There exist two types of commonly studied stochastic elliptic models in literature: (I) � r � (a(x,x)r u(x,x)) = f(x) and (II) � r �ð aðx; xÞ}ruðx;xÞÞ ¼ f ðxÞ, where x indicates randomness, } the Wick product, and a(x,x) is a positive random process. Model (I) is widely used in engineering and physical applications while model (II) is usually studied from the mathematical point of view. The difference between the above two stochastic elliptic models has not been fully clarified. In this work, we discuss the difference between models (I) and (II) when a(x,x) is a log-normal random process. We show that the difference between models (I) and (II) is mainly characterized by a scaling factor, which is an exponential function of the degree of perturbation of a(x,x). We then construct a new stochastic elliptic model (III): � r �ð ða

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Comput. Methods Appl. Mech. Engrg.

This paper presents a new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes. The algorithm—particularly strong for low-dimensional sets embedded in high dimensions—runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates.

/pdf/coreduction-homology-algorithm-dycpr0946a.pdf

Coreduction Homology Algorithm

We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such set-valued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.

/pdf/discrete-morse-theoretic-algorithms-for-computing-homology-29iorl2rbq.pdf

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

We develop a method for measuring homology classes. This involves two problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(@bn^3log^2n) time, where n is the size of the simplicial complex and @b is the Betti number of the homology group. Finally, we prove the stability of our result. The algorithm can be adapted to measure any given class.

Measuring and computing natural generators for homology groups

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.

Homology algorithm based on acyclic subspace

A new algorithm for computing the homology module of a finitely generated chain complex is given. It is based on local one-step reductions of the size of the initial chain complex and it has a clear geometrical interpretation. The complexity of the algorithm is discussed in special cases.

M. Ślusarek

Papers

Homology computation by reduction of chain complexes