Discrete Mathematical Analysis: theory and geophysical applications. DMA definition. Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based on modeling of limit in a finite situation, realized in a series of algorithms.

Download Presentation

Discrete Mathematical Analysis: theory and geophysical applications

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

DMA definition Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based on modeling of limit in a finite situation, realized in a series of algorithms. The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas and judgments. That was reason why they became technical foundation of DMA.

CRYSTAL description • CRYSTAL goal: to identify -dense subsets against a general background • Definition 1. Subset AX is dense against the background X if • Definition 2. Subset AX is -dense against the background X if

CRYSTAL description “Growth I” block: • – the current version of i-th crystal Ki • for • , =1 (1,n) 1 “Struggle” between and in : • If then has “won”, and we proceed to the “Foundation” block • If then has “won”, but Kin+1={Kni, x} should remain dense against the background X in all its points (the “Growth II” block)

CRYSTAL description “Growth II” block (1): • Statement. If the “Growth I” condition is fulfilled then Kn+1={Kn, xn+1} will be -dense only if the “Growth II” condition is fulfilled: • =1 • If there is no such xn+1, we proceed to the “Foundation” block for choosing another foundation for the next crystallization • If there are several such xn+1 ( ), then

CRYSTAL description “Growth II” block (2): • Recalculation of the densities and in for their further “struggle” in the “Growth I” block:

CRYSTAL description “End” block: • Identification of the final crystal versions K1, …, KI, I =I(F,).

RODIN overview • The cluster definition: cluster in X, if A – cluster in X, if

RODIN overview Let A be a cluster in X and xAX: • The cluster quality: • Measure of separability of x in A: where The cluster separability:

RODIN overview Block-scheme of the Global RODIN: – given level of quality, – given level of separability (clusterness), K0 – initial version of cluster, Kn – current version of cluster

Algorithm “Monolith” X – multi-dimensional massif, xX, A – subset inX monAx – monolithnessAinx measure of limitnessAinx MonA X– subsep points inX with large A-monolithness (A-foundations) CrossingAA (1) =MonA X – “multi-dimensional topological” smoothingA in X Algorithm“Monolith”: parameters radius of monolithnessr weight of monolithness parameter of choosing number of iterationsi Algorithm“Monolith”: block-scheme