scholarly journals Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 855
Author(s):  
Vera Angelova ◽  
Mustapha Hached ◽  
Khalide Jbilou
Keyword(s):  
Riccati Equation ◽  
Differential System ◽  
Large Scale ◽  
Integral Form ◽  
Differential Riccati Equation ◽  
Fixed Point Principle ◽  
The Matrix ◽  
Linear Differential ◽  
Low Dimensional ◽  
Differential Matrix

Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative.

A numerical method of solving second-order linear differential equations with two-point boundary conditions

1957 ◽  
Vol 53 (2) ◽  
pp. 442-447 ◽  
Author(s):  
E. Cicely Ridley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Riccati Equation ◽  
Direct Method ◽  
Linear Equations ◽  
Factorization Method ◽  
Order Linear ◽  
First Order ◽  
The Matrix ◽  
Linear Differential

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

scholarly journals Carbon Nanotubes and Polydopamine Modified Poly(dimethylsiloxane) Sponges for Efficient Oil–Water Separation

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2431
Author(s):  
Wen Zhang ◽  
Juanjuan Wang ◽  
Xue Han ◽  
Lele Li ◽  
Enping Liu ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Large Scale ◽  
Absorption Capacity ◽  
Potential Candidate ◽  
Hard Template ◽  
Water Separation ◽  
Industrial Oil ◽  
The Matrix ◽  
Oil Water Separation ◽  
Oil Water

In this paper, effective separation of oil from both immiscible oil–water mixtures and oil-in-water (O/W) emulsions are achieved by using poly(dimethylsiloxane)-based (PDMS-based) composite sponges. A modified hard template method using citric acid monohydrate as the hard template and dissolving it in ethanol is proposed to prepare PDMS sponge composited with carbon nanotubes (CNTs) both in the matrix and the surface. The introduction of CNTs endows the composite sponge with enhanced comprehensive properties including hydrophobicity, absorption capacity, and mechanical strength than the pure PDMS. We demonstrate the successful application of CNT-PDMS composite in efficient removal of oil from immiscible oil–water mixtures within not only a bath absorption, but also continuous separation for both static and turbulent flow conditions. This notable characteristic of the CNT-PDMS sponge enables it as a potential candidate for large-scale industrial oil–water separation. Furthermore, a polydopamine (PDA) modified CNT-PDMS is developed here, which firstly realizes the separation of O/W emulsion without continuous squeezing of the sponge. The combined superhydrophilic and superoleophilic property of PDA/CNT-PDMS is assumed to be critical in the spontaneously demulsification process.

scholarly journals Experimental Research on Bond Behaviour of Fabric Reinforced Cementitious Matrix Composites for Retrofitting Masonry Walls

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Fayu Wang ◽  
Nicholas Kyriakides ◽  
Christis Chrysostomou ◽  
Eleftherios Eleftheriou ◽  
Renos Votsis ◽  
...  
Keyword(s):  
Large Scale ◽  
Tensile Tests ◽  
Seismic Resistance ◽  
Matrix Composites ◽  
Bond Behaviour ◽  
Cementitious Matrix ◽  
The Matrix ◽  
Rc Frame Buildings ◽  
Direct Tensile ◽  
Fabric Reinforced Cementitious Matrix

AbstractFabric reinforced cementitious matrix (FRCM) composites, also known as textile reinforced mortars (TRM), an inorganic matrix constituting fibre fabrics and cement-based mortar, are becoming a widely used composite material in Europe for upgrading the seismic resistance of existing reinforced concrete (RC) frame buildings. One way of providing seismic resistance upgrading is through the application of the proposed FRCM system on existing masonry infill walls to increase their stiffness and integrity. To examine the effectiveness of this application, the bond characteristics achieved between (a) the matrix and the masonry substrate and (b) the fabric and the matrix need to be determined. A series of experiments including 23 material performance tests, 15 direct tensile tests of dry fabric and composites, and 30 shear bond tests between the matrix and brick masonry, were carried out to investigate the fabric-to-matrix and matrix-to-substrate bond behaviour. In addition, different arrangements of extruded polystyrene (XPS) plates were applied to the FRCM to test the shear bond capacity of this insulation system when used on a large-scale wall.

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