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Multi angle  Radiative transport and homogenization for the random Schrödinger equation
Ryzhik, Lenya (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  Some examples of homogenization related rough paths
Friz, Peter K. | CIRM (Editeur )

We discuss the rough path principle and some of its applications to problems of homogenization.

60H15 ; 35B27

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Multi angle  Integrable probability - Lecture 3
Corwin, Ivan (Auteur de la Conférence) | CIRM (Editeur )

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations
A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide ...

82C22 ; 82B23 ; 60H15

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Multi angle  Integrable probability - Lecture 2
Corwin, Ivan (Auteur de la Conférence) | CIRM (Editeur )

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations
A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide ...

82C22 ; 82B23 ; 60H15

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Multi angle  Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation
Tempone, Raul (Auteur de la Conférence) | CIRM (Editeur )

We describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) method for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. These mixed differences yield new and improved complexity results, which are natural generalizations of Giles's MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence. On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that, in the optimal case, the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We propose optimization procedures to select the most effective mixed differences to include in MIMC and MISC. Such optimization is a crucial step that allows us to make MIMC and MISC computationally efficient. We show the effectiveness of MIMC and MISC in some computational tests using the mimclib open source library, including PDEs with random coefficients and Stochastic Interacting Particle Systems. Finally, we will briefly discuss the use of Markovian projection for the approximation of prices in the context of American basket options. We describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) method for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, ...

65C30 ; 65C05 ; 60H15 ; 60H35 ; 35R60 ; 65M70

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Multi angle  Multilevel and multi-index sampling methods with applications - Lecture 1: Adaptive strategies for Multilevel Monte Carlo
Tempone, Raul (Auteur de la Conférence) | CIRM (Editeur )

We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion of adaptivity includes several aspects such as mesh refinements based on either a priori or a posteriori error estimates, the local choice of different time stepping methods and the selection of the total number of levels and the number of samples at different levels. Our Adaptive MLMC estimator uses a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform discretization MLMC method introduced independently by M. Giles and S. Heinrich. In particular, we show that our adaptive MLMC algorithms are asymptotically accurate and have the correct complexity with an improved control of the multiplicative constant factor in the asymptotic analysis. In this context, we developed novel techniques for estimation of parameters needed in our MLMC algorithms, such as the variance of the difference between consecutive approximations. These techniques take particular care of the deepest levels, where for efficiency reasons only few realizations are available to produce essential estimates. Moreover, we show the asymptotic normality of the statistical error in the MLMC estimator, justifying in this way our error estimate that allows prescribing both the required accuracy and confidence level in the final result. We present several examples to illustrate the above results and the corresponding computational savings. We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion ...

65C30 ; 65C05 ; 60H15 ; 60H35 ; 35R60

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Multi angle  Branching for PDEs
Warin, Xavier (Auteur de la Conférence) | CIRM (Editeur )

Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities polynomial in u and Du with convergence results. At last we give some numerical schemes to solve the semi-linear case and even the full non linear case but currently without convergence results. Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities ...

60H15 ; 35R60 ; 60J80