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Recently soft robotics has rapidly become a novel and promising area of research with many designs and applications due to their flexible and compliant structure. However, it is more difficult to derive the nonlinear dynamic model of such…
A simple yet effective numerical method using orthogonal hybrid functions consisting of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal triangular functions is proposed to solve numerically fractional…
We present general series solutions to the Tolman-Oppenheimer-Volkoff equations for compact stellar objects. We develop an algorithm to compute the coefficients of the power series in terms of the equation of state and its derivatives with…
Let $A$ be a real $n\times n$ matrix and $z,b\in \mathbb R^n$. The piecewise linear equation system $z-A\vert z\vert = b$ is called an \textit{absolute value equation}. We consider two solvers for this problem, one direct, one…
This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial…
We adapt the alternating linearization method for proximal decomposition to structured regularization problems, in particular, to the generalized lasso problems. The method is related to two well-known operator splitting methods, the…
Neural networks can approximate solutions to partial differential equations, but they often break the very laws they are meant to model-creating mass from nowhere, drifting shocks, or violating conservation and entropy. We address this by…
It is shown how piecewise differentiable functions $F: \mathbb R^n \mapsto \mathbb R^m $ that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points $\check x$ and $\hat…
We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the…
Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton…
Spectral parameter power series (SPPS) representations for solutions of Sturm-Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized Poisson Boltzmann equation, for molecules represented as non-overlapping spherical cavities.…
Many problems in machine learning can be solved by rounding the solution of an appropriate linear program (LP). This paper shows that we can recover solutions of comparable quality by rounding an approximate LP solution instead of the ex-…
Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
Optimization problems over discrete or quantized variables are very challenging in general due to the combinatorial nature of their search space. Piecewise-affine regularization (PAR) provides a flexible modeling and computational framework…
Numerous physics theories are rooted in partial differential equations (PDEs). However, the increasingly intricate physics equations, especially those that lack analytic solutions or closed forms, have impeded the further development of…
In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of…
A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution…