Iterative Solution Of Nonlinear Equations Research Articles

A New Approach for Estimating the Free Energy Differences among Multiple Thermodynamic States in Statistical Simulations.

In this letter, a new approach to compute free energy differences (FEDs) between multiple thermodynamics states is introduced. The method directly uses energy probability densities, which can be extracted with high accuracy from equilibrium simulations to obtain FEDs. Methods in current use, such as Bennett acceptance ratio (BAR), its multistate generalization (MBAR), or the weighted histogram analysis method (WHAM), require iterative solution of nonlinear equations which are known to be slowly convergent. The equations providing MBAR FEDs are identical to those derived earlier by Souaille and Roux in a method that has become known informally as "binless WHAM". In contrast, we obtain FEDs by solution of linear equations. For the classic two-state problem, the statistical error of our method, solving linear equations, is shown analytically to match that of BAR under common conditions.

The hydrodynamics of low-density particles and optimization of the EMMS calculation process

Different reaction technology determines different catalytic materials and solid density affects the hydrodynamics behavior in a fluidized bed. An efficient hydrodynamics study is, therefore, highly desirable, even for evaluating the assumptions through pilot-scale investigation. To develop new fluidized bed technology that requires low-density particles (e.g., the vapor-phase Beckman rearrangement to produce ε-caprolactam), the authors thoroughly studied the hydrodynamics of low-density particles (true density 750 kg/m3, packing density 450 kg/m3) in a turbulent fluidized bed. Pilot-scale cold experiments were conducted in an ∅800 mm column with optical fiber probes. The TFM was adopted, where the Wen&Yu model, EMMS model, and six filtered models were coupled, respectively. The solution of the EMMS model is often challenging because of the iterative solution of nonlinear equations. In this work, the calculation process of the EMMS model was optimized, saving one global search for every possible state solution, and a set of nonlinear equations can be linearized. Comparisons between experiment and simulation results revealed that the EMMS models gave the best prediction. This work can provide a research model and guidance for the design of industrial fluidized bed reactors using low-density particles.

Importance truncation in non-perturbative many-body techniques

Expansion many-body methods correspond to solving complex tensor networks. The (iterative) solving of the network and the (repeated) storage of the unknown tensors requires a computing power growing polynomially with the size of basis of the one-body Hilbert space one is working with. Thanks to current computer capabilities, ab initio calculations of nuclei up to mass $A\sim100$ delivering a few percent accuracy are routinely feasible today. However, the runtime and memory costs become quickly prohibitive as one attempts (possibly at the same time) (i) to reach out to heavier nuclei, (ii) to employ symmetry-breaking reference states to access open-shell nuclei and (iii) to aim for yet a greater accuracy. The challenge is particularly exacerbated for non-perturbative methods involving the repeated storage of (high-rank) tensors obtained via iterative solutions of non-linear equations. The present work addresses the formal and numerical implementations of so-called importance truncation (IT) techniques within the frame of one particular non-perturbative expansion method, i.e., Gorkov Self-Consistent Green's Function (GSCGF) theory, with the goal to eventually overcome above-mentioned limitations. By a priori truncating irrelevant tensor entries, IT techniques are shown to reduce the storage to less than 1% of its original cost in realistic GSCGF calculations performed at the ADC(2) level while maintaining a 1% accuracy on the correlation energy. The future steps will be to extend the present development to the next, e.g., ADC(3), truncation level and to SCGF calculations applicable to doubly open-shell nuclei.

Energy Contraction and Optimal Convergence of Adaptive Iterative Linearized Finite Element Methods

Abstract We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 2020, 326, 2707–2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 2020, Paper No. 24] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the overall computational costs, preprint 2020]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.

WE‐D‐BRF‐05: Quantitative Dual‐Energy CT Imaging for Proton Stopping Power Computation

Purpose:To extend the two‐parameter separable basis‐vector model (BVM) to estimation of proton stopping power from dual‐energy CT (DECT) imaging.Methods:BVM assumes that the photon cross sections of any unknown material can be represented as a linear combination of the corresponding quantities for two bracketing basis materials. We show that both the electron density (ρe) and mean excitation energy (Iex) can be modeled by BVM, enabling stopping power to be estimated from the Bethe‐Bloch equation. We have implemented an idealized post‐processing dual energy imaging (pDECT) simulation consisting of monogenetic 45 keV and 80 keV scanning beams with polystyrene‐water and water‐CaCl2 solution basis pairs for soft tissues and bony tissues, respectively. The coefficients of 24 standard ICRU tissue compositions were estimated by pDECT. The corresponding ρe, Iex, and stopping power tables were evaluated via BVM and compared to tabulated ICRU 44 reference values.Results:BVM‐based pDECT was found to estimate ρe and Iex with average and maximum errors of 0.5% and 2%, respectively, for the 24 tissues. Proton stopping power values at 175 MeV, show average/maximum errors of 0.8%/1.4%. For adipose, muscle and bone, these errors result range prediction accuracies less than 1%.Conclusion:A new two‐parameter separable DECT model (BVM) for estimating proton stopping power was developed. Compared to competing parametric fit DECT models, BVM has the comparable prediction accuracy without necessitating iterative solution of nonlinear equations or a sample‐dependent empirical relationship between effective atomic number and Iex. Based on the proton BVM, an efficient iterative statistical DECT reconstruction model is under development.

Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator

In this paper, Φ-pseudo-contractive operators and Φ-accretive operators, more general than the strongly pseudo-contractive operators and strongly accretive operators, are introduced. By setting up a new inequality, authors proved that if is a uniformly continuous Φ-pseudo-contractive operator then T has unique fixed point q and the Mann iterative sequence with random errors approximates to q. As an application, the iterative solution of nonlinear equation with Φ-accretive operator is obtained. The results presented in this paper improve and generalize some corresponding results in recent literature.

Approximate fixed point of uniform hemicontractive mapping and to apply to iterative solution of equation with uniform-accretive mapping

The objective of this paper is to introduce the uniform-hemicontractive mapping and to study iterative approximative method for the fixed point of the mappings by Mann iterative sequence with random errors. Let X be a real Banach space and T : X → X an uniformhemicontractive the results presented in this paper show that Mann iterative sequence with random errors converges strongly to an unique fixed point if T is uniformly continuous. Furthermore, if X is uniformly smooth then any continuity of T is unnecessary for the convergence of Mann iterative sequence. As applications, using these results, the iterative solution of nonlinear equation with uniform-accretive mapping is obtained. Mathematics Subject Classification: 47H17, 47H06, 47H10

SGBEM for Cohesive Cracks in Homogeneous Media

In this paper, the Symmetric Galerkin Boundary Element Method for Linear Elastic Fracture Mechanics is extended to non-linear cohesive cracks propagating through homogeneous linear elastic isotropic media. The cohesive model adopted is based on the concept of free energy density per unit undeformed area. The corresponding constitutive cohesive equations present a softening branch which induces a potential instability. Thus, a suitable solution algorithm capable of following the growth of the cohesive zone is needed, and in the present work the numerical simulation is controlled by an arc-length method combined with a Newton-Raphson algorithm for the iterative solution of nonlinear equations. The Boundary ElementMethod is very attractive for modeling cohesive crack problems as all nonlinearities are located on the boundaries of linear elastic domains. Moreover a Galerkin approximation scheme, applied to a suitable symmetric boundary integral equation formulation, ensures an easy and efficient treatment of cracks in homogeneous media and an excellent convergence behavior of the numerical solution. The cohesive zone model is applied to simulate a pure mode I crack propagation in concrete. Numerical results for three-point bending test are used to check the numerical results for mode I and are compared with some numerical results obtained by FEM analysis found in the literature.

On iterative solution of non-linear equation

In solving non – linear equations by iterative method, Ohirhian (1994), (2005) [4],[5] developed a new algorithm based on cubic interpolation for solving non- linear equations of degree 1 to 3. The algorithm was found to be faster than the Regular falsi and the Newton Raphason method. This paper extend the algorithm to solving non linear equations of degree n by deriving a general formular, also some of the iteration procedures are reviewed for ease in computation. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 611-615

On Fixed Point of F-Pseudo-contractive operator in an arbitrary Banach space and an application

The purpose of this paper is to introduce Φ-pseudo-contractive operators, to study the problems of existence, uniqueness and convergence of fixed point, and to set up a new inequality in arbitrary Banach space. As an intimately application, the iterative solution of nonlinear equation with F -accretive operator is obtained. The results presented in this paper improve, generalize and unify some corresponding results in recent literature.

Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A :D(A)=E→2 E be an accretive operator that satisfies the range condition and A −1(0)≠∅. Let { λ n } and { θ n } be two real sequences satisfying appropriate conditions, and for z∈ E arbitrary, let the sequence { x n } be generated from arbitrary x 0∈ E by x n+1 = x n − λ n ( u n + θ n ( x n − z)), u n ∈ Ax n , n⩾0. Assume that { u n } is bounded. It is proved that { x n } converges strongly to some x ∗∈A −1(0) . Furthermore, if K is a nonempty closed convex subset of E and T :K→K is a bounded continuous pseudocontractive map with F( T):={ Tx= x}≠∅, it is proved that for arbitrary z∈ K, the sequence { x n } generated from x 0∈ K by x n+1 = x n − λ n (( I− T) x n + θ n ( x n − z)), n⩾0, where { λ n } and { θ n } are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.

Iterative solutions of nonlinear equations with φ-strongly accretive operators in uniformly smooth Banach spaces

Suppose that X is a uniformly smooth Banach space and T : X → X is a demicontinuous (not necessarily Lipschitz) φ-strongly accretive operator. It is proved that the Ishikawa iterative method with errors converges strongly to the solutions of the equations f = Tx and f = x+ Tx, respectively. A related result deals with the approximation of fixed points of φ-strongly pseudocontractive operators. Our results extend, improve, and unify the recent results obtained by Chidume [1,2] and Zhou [3].

The solution by iteration of nonlinear equations of Hammerstein type

Let E be a real uniformly smooth Banach space. Suppose N is a set-valued (locally) accretive map with open domain D(N) C E , K is a single-valued map with domain D(K) C E such that Im(N) C D(K) and K~ exists and is (locally) strongly accretive on Im(K) . If the equation f e x + KNx has a solution x* E D(N) , the strong convergence of iteration methods of the Ishikawa and Mann types to this solution is established. Related theorems deal with the iterative solution of nonlinear equations of the form f G x + Tx for the case where T is accretive. Explicit convergence rates are established. MIRAMARE TRIESTE November 1996 Permanent address: Department of Mathematics and Computer Science, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria.

Iterative solution of nonlinear equations with $ \phi $-strongly accretive operators

Suppose that X is an arbitrary real Banach space and T : X→X is a Lipschitzian and \( \phi \)-strongly accretive operator. For any given \( f \in X \), define a mapping S : X→X by Sx=f+x−Tx for all \( x \in X \). Define the sequence \( \{x_n\}^{\infty}_{n=0} \) in X iteratively by¶¶\( \cases{x_0 \in X,\cr x_{n+1}=a_nx_n +b_n Sy_n +c_nu_n,\cr y_n = a'_nx_n +b'_nSx_n +{c'}_{\!\!n}v_n, \quad n \geqq 0,} \)¶¶ where \( \{u_n\}^{\infty}_{n=0} \), \( \{v_n\}^{\infty}_{n=0} \) are arbitrary bounded sequences in X and \( \{a_n\}^{\infty}_{n=0} \), \( \{b_n\}^{\infty}_{n=0} \), \( \{c_n\}^{\infty}_{n=0} \),¶\( \{a'_n\}^{\infty}_{n=0} \), \( \{b'_n\}^{\infty}_{n=0} \) and \( \{c'_n\}^{\infty}_{n=0} \) are real sequences in [0,1] satisfying the following conditions:¶¶(i) \( a_n+b_n+c_n=a'_n+b'_n+{c'}_{\!\!n}=1$, $n \geqq 0, \)¶¶ (ii) \( \sum\limits_{n=0}^\infty c_n \) < \( \infty \), \( \sum\limits_{n=0}^\infty b_n b'_n \) < \( \infty \),¶\( \sum\limits_{n=0}^\infty b_n {c'}_{\!\!n} \) < \( \infty \), \( \sum\limits_{n=0}^\infty b_n^2 \) < \(\infty\),¶¶ (iii) \( \sum\limits_{n=0}^\infty b_n =+\infty \).¶¶ Then the sequence \( \{x_n\}^{\infty}_{n=0} \) converges strongly to the unique solution of the equation Tx=f.

Iterative solution of nonlinear equations involving set-valued uniformly accretive operators

Let E be a real normed linear space and let A : E ↦ 2 E be a uniformly continuous and uniformly quasi-accretive multivalued map with nonempty closed values such that the range of ( I – A) is bounded and the inclusion f ϵ Ax has a solution x* ϵ E. It is proved that Ishikawa and Mann type iteration processes converge strongly to x*. Further, if T : E ↦ 2 E is a uniformly continuous and uniformly hemicontractive set-valued map with bounded range and a fixed point x* ϵ E, it is proved that both the Mann and Ishikawa type iteration processes converge strongly to x*. The strong convergence of these iteration processes with errors is also proved.

Iterative solutions of nonlinear equations of the accretive type

Iterative solutions of nonlinear equations of the accretive type

Computation of bifurcation boundaries for power systems: a new Δ-plane method

This paper is devoted to the problems of finding the load flow feasibility saddle node, and Hopf bifurcation boundaries in the space of power system parameters. The first part contains a review of the existing relevant approaches including not-so-well-known contributions from Russia. The second part presents a new robust method for finding the power system load flow feasibility boundary on the plane defined by any three vectors of dependent variables (nodal voltages), called the /spl Delta/ plane. The method exploits some quadratic and linear properties of the load flow equations and state matrices written in rectangular coordinates. An advantage of the method is that it does not require an iterative solution of nonlinear equations (except the eigenvalue problem). In addition to benefits for visualization, the method is a useful tool for topological studies of power system multiple solution structures and stability domains. Although the power system application is developed, the method can be equally efficient for any quadratic algebraic problem.

Iterative solution of nonlinear equations with strongly accretive operators in Banach spaces

Let X be a real Banach space with a uniformly convex dual X*. Let T: X→X be a Lipschitzian and strongly accretive mapping with a Lipschitzian constant L≥1 and a strongly accretive constant k∈(0,1). Let {±n} and {²n} be two real sequences in [0,1] statisfying: $$\alpha _n \to 0 as n \to \infty ;$$ ( i ) $$\beta _n< \frac{{k(1 - k)}}{{L(1 + L)}},for all n \geqslant 0;$$ ( ii ) $$\prod\limits_{n = 0}^\infty {\alpha _n = \infty .} $$ ( iii ) Set Sx = f-Tx+x, ∀x∈X. Assume that {un}n=0∞ and {vn}n=0∞ be two sequences in X satisfying ∥un∥ For arbitrary x0∈X, the iteration sequence {xn} is defined by $$(IS)_1 \left\{ \begin{gathered} x_{n + 1} = (1 - a_n )x_n + a_n Sy_n + u_n , \hfill \\ y_n = (1 - \beta _n )x_n + \beta _n Sx_n + v_n (n \geqslant 0), \hfill \\ \end{gathered} \right.$$ then {xn} converges strongly to the unique solution of the equation Tx=f.

Convergence Theorems for Strongly Pseudo-contractive and Strongly Accretive Maps

SupposeEis an arbitrary real Banach space andKis a nonempty closed convex and bounded subset ofE. SupposeT:K→Kis a uniformly continuous strong pseudo-contraction with constantk∈(0,1). Suppose {an}, {bn}, {cn}, {a′n}, {b′n}, and {c′n} are sequences in (0,1) satisfying the following conditions: (i)an+bn+cn=1=a′n+b′n+c′n∀integersn≥0; (ii) limbn=limb′n=limc′n=0; (iii) Σbn=∞; (iv) Σcn<∞. For arbitraryx0,u0,v0∈K, define the sequence {xn}∞n=0iteratively byxn+1=anxn+bnTyn+cnun;yn=a′nxn+b′nTxn+c′nvn,n≥0, where {un}, {vn} are arbitrary sequences inK. Then {xn} converges strongly to the unique fixed point ofT. Related results deal with the iterative solutions of nonlinear equations involving set-valued, strongly accretive operators.

Iterative Solutions of Nonlinear Equations of the Strongly Accretive Type

Let E be a real q-uniformly smooth Banach space. Suppose T is a strongly pseudo-contractive map with open domain D(T) in E. Suppose further that T has a fixed point in D(T). Under various continuity assumptions on T it is proved that each of the Mann iteration process or the Ishikawa iteration method converges strongly to the unique fixed point of T. Related results deal with iterative solutions of nonlinear operator equations involving strongly accretive maps. Explicit error estimates are also provided.