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While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce…
We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann-Liouville…
In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the $G_{n}^{(1)}$ transformation and Slevinsky-Safouhi formula for differentiation. In the…
A new general Lie-algebraic approach is proposed to solving evolution tasks in some nonlinear problems of quantum physics with polynomially deformed Lie algebras $su_{pd}(2)$ as their dynamic symmetry algebras. The method makes use of an…
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It\^o-Taylor expansion and…
By using a simple observation that the density processes appearing in Ito's martingale representation theorem are invariant under the change of measures, we establish a non-linear version of the Cameron-Martin formula for solutions of a…
Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these…
Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally…
This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance…
An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to…
Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain unknown parameters, say $\theta$ of physical significance which…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with…
Bayesian Optimization (BO) has shown great promise for the global optimization of functions that are expensive to evaluate, but despite many successes, standard approaches can struggle in high dimensions. To improve the performance of BO,…
In this article, we propose a novel discretization method based on numerical integration for discretizing continuous systems, termed the $\alpha\beta$-approximation or Scalable Bilinear Transformation (SBT). In contrast to existing methods,…
Tensors are ubiquitous in science and engineering and tensor factorization approaches have become important tools for the characterization of higher order structure. Factorizations includes the outer-product rank Canonical Polyadic…
We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given…
Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement…