paul Benioff (Benioff, 1980) who proposed a quantum mechanical Turing machine. These ideas were also explored by the likes of Richard Feynman (Feynman, 1982) and Yuri Manin (Manin, 1980) who suggested that quantum computers could provide advantage over classical computers in certain tasks, such as the simulation of physical systems. In 1994, Peter Shor (Shor, 1994) published a groundbreaking paper demonstrating that a quantum algorithm could be used to for large integer factorisation in polynomial time, that is, exponentially faster than the best known classical algorithms. This was followed by Lov K. Grover who proposed a quantum computing algorithm that promised a quadratic speed up over database searches (L. K. Grover, 1996). Advances in Quantum Computing hardware technology in recent years have been accompanied by the acceleration on the development of quantum computing algorithms with applications across many different use cases in different industry sectors: Automotive, Energy, Logistics, Pharma, Chemical/Manufacturing and the Financial Services Industry. One of the use cases in Finance comes from the application of Quantum Computing for Derivative Pricing and Derivative Risk Management. The purpose of this document is toprovide a summary of the “state of the art” for these applications. However, it is important to note that the currently available quantum computers have limited number of qubits and these suffer from high levels of “noise” which limits the depth and length of the quantum circuits that can be implemented in real hardware. Therefore, near-term applications focus on implementations of quantum algorithms in Noisy Intermediate-Scale Quantum (NISQ) computers (Preskill, 2018). Derivatives contract form one of the fundamental pillars of modern financial markets and are routinely traded by both financial institutions and traders with a variety of objectives, such as financial risk hedging. A simple example of financial derivative is a European stock option. This contract provides the derivative holder with the right to purchase or sell the stock at some time in the future for a fixed price agreed today. Hence, providing with potential upside (should the stock increase in price at maturity) while limiting the investor’s downside. Derivatives pricing theory is the branch of financial mathematics that covers the fair valuation of financial derivatives such as options. This framework assumes that the underlying security (e.g. a single stock) follows some random (stochastic) process. The price of the derivative hence depends on the particular realisation of such process at a given point in time (e.g. the option maturity). The best known example of option pricing model is the Black-Scholes model (Black, 1976). This model proves that a fair value of an option can be derived under certain assumptions (e.g. absence of arbitrage, continuous and unlimited long and short trading). However, in general the Black-Sholes model is too simplistic to fit actual quoted prices in the market and other more complex models are used instead, at the cost of requiring numerical approximations to find the fair price of the derivative. Two main numerical approaches are currently used in the industry, Monte Carlo-based simulation techniques and partial differential equation (PDEs) approaches. The key advantage of the former is that it is easy to implement, very general and scales well with the dimension of the problem. On the other hand, Monte Carlo simulation tends to converge slowly to the required solution. It is also difficult to obtain risk sensitivities (i.e. how the derivative price depends on changes to the price underlying) using Monte Carlo. PDEs approaches are generally faster and permit the easy calculation of risk sensitives, however it is usually a difficult problem to solve PDEs for more complex derivatives, specially those that depend on several risk factors (curse of dimensionality). Monte Carlo simulation is therefore the tool of choice for financial risk management where risk metrics need to be estimated at the portfolio level where thousands of derivatives need to be covered. Quantum computing, and in particular the Quantum Amplitude Estimation (QAE) algorithm promises a potential quadratic speed up over classical Monte Carlo approaches but maintaining its main advantages: easy of implementation and linear scaling to higher dimensions. This document covers these topics in more detail and presents some new results, such as the application of the “Quantum Coin” algorithms as an alternative to QAE. Despite the highly promising advantages of quantum computing for derivative pricing and risk management, huge challenges remain open for real-world applications. Some of them are technological, such as the relative small number of qubits currently available and the fact that these are “noisy” (i.e. not always reliable). Others are more theoretical, such as the lack of understanding on how to load classical information (such as probability distributions) to quantum registers, or how to represent relatively complex pay-off functions with quantum circuits that are as small as possible., {"references":["Aaronson, S., & Rall, P. (2020). Quantum approximate counting, simplified. Symposium on simplicity in algorithms (pp. 24–32). 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We present a family of preconditioners based on the auxiliary space method for a discontinuous Galerkin discretization on cubical meshes of H(curl;Ω)-elliptic problems with possibly discontinuous coefficients. We address the influence of possible discontinuities in the coefficients on the asymptotic performance of the proposed solvers and present numerical results in two dimensions.

We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric interior penalty discontinuous Galerkin discretization of H(curl,Ω)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners rely on the auxiliary space method (ASM) employing an auxiliary space of H(curl,Ω)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ν and β in the second- and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,Ω)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners., IMA Journal of Numerical Analysis, 37 (2), ISSN:0272-4979, ISSN:1464-3642

SUMMARY The plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the h-version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so-called ultra-weak variational formulation, see [O. CESSENAT AND B. DESPReS, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. The method relies on Trefftz-type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of DG methods. We conduct a partly empirical dispersion analysis of the method in a discrete translation-invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions, the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as a pollution effect haunting the h-version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves. Copyright © 2014 John Wiley & Sons, Ltd.