Frances Y. Kuo

Frances Y. Kuo
UNSW Sydney | UNSW · School of Mathematics and Statistics

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136
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Introduction
Skills and Expertise

Publications

Publications (136)
Article
Full-text available
We approximate d-variate periodic functions in weighted Korobov spaces with general weight parameters using n function values at lattice points. We do not limit n to be a prime number, as in currently available literature, but allow any number of points, including powers of 2, thus providing the fundamental theory for construction of embedded latti...
Article
Full-text available
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for th...
Article
Full-text available
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic function...
Preprint
We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the $d$-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a gi...
Preprint
The kernel interpolant in a reproducing kernel Hilbert space is optimal in the worst-case sense among all approximations of a function using the same set of function values. In this paper, we compare two search criteria to construct lattice point sets for use in lattice-based kernel approximation. The first candidate, $\calP_n^*$, is based on the p...
Preprint
Full-text available
We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random...
Preprint
In this paper we propose and analyse a method for estimating three quantities related to an Asian option: the fair price, the cumulative distribution function, and the probability density. The method involves preintegration with respect to one well chosen integration variable to obtain a smooth function of the remaining variables, followed by the a...
Chapter
Preintegration is a technique for high-dimensional integration over the d-dimensional Euclidean space, which is designed to reduce an integral whose integrand contains kinks or jumps to a \((d-1)\)-dimensional integral of a smooth function. The resulting smoothness allows efficient evaluation of the \((d-1)\)-dimensional integral by a Quasi-Monte C...
Preprint
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic...
Preprint
We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow any number of points, including powers of $2$, thus providing the fundamental theory for construction of embedd...
Article
Full-text available
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast...
Preprint
Preintegration is a technique for high-dimensional integration over $d$-dimensional Euclidean space, which is designed to reduce an integral whose integrand contains kinks or jumps to a $(d-1)$-dimensional integral of a smooth function. The resulting smoothness allows efficient evaluation of the $(d-1)$-dimensional integral by a Quasi-Monte Carlo o...
Preprint
The cumulative distribution or probability density of a random variable, which is itself a function of a high number of independent real-valued random variables, can be formulated as high-dimensional integrals of an indicator or a Dirac $\delta$ function, respectively. To approximate the distribution or density at a point, we carry out preintegrati...
Preprint
Full-text available
We investigate the application of efficient recursive numerical integration strategies to models in lattice gauge theory from quantum field theory. Given the coupling structure of the physics problems and the group structure within lattice cubature rules for numerical integration, we show how to approach these problems efficiently by means of Fast...
Article
High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product structure with low order couplings, motivated by models in lattice gauge theory from quantum field theory. A novel e...
Preprint
We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on R d \mathbb {R}^d , for d ≥ 1 d \geq 1 . Both spaces are Hilbert spaces involving weight functions , which determine the behaviour as different variables tend to ± ∞ \pm \infty...
Preprint
Full-text available
High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product structure with low order couplings, motivated by models in lattice gauge theory from quantum field theory. A novel e...
Preprint
Full-text available
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice---a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need...
Preprint
Full-text available
We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of th...
Article
We correct the expression for the worst-case error derived in [Kuo, Wasilkowski, Woźniakowski, Construct. Approx. 30 (2009), 475–493] and explain that the main theorem of the paper holds with enlarged constants.
Article
We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specific...
Preprint
We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known...
Preprint
This paper provides the theoretical foundation for the construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. Our construction leads to an error bound that achieves the best possible rate of convergence for lattice algorithms. This work is...
Preprint
In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible r...
Preprint
We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into...
Article
Full-text available
We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fie...
Article
Full-text available
The corrected version states the parameter range as p ∈ 2ℕ instead of p ∈ ℕ. The effect is to disallow non-smooth norms such as the ℓ1-norm for the distance measure.
Preprint
Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each uniformly distributed on an interval, with these random variables entering linearly in the input random field (the so-called "affine" model). In this paper we consider an...
Preprint
We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specific...
Preprint
An existence result is presented for the worst-case error of lattice rules for high dimensional integration over the unit cube, in an unanchored weighted space of functions with square-integrable mixed first derivatives. Existing studies rely on random shifting of the lattice to simplify the analysis, whereas in this paper neither shifting nor any...
Article
Full-text available
We use the $H$-matrix technology to compute the approximate square root of a covariance matrix in linear complexity. This allows us to generate normal and log-normal random fields on general point sets with optimal complexity. We derive rigorous error estimates which show convergence of the method. Our approach requires only mild assumptions on the...
Preprint
The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the "weighted" function space setting introduced by Sloan and Wo\'zniakowski. The "weights" that define such spaces are needed as inputs into the CBC algorithm, and so a nat...
Article
Full-text available
In a previous paper (J. Comp. Phys. \textbf{230} (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expe...
Preprint
Full-text available
We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fie...
Chapter
This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tuto...
Chapter
This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube \([0,1]^s\) and in \({\mathbb {R}}^s\), and higher order QMC methods in the unit cube. One important feature is that their error bounds can...
Chapter
Building on previous research which generalized multilevel Monte Carlo methods using either sparse grids or Quasi-Monte Carlo methods, this paper considers the combination of all these ideas applied to elliptic PDEs with finite-dimensional uncertainty in the coefficients. It shows the potential for the computational cost to achieve an \(O(\varepsil...
Book
This book is a tribute to Professor Ian Hugh Sloan on the occasion of his 80th birthday. It consists of nearly 60 articles written by international leaders in a diverse range of areas in contemporary computational mathematics. These papers highlight the impact and many achievements of Professor Sloan in his distinguished academic career. The book a...
Article
Full-text available
We show how simple kinks and jumps of otherwise smooth integrands over $\mathbb{R}^d$ can be dealt with by a preliminary integration with respect to a single well chosen variable. It is assumed that this preintegration, or conditional sampling, can be carried out with negligible error, which is the case in particular for option pricing problems. It...
Article
In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of $\infty$-variate functions. Such $\infty$-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition $f = \sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{...
Preprint
In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of $\infty$-variate functions. Such $\infty$-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition $f = \sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{...
Preprint
We show how simple kinks and jumps of otherwise smooth integrands over $\mathbb{R}^d$ can be dealt with by a preliminary integration with respect to a single well chosen variable. It is assumed that this preintegration, or conditional sampling, can be carried out with negligible error, which is the case in particular for option pricing problems. It...
Article
Building on previous research which generalized multilevel Monte Carlo methods using either sparse grids or Quasi-Monte Carlo methods, this paper considers the combination of all these ideas applied to elliptic PDEs with finite-dimensional uncertainty in the coefficients. It shows the potential for the computational cost to achieve an $O(\varepsilo...
Article
This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be ind...
Article
This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tuto...
Preprint
In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected valu...
Article
In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, pos...
Preprint
In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, pos...
Article
Full-text available
We analyze a random algorithm for numerical integration of $d$-variate functions from weighted Sobolev spaces with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. A variant of this algorithm was first introduced b...
Preprint
We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where the functions are continuous and periodic when $\alpha>1/2$. The algorithm is based on rank-$1$ lattice rules...
Article
It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the error. In particular, we show for geometric Brownian motion that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the expected...
Preprint
Full-text available
It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the uniform error. In particular, we show constructively that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error a...
Preprint
We use the $H$-matrix technology to compute the approximate square root of a covariance matrix in linear cost. This allows us to generate normal and log-normal random fields on general point sets with optimal cost. We derive rigorous error estimates which show convergence of the method. Our approach requires only mild assumptions on the covariance...
Article
This is a note on Math. Comp. 82 (2013), 383-400. We first report a mistake, in that the main result Theorem 3.1, though correct, does not as claimed apply to the Asian option pricing problem. This is because assumption (3.3) in the theorem is not satisfied by the Asian option pricing problem. In this note we present a strengthened theorem, which r...
Article
This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first order QMC rules ve...
Preprint
This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first order QMC rules ve...
Preprint
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connectio...
Article
The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the “weighted” function space setting introduced by Sloan and Woźniakowski. The “weights” that define such spaces are needed as inputs into the CBC algorithm, and so a natur...
Article
We study multivariate integration of analytic functions defined on Rd. These functions are assumed to belong to a reproducing kernel Hilbert space whose kernel is Gaussian, with nonincreasing shape parameters. We prove that a tensor product algorithm based on the univariate Gauss-Hermite quadrature rules enjoys exponential convergence and computes...
Article
The pricing problem for a continuous path-dependent option results in a path integral which can be recast into an infinite-dimensional integration problem. We study ANOVA decomposition of a function of infinitely many variables arising from the Brownian bridge formulation of the continuous option pricing problem. We show that all resulting ANOVA te...
Article
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank- lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection...
Article
We develop a convergence analysis of a multilev el algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic types, extending both the multilevel first order analysis in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Found....
Article
We consider a ρ{variant}-weighted Lq approximation in the space of univariate functions f:R+→R with finite ‖f(r)ψ‖Lp. Let α=r-1/p+1/q and ω=ρ{variant}/ψ. Assuming that ψ and ω are non-increasing and the quasi-norm ‖ω‖L1/α is finite, we construct algorithms using function/derivatives evaluations at n points with the worst case errors proportional to...
Article
Full-text available
In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to mu...
Article
Full-text available
Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general c...
Article
We develop the generic "Multivariate Decomposition Method" (MDM) for weighted integration of functions of infinitely many variables $(x_1,x_2,x_3,\ldots)$. The method works for functions that admit a decomposition $f=\sum_{\mathfrak{u}} f_{\mathfrak{u}}$, where $\mathfrak{u}$ runs over all finite subsets of positive integers, and for each $\mathfra...
Article
This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over RsRs arising from practical applications. For an integral of the form ∫Rsf(y)∏j=1sϕ(yj)dy with a univariate probability density ϕϕ, our general strategy is to first map the integral i...
Chapter
Evaluation of the likelihood of generalised response models in statistics leads to integrals over unbounded regions in high dimensions. In order to apply a quasi-Monte Carlo (QMC) method to approximate such integrals, one has to transform the original integral into an equivalent integral over the unit cube. From the point of view of QMC, this leads...
Article
Full-text available
We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regular...
Article
Full-text available
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z(x, ω)) where x is the spatial variable and Z(x, ·) is a Gaussian random field. The analysis presents particular challenges since the corr...
Article
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related them...
Article
Full-text available
This paper studies the ANOVA decomposition of a d-variate function f defined on the whole of ℝd, where f is the maximum of a smooth function and zero (or f could be the absolute value of a smooth function). Our study is motivated by option pricing problems. We show that under suitable conditions all terms of the ANOVA decomposition, except the one...
Article
The topics of the workshop were recent progress in the theory of uniform distribution theory (also known as discrepancy theory) and new developments in its applications in analysis, approximation theory, computer science, numerics, pseudo-randomness and stochastic simulation.
Article
CORRECTION TO “QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND” - Volume 54 Issue 3 - F. Y. KUO, CH. SCHWAB, I. H. SLOAN
Book
This book represents the refereed proceedings of the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held at the University of New South Wales (Australia) in February 2012. These biennial conferences are major events for Monte Carlo and the premiere event for quasi-Monte Carlo research. T...
Article
Full-text available
Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient, to estimate expected values of linear functionals of the solution. The expected value is considered as an infinite-dimensional integral in the parameter space corresponding to th...
Article
We study univariate integration with the Gaussian weight for a positive variance α. This is done for the reproducing kernel Hilbert space with the Gaussian kernel for a positive shape parameter γ. We study Gauss-Hermite quadratures, although this choice of quadratures may be questionable since polynomials do not belong to this space of functions. N...
Article
This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube \([0,1]^s\). It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates al...
Article
Full-text available
Quasi-Monte Carlo integration rules, which are equal-weight sample averages of function values, have been popular for approximating multivariate integrals due to their superior convergence rate of order close to 1/N or better, compared to the order of simple Monte Carlo algorithms. For practical applications, it is desirable to be able to increase...
Article
In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in unc...
Article
This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alterna...