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Metric Entropy and Nonlinear Partial Differential Equations.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Metric Entropy and Nonlinear Partial Differential Equations./
作者:	Dutta, Prerona.
面頁冊數:	1 online resource (98 pages)
附註:	Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
Contained By:	Dissertations Abstracts International83-02B.
標題:	Mathematical functions. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28552737click for full text (PQDT)
ISBN:	9798522944896

Metric Entropy and Nonlinear Partial Differential Equations.
Dutta, Prerona.

Metric Entropy and Nonlinear Partial Differential Equations. - 1 online resource (98 pages)

Source: Dissertations Abstracts International, Volume: 83-02, Section: B.

Thesis (Ph.D.)--North Carolina State University, 2021.

Includes bibliographical references

The metric entropy (or "-entropy) has been studied extensively in a variety of literature and disciplines. This notion was introduced by Kolmogorov and Tikhomirov in 1959 as the minimum number of bits needed to represent a point in a given subset K of a metric space (E,ρ), up to an accuracy " with respect to the metric ρ. Recently, the "-entropy has also been used to measure the set of solutions of various nonlinear partial differential equations (PDEs). In this context, it provides a measure of the order of "resolution" and the "complexity" of a numerical scheme.This dissertation is motivated by the above points of view and demonstrates techniques emerging from the field of nonlinear analysis, that are used to estimate the metric entropy for different classes of bounded variation (BV) functions. Since Helly's theorem states that a set of uniformly bounded total variation functions is compact in L 1 -space, a natural question arises on how to quantify the degree of compactness and we answer this using the concept of "-entropy. Subsequently, we apply the obtained results to measure solution sets of conservation laws and Hamilton-Jacobi equationsIn the first half of this thesis, we elucidate the foundational principles involved in our work and show that the minimal number of functions needed to represent a bounded total variation function in L 1 ([0,L] d,R) up to an error " with respect to L 1 -distance, is of the order " −d. We use this outcome to examine the metric entropy for sets of viscosity solutions to the Hamilton-Jacobi equation. Earlier works on this topic considered a uniformly convex Hamiltonian. We extend the analysis to the case when the Hamiltonian H ∈ C 1 (Rd) is strictly convex, coercive and a uniformly directionally convex function. Under these assumptions, we establish sharp estimates on the "-entropy for sets of viscosity solutions to the Hamilton-Jacobi equation with respect to W1,1 -distance in multi-dimensional casesThe second half of the thesis focuses on finding sharp estimates for the metric entropy of a class of bounded total generalized variation functions taking values in a general totally bounded metric space (E,ρ) upto an accuracy of " with respect to the L 1-distance. We rely on the ideas of covering and packing in (E,ρ) to derive such bounds and utilize them to study a scalar conservation law in one-dimensional space, which yields an upper bound on the "-entropy of a set of entropy admissible weak solutions to it, in case of weakly genuinely nonlinear fluxes, i.e., for fluxes with no affine parts. In particular, for fluxes admitting finitely many inflection points with a polynomial degeneracy, this estimate is sharp.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798522944896Subjects--Topical Terms:

3564295
Mathematical functions.
Index Terms--Genre/Form:

542853
Electronic books.

Metric Entropy and Nonlinear Partial Differential Equations.
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100 1 $a Dutta, Prerona. $3 3697742
245 1 0 $a Metric Entropy and Nonlinear Partial Differential Equations.
264 0 $c 2021
300 $a 1 online resource (98 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
500 $a Advisor: Nguyen, Tien Khai.
502 $a Thesis (Ph.D.)--North Carolina State University, 2021.
504 $a Includes bibliographical references
520 $a The metric entropy (or "-entropy) has been studied extensively in a variety of literature and disciplines. This notion was introduced by Kolmogorov and Tikhomirov in 1959 as the minimum number of bits needed to represent a point in a given subset K of a metric space (E,ρ), up to an accuracy " with respect to the metric ρ. Recently, the "-entropy has also been used to measure the set of solutions of various nonlinear partial differential equations (PDEs). In this context, it provides a measure of the order of "resolution" and the "complexity" of a numerical scheme.This dissertation is motivated by the above points of view and demonstrates techniques emerging from the field of nonlinear analysis, that are used to estimate the metric entropy for different classes of bounded variation (BV) functions. Since Helly's theorem states that a set of uniformly bounded total variation functions is compact in L 1 -space, a natural question arises on how to quantify the degree of compactness and we answer this using the concept of "-entropy. Subsequently, we apply the obtained results to measure solution sets of conservation laws and Hamilton-Jacobi equationsIn the first half of this thesis, we elucidate the foundational principles involved in our work and show that the minimal number of functions needed to represent a bounded total variation function in L 1 ([0,L] d,R) up to an error " with respect to L 1 -distance, is of the order " −d. We use this outcome to examine the metric entropy for sets of viscosity solutions to the Hamilton-Jacobi equation. Earlier works on this topic considered a uniformly convex Hamiltonian. We extend the analysis to the case when the Hamiltonian H ∈ C 1 (Rd) is strictly convex, coercive and a uniformly directionally convex function. Under these assumptions, we establish sharp estimates on the "-entropy for sets of viscosity solutions to the Hamilton-Jacobi equation with respect to W1,1 -distance in multi-dimensional casesThe second half of the thesis focuses on finding sharp estimates for the metric entropy of a class of bounded total generalized variation functions taking values in a general totally bounded metric space (E,ρ) upto an accuracy of " with respect to the L 1-distance. We rely on the ideas of covering and packing in (E,ρ) to derive such bounds and utilize them to study a scalar conservation law in one-dimensional space, which yields an upper bound on the "-entropy of a set of entropy admissible weak solutions to it, in case of weakly genuinely nonlinear fluxes, i.e., for fluxes with no affine parts. In particular, for fluxes admitting finitely many inflection points with a polynomial degeneracy, this estimate is sharp.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2023
538 $a Mode of access: World Wide Web
650 4 $a Mathematical functions. $3 3564295
650 4 $a Convex analysis. $3 3681761
650 4 $a Conservation laws. $3 3682569
650 4 $a Mathematics. $3 515831
650 4 $a Collaboration. $3 3556296
650 4 $a Partial differential equations. $3 2180177
650 4 $a Viscosity. $3 1050706
650 4 $a Artificial intelligence. $3 516317
650 4 $a Accuracy. $3 3559958
655 7 $a Electronic books. $2 lcsh $3 542853
690 $a 0405
690 $a 0800
710 2 $a ProQuest Information and Learning Co. $3 783688
710 2 $a North Carolina State University. $3 1018772
773 0 $t Dissertations Abstracts International $g 83-02B.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28552737 $z click for full text (PQDT)