5,534 research outputs found
A stochastic maximum principle for general mean-field backward doubly stochastic control
In this paper we study the optimal control problems of general MckeanVlasov for backward doubly stochastic differential equations (BDSDEs), in which the coefficients depend on the state of the solution process as well as of its law. We establish a stochastic maximum principle on the hypothesis that the control field is convex. For example, an example of a control problem is offered and solved using the primary result.Publisher's Versio
Uncertainty quantification for random domains using periodic random variables
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates
Unification of Adomian decomposition method and ZZ transformation for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems
This paper presents a novel approach for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems by using the unification of the Adomian decomposition method and ZZ transformation. The suggested method combines the Aboodh transform and the Adomian decomposition method, both of which are trustworthy and efficient mathematical tools for solving fractional differential equations (FDEs). This method's theoretical analysis is addressed for nonlinear FDE systems. To find exact solutions to the equations, the method is applied to fractional Kersten-Krasil'shchik linked KdV-mKdV systems. The results show that the suggested method is efficient and practical for solving fractional Kersten-Krasil'shchik linked KdV-mKdV systems and that it may be applied to other nonlinear FDEs. The suggested method has the potential to provide new insights into the behavior of nonlinear waves in fluid and plasma environments, as well as the development of new mathematical tools for modeling and studying complicated wave phenomena
Staircasing effect for minimizers of the one-dimensional discrete Perona-Malik functional
We consider the one-dimensional Perona-Malik functional, that is the energy
associated to the celebrated forward-backward equation introduced by P. Perona
and J. Malik in the context of image processing, with the addition of a forcing
term. We discretize the functional by restricting its domain to a finite
dimensional space of piecewise constant functions, and by replacing the
derivative with a difference quotient.
We investigate the asymptotic behavior of minima and minimizers as the
discretization scale vanishes. In particular, if the forcing term has bounded
variation, we show that any sequence of minimizers converges in the sense of
varifolds to the graph of the forcing term, but with tangent component which is
a combination of the horizontal and vertical directions.
If the forcing term is more regular, we also prove that minimizers actually
develop a microstructure that looks like a piecewise constant function at a
suitable scale, which is intermediate between the macroscopic scale and the
scale of the discretization.Comment: 43 pages. In the second version some typos have been correcte
A modern analytic method to solve singular and non-singular linear and non-linear differential equations
This article circumvents the Laplace transform to provide an analytical solution in a power series form for singular, non-singular, linear, and non-linear ordinary differential equations. It introduces a new analytical approach, the Laplace residual power series, which provides a powerful tool for obtaining accurate analytical and numerical solutions to these equations. It demonstrates the new approach’s effectiveness, accuracy, and applicability in several ordinary differential equations problem. The proposed technique shows the possibility of finding exact solutions when a pattern to the series solution obtained exists; otherwise, only rough estimates can be given. To ensure the accuracy of the generated results, we use three types of errors: actual, relative, and residual error. We compare our results with exact solutions to the problems discussed. We conclude that the current method is simple, easy, and effective in solving non-linear differential equations, considering that the obtained approximate series solutions are in closed form for the actual results. Finally, we would like to point out that both symbolic and numerical quantities are calculated using Mathematica software
Iterative algorithms for partitioned neural network approximation to partial differential equations
To enhance solution accuracy and training efficiency in neural network
approximation to partial differential equations, partitioned neural networks
can be used as a solution surrogate instead of a single large and deep neural
network defined on the whole problem domain. In such a partitioned neural
network approach, suitable interface conditions or subdomain boundary
conditions are combined to obtain a convergent approximate solution. However,
there has been no rigorous study on the convergence and parallel computing
enhancement on the partitioned neural network approach. In this paper,
iterative algorithms are proposed to address these issues. Our algorithms are
based on classical additive Schwarz domain decomposition methods. Numerical
results are included to show the performance of the proposed iterative
algorithms
AI: Limits and Prospects of Artificial Intelligence
The emergence of artificial intelligence has triggered enthusiasm and promise of boundless opportunities as much as uncertainty about its limits. The contributions to this volume explore the limits of AI, describe the necessary conditions for its functionality, reveal its attendant technical and social problems, and present some existing and potential solutions. At the same time, the contributors highlight the societal and attending economic hopes and fears, utopias and dystopias that are associated with the current and future development of artificial intelligence
Mechanical characterization, constitutive modeling and applications of ultra-soft magnetorheological elastomers
Mención Internacional en el título de doctorSmart materials are bringing sweeping changes in the way humans interact with engineering devices. A myriad of state-of-the-art applications are based on novel ways to actuate on structures that respond under different types of stimuli. Among them, materials that respond to magnetic fields allow to remotely modify their mechanical properties and macroscopic
shape. Ultra-soft magnetorheological elastomers (MREs) are composed of a highly stretchable soft elastomeric matrix in the order of 1 kPa and magnetic particles embedded in it. This combination allows large deformations with small external actuations.
The type of the magnetic particles plays a crucial role as it defines the reversibility or remanence of the material magnetization. According to the fillers used, MREs are referred to as soft-magnetic magnetorheological elastomers (sMREs) and hard-magnetic magnetorheological elastomers (hMREs). sMREs exhibit strong changes in their mechanical properties
when an external magnetic field is applied, whereas hMREs allow sustained magnetic effects along time and complex shape-morphing capabilities. In this regard, end-of-pipe applications of MREs in the literature are based on two major characteristics: the modification of their mechanical properties and macrostructural shape changes. For instance, smart actuators,
sensors and soft robots for bioengineering applications are remotely actuated to perform functional deformations and autonomous locomotion. In addition, hMREs have been used for industrial applications, such as damping systems and electrical machines.
From the analysis of the current state of the art, we identified some impediments to advance in certain research fields that may be overcome with new solutions based on ultrasoft MREs. On the mechanobiology area, we found no available experimental methodologies to transmit complex and dynamic heterogeneous strain patterns to biological systems in a reversible manner. To remedy this shortcoming, this doctoral research proposes
a new mechanobiology experimental setup based on responsive ultra-soft MRE biological substrates. Such an endeavor requires deeper insights into the magneto-viscoelastic and microstructural mechanisms of ultra-soft MREs. In addition, there is still a lack of guidance for the selection of the magnetic fillers to be used for MREs and the final properties provided
to the structure. Eventually, the great advances on both sMREs and hMREs to date pose a timely question on whether the combination of both types of particles in a hybrid MRE may optimize the multifunctional response of these active structures.
To overcome these roadblocks, this thesis provides an extensive and comprehensive experimental characterization of ultra-soft sMREs, hMREs and hybrid MREs. The experimental methodology uncovers magneto-mechanical rate dependences under numerous loading and manufacturing conditions. Then, a set of modeling frameworks allows to delve into such
mechanisms and develop three ground-breaking applications. Therefore, the thesis has lead to three main contributions. First and motivated on mechanobiology research, a computational framework guides a sMRE substrate to transmit complex strain patterns in vitro to biological systems. Second, we demonstrate the ability of remanent magnetic fields in hMREs to arrest cracks propagations and improve fracture toughness. Finally, the combination of soft- and hard-magnetic particles is proved to enhance the magnetorheological and magnetostrictive effects, providing promising results for soft robotics.Los materiales inteligentes están generando cambios radicales en la forma que los humanos interactúan con dispositivos ingenieriles. Distintas aplicaciones punteras se basan en formas novedosas de actuar sobre materiales que responden a diferentes estímulos. Entre ellos, las estructuras que responden a campos magnéticos permiten la modificación de manera remota tanto de sus propiedades mecánicas como de su forma. Los elastómeros magnetorreológicos (MREs) ultra blandos están compuestos por una matriz elastomérica con gran ductilidad y una rigidez en torno a 1 kPa, reforzada con partículas magnéticas. Esta combinación permite
inducir grandes deformaciones en el material mediante la aplicación de campos magnéticos pequeños.
La naturaleza de las partículas magnéticas define la reversibilidad o remanencia de la magnetización del material compuesto. De esta manera, según el tipo de partículas que contengan, los MREs pueden presentar magnetización débil (sMRE) o magnetización fuerte (hMRE). Los sMREs experimentan grandes cambios en sus propiedades mecánicas al aplicar
un campo magnético externo, mientras que los hMREs permiten efectos magneto-mecánicos sostenidos a lo largo del tiempo, así como programar cambios de forma complejos. En este sentido, las aplicaciones de los MREs se basan en dos características principales: la modificación de sus propiedades mecánicas y los cambios de forma macroestructurales. Por
ejemplo, los campos magnéticos pueden emplearse para inducir deformaciones funcionales en actuadores y sensores inteligentes, o en robótica blanda para bioingeniería. Los hMREs también se han aplicado en el ámbito industrial en sistemas de amortiguación y máquinas eléctricas.
A partir del análisis del estado del arte, se identifican algunas limitaciones que impiden el avance en ciertos campos de investigación y que podrían resolverse con nuevas soluciones basadas en MREs ultra blandos. En el área de la mecanobiología, no existen metodologías experimentales para transmitir patrones de deformación complejos y dinámicos a sistemas
biológicos de manera reversible. En esta investigación doctoral se propone una configuración experimental novedosa basada en sustratos biológicos fabricados con MREs ultra blandos. Dicha solución requiere la identificación de los mecanismos magneto-viscoelásticos y microestructurales de estos materiales, según el tipo de partículas magnéticas, y las consiguientes
propiedades macroscópicas del material. Además, investigaciones recientes en sMREs y hMREs plantean la pregunta sobre si la combinación de distintos tipos de partículas magnéticas en un MRE híbrido puede optimizar su respuesta multifuncional.
Para superar estos obstáculos, la presente tesis proporciona una caracterización experimental completa de sMREs, hMREs y MREs híbridos ultra blandos. Estos resultados muestran las dependencias del comportamiento multifuncional del material con la velocidad de aplicación de cargas magneto-mecánicas. El desarrollo de un conjunto de modelos
teórico-computacionales permite profundizar en dichos mecanismos y desarrollar aplicaciones innovadoras. De este modo, la tesis doctoral ha dado lugar a tres bloques de aportaciones principales. En primer lugar, este trabajo proporciona un marco computacional para guiar el diseño de sustratos basados en sMREs para transmitir patrones de deformación complejos in vitro a sistemas biológicos. En segundo lugar, se demuestra la capacidad de los campos magnéticos remanentes en los hMRE para detener la propagación de grietas y mejorar la tenacidad a la fractura. Finalmente, se establece que la combinación de partículas magnéticas de magnetización débil y fuerte mejora el efecto magnetorreológico y magnetoestrictivo, abriendo nuevas posibilidades para el diseño de robots blandos.I want to acknowledge the support from the Ministerio de Ciencia, Innovación y Universidades, Spain (FPU19/03874), and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947723, project: 4D-BIOMAP).Programa de Doctorado en Ingeniería Mecánica y de Organización Industrial por la Universidad Carlos III de MadridPresidente: Ramón Eulalio Zaera Polo.- Secretario: Abdón Pena Francesch.- Vocal: Laura de Lorenzi
국소 및 비국소 측도 데이터 문제의 정칙성 이론
학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 2. 변순식.In this thesis, we establish various regularity results for nonlinear measure data problems. The results obtained are part of a program devoted to nonlinear Calderón-Zygmund theory and nonlinear potential theory. Firstly, we obtain maximal integrability and fractional differentiability results for elliptic measure data problems with Orlicz growth and borderline double phase growth, respectively. We also obtain fractional differentiability results for parabolic measure data problems under a minimal assumption on the coefficients. Secondly, we obtain gradient potential estimates and fractional differentiability results for elliptic obstacle problems with measure data, by using linearization techniques. In particular, we develop a new method to obtain potential estimates for irregular obstacle problems. For the case of single obstacle problems with L¹-data, we further obtain uniqueness results and comparison principles in order to improve such regularity results. Lastly, we show existence, regularity and potential estimates for mixed local and nonlocal equations with measure data. Also, as a first step to the regularity theory for anisotropic nonlocal problems with nonstandard growth, we establish Hölder regularity for nonlocal double phase problems by identifying sharp assumptions analogous to those for local double phase problems.이 학위논문에서는 비선형 측도 데이터 문제들에 대하여 다양한 정칙성 결과들을 얻는다. 해당 결과들은 비선형 칼데론-지그문트 이론 및 비선형 퍼텐셜 이론을 다루는 과정의 일부이다. 첫 번째로, 오를리츠 성장조건 및 경계선 이중위상 성장조건을 가지는 타원형 측도 데이터 문제에 대하여 각각 최대 적분성 및 분수차수 미분성 결과를 얻는다. 또한 포물형 측도 데이터 문제에 대하여 분수차수 미분성을 계수에 대한 최소한의 가정 하에서 증명한다. 두 번째로, 측도 데이터를 가지는 타원형 장애물 문제에 대하여 선형화 기법을 이용함으로써 그레이디언트 퍼텐셜 가늠 및 분수차수 미분성을 증명한다. 특히 비정칙 장애물 문제에 대해 퍼텐셜 가늠을 얻기 위한 새로운 방법을 개발한다. 더 나아가, L¹ 데이터를 가지는 단일 장애물 문제에 대하여는 해의 유일성 및 비교 원리를 증명하여 이러한 정칙성 결과들을 개선한다. 마지막으로, 측도 데이터를 가지는 국소 및 비국소 혼합 방정식에 대하여 해의 존재성, 정칙성 및 퍼텐셜 가늠을 증명한다. 또한, 비표준 성장조건을 가지는 비등방적 비국소 문제에 대한 정칙성 이론의 첫걸음으로서, 비국소 이중위상 문제에 대한 횔더 정칙성을 국소 이중위상 문제의 경우과 유사한 최적의 조건 하에서 증명한다.1 Introduction 1
1.1 Measure data problems 1
1.1.1 Nonlinear Calderón-Zygmund theory 2
1.1.2 Nonlinear potential theory 4
1.2 Elliptic measure data problems with nonstandard growth 7
1.3 Elliptic obstacle problems with measure data 8
1.4 Nonlocal equations, mixed local and nonlocal equations 9
1.5 Nonlocal operators and measure data 10
1.6 Nonlocal operators with nonstandard growth 11
2 Preliminaries 13
2.1 General notations 13
2.2 Function spaces 15
2.2.1 Musielak-Orlicz spaces 15
2.2.2 Fractional Sobolev spaces 18
2.2.3 Lorentz spaces, Marcinkiewicz spaces 21
2.3 Auxiliary results 22
2.3.1 Basic properties of the vector fields V(·) and A(·) 22
2.3.2 Regularity for homogeneous equations 24
2.3.3 Technical lemmas 34
3 Elliptic and parabolic equations with measure data 35
3.1 Maximal integrability for elliptic measure data problems with Orlicz growth 35
3.1.1 Main results 35
3.1.2 Some technical results 37
3.1.3 Proof of Theorem 3.1.2 43
3.2 Fractional differentiability for elliptic measure data problems with double phase in the borderline case 53
3.2.1 Main results 53
3.2.2 Preliminaries 55
3.2.3 Regularity for homogeneous problems 56
3.2.4 Comparison estimates 61
3.2.5 Proof of Theorem 3.2.2 66
3.3 Fractional differentiability for parabolic measure data problems 71
3.3.1 Main results 71
3.3.2 Preliminaries 73
3.3.3 Some technical results 75
3.3.4 Proof of Theorem 3.3.3 79
4 Elliptic obstacle problems with measure data 83
4.1 Potential estimates for obstacle problems with measure data 84
4.1.1 Main results 85
4.1.2 Reverse Hölders inequalities for homogeneous obstacle problems 88
4.1.3 Basic comparison estimates 93
4.1.4 Linearized comparison estimates 109
4.1.5 The two-scales degenerate alternative 109
4.1.6 The two-scales non-degenerate alternative 111
4.1.7 Combining the two alternatives 126
4.1.8 Proof of Theorem 4.1.2 128
4.1.9 Proof of Theorem 4.1.3 132
4.2 Fractional differentiability for double obstacle problems with measure data 138
4.2.1 Main results 139
4.2.2 Comparison estimates 141
4.2.3 Proof of Theorem 4.2.2 156
4.2.4 Proof of Theorem 4.2.4 158
4.3 Comparison principle for obstacle problems with L¹-data 162
4.3.1 Comparison principles 163
4.3.2 Applications to regularity results 166
5 Mixed local and nonlocal equations with measure data 171
5.1 Main results 171
5.2 Preliminaries 177
5.3 Regularity for homogeneous equations 178
5.4 Comparison estimates 184
5.5 Existence of SOLA 189
5.6 Potential estimates 194
5.6.1 Proof of Theorems 5.1.4 and 5.1.7 194
5.6.2 Proof of Theorem 5.1.5 197
5.7 Continuity criteria for SOLA 204
5.7.1 Proof of Theorem 5.1.8 204
5.7.2 Proof of Theorem 5.1.10 205
6 Nonlocal double phase problems 207
6.1 Main results 208
6.2 Preliminaries 211
6.2.1 Function spaces 211
6.2.2 Inequalities 212
6.3 Existence of weak solutions 215
6.4 Caccioppoli estimates and local boundedness 217
6.5 Hölder continuity 225
6.5.1 Logarithmic estimates 225
6.5.2 Proof of Theorem 6.1.2 235
Abstract (in Korean) 261박
A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions
In this paper, we develop a Galerkin-type approximation, with quantitative
error estimates, for weak solutions to the Cauchy problem for kinetic
Fokker-Planck equations in the domain ,
where is either or . Our approach is based on
a Hermite expansion in the velocity variable only, with a hyperbolic system
that appears as the truncation of the Brinkman hierarchy, as well as ideas from
and additional energy-type estimates that
we have developed. We also establish the regularity of the solution based on
the regularity of the initial data and the source term.Comment: 24 pages, corrected the references, version submitted to Kinet.
Relat. Models, to appea
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