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High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the…
This paper provides a comparative study of modern uncertainty quantification (UQ) methods. To greatly enhance real-time performance, both differential algebra (DA) and a directional differential algebra (DDA) approach are employed. This can…
A novel algorithm for producing smooth nonlinearities on digital hardware is presented. The non-linearities are inherently quadratic and have both symmetrical and asymmetrical variants. The integer (and fixed point) implementation is highly…
The convolution quadrature method originally developed for the Riemann-Liouville fractional calculus is extended in this work to the Hadamard fractional calculus by using the exponential type meshes. Local truncation error analysis is…
For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…
Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry and statistics. Traditional solution methods such as Gaussian elimination become very time…
Maximum simulated likelihood estimation of mixed multinomial logit (MMNL) or probit models requires evaluation of a multidimensional integral. Quasi-Monte Carlo (QMC) methods such as shuffled and scrambled Halton sequences and modified…
Interface problems depict many fundamental physical phenomena and widely apply in the engineering. However, it is challenging to develop efficient fully decoupled numerical methods for solving degenerate interface problems in which the…
In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the…
We present a new discretization for the Hadamard fractional derivative, that simplifies the computations. We then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the…
This paper proposes new proximal Newton-type methods with a diagonal metric for solving composite optimization problems whose objective function is the sum of a twice continuously differentiable function and a proper closed directionally…
The growing prevalence of near-term intermediate-scale quantum (NISQ) systems has brought forth a heightened focus on the issue of circuit reliability. Several quantum computing activities, such as circuit design and multi-qubit mapping,…
This letter presents a method to reduce the computational demands of including second-order dynamics sensitivity information into the Differential Dynamic Programming (DDP) trajectory optimization algorithm. An approach to DDP is developed…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of…
We introduce a new overlapping Domain Decomposition Method (DDM) to solve the fully nonlinear Monge-Amp\`ere equation. While DDMs have been extensively studied for linear problems, their application to fully nonlinear partial differential…
The paper considers the numerical solution of nonlinear integral equations using the Newton-Kantorovich method with the mpmath library. High-precision quadrature of the kernel K(t, s, u) with respect to the variable s for fixed t increases…
The objective of this study is to present a novel, efficient, and fast direct method for solving linear systems of equations whose coefficient matrix is a tridiagonal Quasi-Toeplitz matrix. Such matrices are frequently encountered in the…
Differential transformation (DT) method has shown to be promising for power system simulation in our recent works. This letter applies the DT method to nonlinear power flow equations and proves that the nonlinear power flow equations are…