Stochastic homogenization of Λ\Lambda-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1,\infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($\Lambda$-)convex functionals.Comment: arXiv admin note: text overlap with arXiv:1805.0954

Evaluating Value-Based Frameworks Used for Relapsed or Refractory Multiple Myeloma Regimens: ICER Report, ASCO Value Framework, and NCCN Evidence Blocks

BACKGROUND: With the continuous rise in costs for oncology drugs, the Institute for Clinical and Economic Review (ICER), the American Society of Clinical Oncology (ASCO) and the National Comprehensive Cancer Network (NCCN) have developed value-based frameworks (VBFs) to assist stakeholders in formulary and treatment decisionmaking. While emerging VBFs have the potential to significantly impact therapeutic options for patients, it is important to understand the differences associated with those VBFs within a therapeutic area. OBJECTIVE: To compare ICER, ASCO, and NCCN VBFs across three therapeutic options for relapsed or refractory multiple myeloma (RRMM). METHODS: The values of carfilzomib (CFZ), elotuzumab (ELO), and ixazomib (IX) were generated using ICER, ASCO, and NCCN VBFs. Those regimens, used for second or third line treatment of RRMM, were chosen because they share a common comparator in clinical trials, lenalidomide + dexamethasone (LEN + DEX). The ICER 2016 report of treatment options for RRMM was used to obtain results of the comparative clinical effectiveness and the cost effectiveness analysis for those regimens compared to LEN + DEX. ASCO’s 2016 VBF, which incorporates clinical benefit, toxicity and bonus points was used to generate a net health benefit (NHB) score without a scale along with the drug wholesale acquisition cost (WAC) for each regimen compared to LEN + DEX. The NCCN VBF uses a score ranging from 1 to 5, with 1 as the least favorable and 5 as the most favorable, for each of five evidence blocks: efficacy, safety, quality, consistency, and affordability. The 2016 Multiple Myeloma NCCN evidence blocks was used to obtain the value of CFZ, ELO, and IX. RESULTS: The ICER VBF suggested with moderate certainty that CFZ, ELO, and IX provide a better NHB in patients with RRMM compared to LEN + DEX. Second-line and third-line treatment costs per QALY for CFZ, ELO, and IX were $199,982,$427,607 and $433,794, and$238,560, $481,244, and$484,582, respectively. The ASCO VBF generated a total NHB of 28.8, 23.7 and 23.0 with a monthly WAC of $17,364,$16,032 and $20,607 for CFZ, ELO, and IX, respectively. The monthly cost of LEN + DEX was$11,616. The NCCN VBF had an efficacy score of 5, 3, and 4 for CFZ, ELO, IX, respectively. Safety, quality, consistency, and affordability scores of 3, 4, 4, and 1, respectively, were the same across regimens. CONCLUSIONS: ICER, ASCO and NCCN VBFs suggest CFZ may be the most valued treatment out of the three regimens. However, their applicability in stakeholder’s decision-making remains unclear due to uncertainty and challenges associated with them. SPONSORSHIP: None.https://jdc.jefferson.edu/jcphposters/1012/thumbnail.jp

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends extitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition

Stochastic homogenization of Lambda-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals

Stochastic two-scale convergence and Young measures

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere weak convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a new result on stochastic homogenization for a non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method described in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.Comment: 46 page