System Of Linear ODEs Research Articles

The basic set of test problems for ODE system solvers

Known mathematical packages, MATLAB, Maple, Mathematica, MathCAD and others may get wrong, often plausible, the result of numerical solution of ODE systems with low, given by default, the requirements for mathematical accuracy of the results of the numerical solution of ODE systems. Since ODE system parameters obtained usually experimental, with a low mathematical precision, so the requirements for the precision of the results of mathematical solutions of ODE systems is low (for example, in MATLAB package required precision is 0.001). The article offers a basic set of tests to assess the range of applicability of the relevant solvers. The basic set of test problems for ODE solvers systems include linear ODE systems with the known analytic solution and nonlinear systems with known graphics solution. Presented comparative results of solutions for proposed problems using MATLAB solvers and manzhuk program from a library of standard mathematical programs SADEL (Sets of Algebraic and Differential Equations solvers Library), which has been designed for reliable and accurate solving of systems of linear algebraic equations (LAE) and ODE systems. The results can be used in mathematical modeling of dynamic systems described by ODE systems.

A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equations

Approximate analytical solution of a fractional Black–Scholes pricing model is much relevant due to its practical importance to financial markets. In this article, a powerful approximate iterative mathematical scheme based on residual power series (RPS) algorithm is presented to achieve the numerical results of the nonlinear time fractional Black–Scholes (BS) equations based on European options. The Caputo-type fractional derivatives are considered in the present article. The residual power series method (RPSM) developed by Arqub (2013) is the novel technique for finding the analytical Taylor series solutions of systems of linear and nonlinear ODEs and PDEs. The residual power series technique supplies the approximate analytical solutions of the problem in truncated series form using residual error concept along with given initial conditions. The numerical procedures reveal that only a few iterations are sufficient for better approximations of the solutions, which clearly exhibits the reliability and effectiveness of this iterative scheme. This article shows that the adopted scheme is quite systematic as well as computationally attractive regarding solution procedure. In addition, effects of fractional order of time derivatives on the solutions for various particular cases are also depicted through graphs.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

Optimal Dynamic Futures Portfolio in a Regime-Switching Market Framework

We study the problem of dynamically trading futures in a regime-switching market. Modeling the underlying asset price as a Markov-modulated diffusion process, we present a utility maximization approach to determine the optimal futures trading strategy. This leads to the analysis of the associated system of Hamilton-Jacobi-Bellman (HJB) equations, which are reduced to a system of linear ODEs. We apply our stochastic framework to two models, namely, the Regime-Switching Geometric Brownian Motion (RS-GBM) model and Regime-Switching Exponential Ornstein-Uhlenbeck (RS-XOU) model. Numerical examples are provided to illustrate the investor's optimal futures positions and portfolio value across market regimes.

Acceleration of the Two-Level MGRIT Algorithm via the Diagonalization Technique

The multigrid-reduction-in-time (MGRIT) algorithm is an efficient parallel-in-time algorithm for solving dynamic problems. The goal of this paper is to accelerate this algorithm via two strategies. The first strategy is to improve the convergence rate by using the 2nd-order Lobatto IIIC (LIIIC-2) method as the $\mathcal{G}$-propagator, instead of using the backward-Euler method which is the common choice in this field. For a system of linear ODEs with a symmetric positive definite (SPD) coefficient matrix, we show that such a choice reduces the convergence factor of the MGRIT algorithm from 0.1 to 0.02. We prove a robust convergence factor for the MGRIT algorithm, which is independent of the eigenvalues of the coefficient matrix and the ratio $J=\Delta T/\Delta t$. The second strategy is to make the coarse-grid-correction (CGC) parallel by using the diagonalization technique. By properly choosing the involved parameter, we show that the new MGRIT algorithm has the same convergence rate as that of the original algorithm. Moreover, we show that within the framework of the parallel CGC the cost of the LIIIC-2 method, which is an implicit two-stage Runge--Kutta method, can be reduced to the same cost of the backward-Euler method. The idea toward this goal is still a suitable application of the diagonalization technique. Numerical experiments for the advection-diffusion equations with uncertain coefficients and the Gray--Scott model are given to support our findings.

A waveform relaxation Newmark method for structural dynamics problems

In the conventional Newmark family for time integration of hyperbolic problems, both explicit and implicit methods are inherently sequential in the time domain and not well suited for parallel implementations due to unavoidable processor communication at every time step. In this work we propose a Waveform Relaxation Newmark (WRN $$_\beta $$ ) algorithm for the solution of linear second-order hyperbolic systems of ODEs in time, which retains the unconditional stability of the implicit Newmark scheme with the advantage of the lower computational cost of explicit time integration schemes. This method is unstructured in the time domain and is well suited for parallel implementation. We consider a Jacobi and Gauss–Seidel type splitting and study their convergence and stability. The performance of the WRN $$_\beta $$ algorithm is compared to a standard implicit Newmark method and the obtained results confirm the effectiveness of the Waveform Relaxation Newmark algorithm as a new class of more efficient time integrators, which is applicable, as shown in the numerical examples, to both the finite element method and meshfree methods (e.g. the reproducing kernel particle method).

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.

High-order, linearly stable, partitioned solvers for general multiphysics problems based on implicit–explicit Runge–Kutta schemes

This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit–explicit Runge–Kutta (IMEX-RK) discretization of the semi-discrete equations. The generic multiphysics problem is modeled as a system of n systems of partial differential equations where the ith subsystem is coupled to the other subsystems through a coupling term that can depend on the state of all the other subsystems. This coupled system of partial differential equations reduces to a coupled system of ordinary differential equations via the method of lines where an appropriate spatial discretization is applied to each subsystem. The coupled system of ordinary differential equations is taken as a monolithic system and discretized using an IMEX-RK discretization with a specific implicit–explicit decomposition that introduces the concept of a predictor for the coupling term. We propose four coupling predictors that enable the monolithic system to be solved in a partitioned manner, i.e., subsystem-by-subsystem, and preserve the IMEX-RK structure and therefore the design order of accuracy of the monolithic scheme. The four partitioned solvers that result from these predictors are high-order accurate, allow for maximum re-use of existing single-physics software, and two of the four solvers allow the subsystems to be solved in parallel at a given stage and time step. We also analyze the stability of a coupled, linear model problem with a specific coupling structure and show that one of the partitioned solvers achieves unconditional linear stability for this problem, while the others are unconditionally stable only for certain values of the coupling strength. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems including a simple linear system of ODEs, advection–diffusion–reaction systems, fluid–structure interaction problems, and particle-laden flows, where we verify the design order of the IMEX schemes and study various stability properties.

Asymptotic expansion for forward-backward SDEs with jumps

ABSTRACTThis work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.

Parameter Estimation and Variable Selection for Big Systems of Linear Ordinary Differential Equations: A Matrix-Based Approach

ABSTRACTOrdinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a “Big System” with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector-based two-stage method that are currently available. We applied this new method to two real datasets—a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters—to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice. Supplementary materials for this article are available online.

Minimal time splines on the sphere

An interesting problem in geometric control theory arises from robotics and space science: find a smooth curve, controlled by a bounded acceleration, connecting in minimum time two prescribed tangent vectors of a Riemannian manifold Q. The state equation is $$ \nabla _{\dot{\gamma }} \dot{\gamma } = u \in TQ, |u| \le A$$ . Applying Pontryagin’s principle one gets a Hamiltonian system in $$T^*(TQ)$$ . We consider this problem in $$S^2(r)$$ . Seemingly, it has not been addressed before. Via the SO(3) symmetry, we reduce the four degrees of freedom system to the five variables $$(a,v, M_1,M_2,M_3),$$ where v is the scalar velocity, conjugated to a costate variable a and $$(M_1,M_2,M_3)$$ are costate variables that satisfy $$\{ M_i, M_j\} = \epsilon _{ijk} M_k$$ . We derive the reduced equations and find special analytical solutions, that are organizing centers for the dynamics. Reconstruction of the curve $$\gamma (t)$$ is achieved by a time dependent linear system of ODEs for the orthogonal matrix R whose first column is the unit tangent vector of the curve and whose last column is the unit normal vector to the sphere.

A Parallel Iterative Algorithm for Differential Linear Complementarity Problems

We propose a parallel iterative algorithm for solving the differential linear complementarity problems consisting of two systems, a linear ODE system and a linear complementarity system (LCS). At each iteration we proceed in a system decoupling way: by using a rough approximation of the state variable obtained from the previous iteration, we solve the LCS; then we solve the ODE system and update the state variable for preparing for the next iteration, by using the obtained constraint variable as a known source term. The algorithm is highly parallelizable, because at each iteration the computations of both the LCS and the ODE system at all the time points of interest can start simultaneously. The parallelism for solving the LCS is natural and for the ODE system it is achieved by using the Laplace inversion technique. For the P-matrix LCS, we prove that the algorithm converges superlinearly with arbitrarily chosen initial iterate and for the Z-matrix LCS the algorithm still converges superlinearly if we use...

Characterization of canonical classes for systems of linear ODEs

The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical form y(n)=0 consists of copies of the same iterative scalar equation. It is also shown that contrary to the scalar case, an iterative vector equation need not be reducible to the canonical form by an invertible point transformation. Other properties of iterative linear systems are also derived, as well as a simple algebraic formula for their general solution. Copyright © 2017 John Wiley & Sons, Ltd.

Simple Lie Algebras and Topological ODEs

For a simple Lie algebra $\mathfrak g$ we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of $\mathfrak g$-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example $\mathfrak g=sl_2(\mathbb C)$ the regular solution can be expressed via products of Airy functions and their derivatives; this matrix valued function was used in our previous work for computing logarithmic derivatives of the Witten - Kontsevich tau-function. For an arbitrary simple Lie algebra we construct a basis in the space of regular solutions to the topological equation called generalized Airy resolvents. We also outline applications of the generalized Airy resolvents to computing the Witten and Fan - Jarvis - Ruan invariants of the Deligne - Mumford moduli spaces of stable algebraic curves.

Block Krylov subspace methods for approximating the linear combination of [formula omitted]-functions arising in exponential integrators

In this paper we present numerical methods for computing the matrix functions arising in exponential integrators. The matrix functions are the linear combination of the form ∑j=0pφj(A)bj, where φj is related to exponential function. By reducing the corresponding systems of linear ODEs with polynomial inhomogeneity based on block Krylov subspace methods, two effective exponentials of low dimension matrix are derived to approximate the objective functions. Some useful a posteriori error estimates are established. The algorithms can be combined with time-stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block Krylov subspace. Several numerical experiments are presented to demonstrate the efficiency of the methods.

Ward identities and relations between conductivities and viscosities in holography

We derive relations between viscosities and momentum conductivity in $2+1$ dimensions by finding a generalization of holographic Ward identities for the energy-momentum tensor. The generalization is novel in the sense that it goes beyond the usual identities obtained from holographic renormalization. Our results are consistent with previous field theory analysis. The main tools we use are a constant `probability current' in the gravity dual, that we are able to define for any system of linear ODEs, and parity symmetry.

On Stokes Matrices in terms of Connection Coefficients

The classical problem of computing a complete system of Stokes multipliers of a linear system of ODEs of rank one in terms of some connection coefficients of an associated hypergeometric system of ODEs, is solved with no genericness assumptions on the residue matrix at zero, by an extension of the method of [3].

Analytic enclosure of the fundamental matrix solution

This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the Cs category.

Differentiability of Palmer's linearization theorem and converse result for density functions

We study differentiability properties in a particular case of the Palmer's linearization theorem, which states the existence of a homeomorphism H between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that H is a C2 preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.

Optimal Position Management for a Market Maker with Stochastic Price Impacts

This paper deals with an optimal position management problem for a market maker who has to face uncertain customer order flows in an illiquid market, where the market maker’s continuous trading incurs a stochastic linear price impact. Although the execution timing is uncertain, the market maker can also ask its OTC counterparties to transact a block trade without causing a direct price impact. We adopt quite generic stochastic processes of the securities, order flows, price impacts, quadratic penalties as well as security borrowing/lending rates. The solution of the market maker’s optimal position-management strategy is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we make use of the connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties were studied by Bismut (1976). We also propose a perturbative approximation scheme for the resultant BSRDE, which only requires a system of linear ODEs to be solved at each expansion order. Its justification and the convergence rate are also given.