Toeplitz Matrix-vector Multiplication Research Articles

A quantum algorithm for Toeplitz matrix-vector multiplication

Toeplitz matrix-vector multiplication is widely used in various fields, including optimal control, systolic finite field multipliers, multidimensional convolution, etc. In this paper, we first present a non-asymptotic quantum algorithm for Toeplitz matrix-vector multiplication with time complexity , where κ and 2n are the condition number and the dimension of the circulant matrix extended from the Toeplitz matrix, respectively. For the case with an unknown generating function, we also give a corresponding non-asymptotic quantum version that eliminates the dependency on the L 1-norm ρ of the displacement of the structured matrices. Due to the good use of the special properties of Toeplitz matrices, the proposed quantum algorithms are sufficiently accurate and efficient compared to the existing quantum algorithms under certain circumstances.

Algorithms for solving a class of real quasi-symmetric Toeplitz linear systems and its applications

<abstract><p>In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.</p></abstract>

Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations

The solution of d-dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d-dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT.

Robust barrier option pricing by frame projection under exponential Lévy dynamics

ABSTRACTWe present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.

A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics

The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from $$O(N^{3})$$ required by traditional methods to O(Nlog $$^{2}N)$$ and the memory requirement from $$O(N^{2})$$ to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

A fast solution technique for finite element discretization of the space–time fractional diffusion equation

In this paper, we study fast Galerkin finite element methods to solve a space–time fractional diffusion equation. We develop an optimal piecewise-linear and piecewise-quadratic finite element methods for solving this problem and give optimal error estimates. Furthermore, we develop piecewise-constant discontinuous finite element method for discontinuous problem of this model. Importantly, a fast solution technique to accelerate non-square Toeplitz matrix–vector multiplications which arise from both continuous and discontinuous Galerkin finite element discretization respectively is considered. This fast solution technique is based on fast Fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from O(N3) required by the traditional methods to O(Nlog2⁡N), where N is the size (number of spatial grid points) of the coefficient matrices for every time step. Moreover, the applicability and accuracy of the method are demonstrated by numerical experiments to support our theoretical analysis.

An [formula omitted]-Galerkin method with fast solution for linear peridynamic models in one dimension

The computational work and memory requirement are bottlenecks for Galerkin finite element methods for peridynamic models because of their non-locality. In this paper, fast Galerkin and hp-Galerkin finite element methods are introduced and analyzed to solve a steady-state peridynamic model. We present a fast solution technique to accelerate non-square Toeplitz matrix–vector multiplications arising from piecewise-linear, piecewise-quadratic and piecewise-cubic Galerkin methods. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O(N3) required by traditional methods to O(Nlog2N) and the memory requirement from O(N2) to O(N) without using any lossy compression, where N is the number of unknowns. The peridynamic model admits solutions having jump discontinuities. For problems with discontinuous solutions, we therefore introduce a piecewise-constant Galerkin method and give an h- and p-refinement algorithm. Then, we develop fast hp-Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewise-constant/piecewise-quadratic finite element approximations. The new method reduces the computational work from O(N3) required by the traditional methods to O(Nlog2N) and the memory requirement from O(N2)to O(N).

A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

In this paper, we study the block lower triangular Toeplitz-like with tri-diagonal blocks system which arises from the time-fractional partial differential equation. Existing fast numerical solver (e.g., fast approximate inversion method) cannot handle such linear system as the main diagonal blocks are different. The main contribution of this paper is to propose a fast direct method for solving this linear system, and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system. Our idea is based on the divide-and-conquer strategy and together with the fast Fourier transforms for calculating Toeplitz matrix–vector multiplication. The complexity needs O(MNlog2⁡M) arithmetic operations, where M is the number of blocks (the number of time steps) in the system and N is the size (number of spatial grid points) of each block. Numerical examples from the finite difference discretization of time-fractional partial differential equations are also given to demonstrate the efficiency of the proposed method.

Robust Barrier Option Pricing by Frame Projection Under Exponential Levy Dynamics

We present an efficient method for robustly pricing discretely monitored single and double barrier/Parisian options with an exponential Levy model for the transition density between monitoring dates. This includes ordinary barrier options, as well as (resetting) Parisian options and delayed barrier options (also known as cumulative Parisian or Parasian options) which have yet to be priced with more general Levy processes, including KoBoL (CGMY), Merton's jump diffusion and NIG. The method's efficiency is derived in part from the use of frame projected transition densities, which transform the problem into the Fourier domain, and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method's robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method's efficiency and accuracy.

A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix $T$ into a Cauchy-like matrix $\mathcal{C}$, which is known to have low-numerical-rank off-diagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to $\mathcal{C}$, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrix-vector multiplications, and thus converts the direct compression of the off-diagonal blocks of $\mathcal{C}$ into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size $n$. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes $O(n)$ flops for HSS operations and $O(n\log^{2} n)$ flops for matrix multiplications via FFTs, where $n$ is the order of $T$. Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

Multigrid method for fractional diffusion equations

The fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted Grünwald formula. The scheme is unconditionally stable and the coefficient matrix possesses the Toeplitz-like structure. A multigrid method is proposed to solve the resulting system. Meanwhile, the fast Toeplitz matrix–vector multiplication is utilized to lower the computational cost with only O ( N log N ) complexity, where N is the number of the grid points. Numerical experiments are given to demonstrate the efficiency of the method.

Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential

The shift-invert Arnoldi method is employed to generate an orthonormal basis from the Krylov subspace corresponding to a real Toeplitz matrix and an initial vector. The vectors and recurrence coefficients produced by this method are exploited to approximate the Toeplitz matrix exponential. Toeplitz matrix inversion formula and rapid Toeplitz matrix-vector multiplications are utilized to lower the computational costs. For convergence analysis, a sufficient condition is established to guarantee that the error bound is independent of the norm of the matrix. Numerical results are given to demonstrate the efficiency of the method.

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Levy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Levy process. It can be discretized with exponentially decaying errors of the form O(exp (−aM b )) for some a,b>0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix–vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NMlog (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Levy models.

PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH

This paper presents a novel method to price discretely monitored single‐ and double‐barrier options in Lévy process‐based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT‐based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.

Evaluating recursive filters on distributed memory parallel computers

The aim of this paper is to show that the recently developed high performance divide and conquer algorithm for solving linear recurrence systems with constant coefficients together with the new BLAS-based algorithm for narrow-banded triangular Toeplitz matrix–vector multiplication, allow to evaluate linear recursive filters efficiently on distributed memory parallel computers. We apply the BSP model of parallel computing to predict the behaviour of the algorithm and to find the optimal values of the method's parameters. The results of experiments performed on a cluster of twelve dual-processor Itanium 2 computers and Cray X1 are also presented and discussed. The algorithm allows to utilize up to 30% of the peak performance of 24 Itanium processors, while a simple scalar algorithm can only utilize about 4% of the peak performance of a single processor. Copyright © 2006 John Wiley & Sons, Ltd.

Evaluating linear recursive filters on clusters of workstations

The aim of this paper is to show that the recently developed high performance algorithm for solving linear recurrence systems with constant coefficients together with the new BLAS-based algorithm for narrow-banded triangular Toeplitz matrix-vector multiplication allow to evaluate linear recursive filters efficiently, even on clusters of workstations. The results of experiments performed on a cluster of twelve Linux workstations are also presented. The performance of the algorithm is comparable with the performance of two processors of Cray SV-1 for such kind of recursive problems.

Fast Recursive Least Squares Adaptive Filtering by Fast Fourier Transform-Based Conjugate Gradient Iterations

Recursive least squares (RLS) estimations are used extensively in many signal processing and control applications. In this paper we consider RLS with sliding data windows involving multiple (rank k) updating and downdating computations. The least squares estimator can be found by solving a near-Toeplitz matrix system at each step. Our approach is to employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Here we iterate in the time domain (using Toeplitz matrix-vector multiplications) and precondition in the Fourier domain, so that the fast Fourier transform (FFT) is used throughout the computations. The circulant preconditioners are derived from the spectral properties of the given input stochastic process. When the input stochastic process is stationary, we prove that with probability 1, the spectrum of the preconditioned system is clustered around 1 and the method converges superlinearly provided that a sufficient number of data samples are taken, i.e.,...

Extensions to the distributed approximating functional: The harmonic propagator.

The formalism of the distributed approximating functional for integrating the time-dependent Schr\odinger equation is extended to allow an analytic treatment of all Hamiltonians of quadratic order. The resulting propagator matrix is highly banded in both position and momentum representations. For certain choices of the parameters, the action of this matrix can be cast in the form of a Toeplitz operator. We demonstrate that banded Toeplitz matrix-vector multiplication can be performed in significantly fewer numerical operations than is required by conventional split-operator fast Fourier transform algorithms. Numerical examples verify the efficiency of this approach in both storage and speed. Extensions of the method to Monte Carlo path integration, and the implementation on massively parallel computer architectures, are discussed.