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A priori and a posteriori analysis of the quasinonlocal quasicontinuum method
"... Abstract. For a nextnearest neighbour pair interaction model in a periodic domain, a priori and a posteriori analyses of the quasinonlocal quasicontinuum method (QNLQC) are presented. The results are valid for large deformations and essentially guarantee a onetoone correspondence between atomi ..."
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Cited by 45 (28 self)
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Abstract. For a nextnearest neighbour pair interaction model in a periodic domain, a priori and a posteriori analyses of the quasinonlocal quasicontinuum method (QNLQC) are presented. The results are valid for large deformations and essentially guarantee a onetoone correspondence between atomistic solutions and QNLQC solutions. The analysis is based on consistency error estimates in negative norms, novel a priori and a posteriori stability estimates, and a quantitative inverse function theorem.
A Posteriori Existence in Numerical Computations
, 2008
"... A typical a posteriori error analysis for numerical methods for differential equations consists of two elements: the stability of the continuous problem and an a posteriori computed residual estimate. If the problem is nonlinear, then this is only possible if it is strictly monotone. In this paper, ..."
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Cited by 17 (12 self)
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A typical a posteriori error analysis for numerical methods for differential equations consists of two elements: the stability of the continuous problem and an a posteriori computed residual estimate. If the problem is nonlinear, then this is only possible if it is strictly monotone. In this paper, we will estimate the stability constant a posteriori as well, in order to apply the Inverse Function Theorem to derive the existence of solutions from the numerical computation, and thus obtain rigorous a posteriori error estimates for nonmonotone equations.
A priori error analysis of two forcebased atomistic/continuum models of a periodic chain
 Numer. Math. 119
, 2011
"... A priori error analysis of two forcebased atomistic/continuum models of a periodic chain ..."
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Cited by 11 (9 self)
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A priori error analysis of two forcebased atomistic/continuum models of a periodic chain
GLOBAL SMOOTH SOLUTION CURVES USING RIGOROUS BRANCH FOLLOWING
, 2010
"... In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f: R l1 ×B1 → R l2 ×B2, where B1 and B2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudoarclength continuation. Examples in th ..."
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Cited by 6 (4 self)
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In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f: R l1 ×B1 → R l2 ×B2, where B1 and B2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudoarclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.
PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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Cited by 2 (1 self)
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
APOSTERIORI VERIFICATION OF OPTIMALITY CONDITIONS FOR CONTROL PROBLEMS WITH FINITEDIMENSIONAL CONTROL SPACE ∗
"... Abstract. In this paper we investigate nonconvex optimal control problems. We are concerned with aposteriori verification of sufficient optimality conditions. If the proposed verification method confirms the fulfillment of the sufficient condition then aposteriori error estimates can be computed. ..."
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Abstract. In this paper we investigate nonconvex optimal control problems. We are concerned with aposteriori verification of sufficient optimality conditions. If the proposed verification method confirms the fulfillment of the sufficient condition then aposteriori error estimates can be computed. A special ingredient of our method is an error analysis for the Hessian of the underlying optimization problem. We derive conditions under which positive definiteness of the Hessian of the discrete problem implies positive definiteness of the Hessian of the continuous problem. The article is complemented with numerical experiments. Key words. nonconvex optimal control problems, sufficient optimality conditions, aposteriori error estimates AMS subject classifications. 49M25, 49K20, 65N15, 65N25
A Computer–assisted Proof of the Existence of Solutions to a Boundary Value Problem with an Integral Boundary Condition
, 2008
"... In this paper, we present a computer–aided method (based on [Ya98]) that establishes the existence and local uniqueness of a stationary solution to the viscous Burgers’ equation. The problem formulation involves a left boundary condition and one integral boundary condition, which is a variation of t ..."
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In this paper, we present a computer–aided method (based on [Ya98]) that establishes the existence and local uniqueness of a stationary solution to the viscous Burgers’ equation. The problem formulation involves a left boundary condition and one integral boundary condition, which is a variation of the approach taken in [Si04]. Key words: computerassisted proof, numerical verification, viscous Burgers ’ equation, enclosure, existence, twopoint boundary value problems, fixedpoint problems
WHITHER MATHEMATICS?
, 2004
"... whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1 ..."
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whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1
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"... How to check numerically the sufficient optimality conditions for infinitedimensional optimization problems ..."
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How to check numerically the sufficient optimality conditions for infinitedimensional optimization problems
Preprint no. 25/2009 A Priori and A Posteriori Analysis of the QuasiNonlocal Quasicontinuum Method in 1D
"... For a nextnearest neighbour pair interaction model in a periodic domain, a priori and a a posteriori analyses of the quasinonlocal quasicontinuum method (QNLQC) are presented. The results are valid for large deformations and essentially guarantee a onetoone correspondence between atomistic solut ..."
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For a nextnearest neighbour pair interaction model in a periodic domain, a priori and a a posteriori analyses of the quasinonlocal quasicontinuum method (QNLQC) are presented. The results are valid for large deformations and essentially guarantee a onetoone correspondence between atomistic solutions and QNLQC solutions. The analysis is based on truncation error and residual estimates in negative norms and novel a priori and a posteriori stability estimates. Key words and phrases: quasicontinuum method, quasinonlocal coupling, finite deformations, sharp stability, error analysis