Cleansing the Common: Narrative-Intertextual Study of Mark 7:1-23

The conflict story of Mark 7:1–23 between Jesus and the religious leaders over the issue of defilement is the meeting point of a variety of disciplines: Purity studies, Jewish studies, exegetical studies, Historical Jesus studies, and studies on Jesus and the law. The crux of the passage, the meaning of the parable in v. 15 and the ensuing “cleansing” in v. 19, has been interpreted very differently. Scholars doing exegetical studies and studies on the relationship between Jesus and the law have maintained that the Gospel writer correctly reflects in 7:19 the meaning of Jesus’ parable (7:15), abrogating the clean/unclean categories of Lev 11. Scholars doing purity, Jewish, and recent Historical Jesus studies have generally argued that Jesus could not have abrogated these food laws in the social and religious setting of his day. The controversial remark in a narrative aside must be Mark’s comment on Jesus’ saying to accommodate the Christian community in the later part of the first century. Chapter 1 introduces the narrative-intertextual methodology used in the subsequent chapters. This methodology allows a careful examination of the literary material in Mark’s Gospel in the first part of the dissertation and a careful examination of purity issues arising out of the Hebrew Scriptures and the Second Temple period in the later part. The narrative analysis in chapters 2–3 reveals that Mark uses space, time, props, movement, prefixes, verb tenses, and technical terminology meticulously and astutely to develop the themes in the pericope and build a cohesive literary unit. The central theme of the entire pericope is “touch defilement,” which is first introduced in the observation that the disciples eat with defiled (unwashed) hands. It is augmented with a conflict over authority. Chapter 4 examines the interrelationship of purity terms in biblical literature of the later Second Temple period. In the major reference works predating the 1970s, the purity terms κοινός (“defiled”), ἀκάθαρτος (“unclean”), and βέβηλος (“profane”) were more or less used interchangeably. Since the 1970s though, studies examining the topic of purity have differentiated these terms. An assessment of 1 Macc 1:47, 62; Mark 7:1–23; Acts 10–11; and the parallel passages of Acts 21:28 and 24:6 leads to the conclusion that κοινός/κοινόω is a term unique to the Second Temple period and distinct from other purity terminology. It is best defined as an intermediary defilement that a clean person/object acquires by coming in contact with an unclean person/object. Since κοινός impurity is unknown in the Hebrew Scriptures, Mark is correct in attributing it to the “tradition of the elders.” Scholarship has generally connected allusions in Mark 7:1–23 to the clean/unclean animals of Lev 11. Chapter 5 assesses the intertextual allusions based on literary, thematic, and logical parallels. In each category Mark indeed refers to Lev 11, but not to the section on clean/unclean animals (Lev 11:1–23, 41–43). Instead, the allusions always point to the section on touch contamination by a carcass (Lev 11:24– 40) or the section containing holiness language (Lev 11:44–45). Mark underlines the topic of touch defilement and ethical purity by means of these allusions to Lev 11. A concluding chapter summarizes the findings. In Mark 7:1–23 neither Mark nor Jesus abrogates the clean/unclean distinction of Leviticus. Instead, Mark in v. 19 correctly summarizes Jesus’ position that new “traditions,” established during the Second Temple period, overextended God’s requirements and are hence invalid. In the larger context (Mark 6–8 and particularly Mark 7:24–30), κοινός defilement from Gentiles is therefore an invalid expansion of God’s law and, instead, mission to all people is a divine imperative (Gen 12:1–3; Mark 7:24–30; Acts 10–11). Mark 7:1–23 is shown to be a coherent whole illustrated in four steps. The narrative data demonstrate the unity of the pericope. Jesus’ support of the law against Second Temple period additions is found in both vv. 1–13 and 14–23. The passage’s marked parallelism to the defilement and holiness theology of Lev 11 exhibits the Evangelist’s sensitivity to purity issues. And the congruence of the passage’s teaching with the trajectory of mission in Acts 10 demonstrates the heuristic power of this explanation of Mark 7

Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs

Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure correction has to be solved at every time step in the dynamical core of many numerical weather prediction models, and equations of a very similar structure arise in global ocean models, subsurface flow simulations and gas and oil reservoir modelling. The elliptic solve is often the bottleneck of the forecast, and an algorithmically optimal method has to be used and implemented efficiently. Graphics Processing Units have been shown to be highly efficient for a wide range of applications in scientific computing, and recently iterative solvers have been parallelised on these architectures. We describe the GPU implementation and optimisation of a Preconditioned Conjugate Gradient (PCG) algorithm for the solution of a three dimensional anisotropic elliptic PDE for the pressure correction in NWP. Our implementation exploits the strong vertical anisotropy of the elliptic operator in the construction of a suitable preconditioner. As the algorithm is memory bound, performance can be improved significantly by reducing the amount of global memory access. We achieve this by using a matrix-free implementation which does not require explicit storage of the matrix and instead recalculates the local stencil. Global memory access can also be reduced by rewriting the algorithm using loop fusion and we show that this further reduces the runtime on the GPU. We demonstrate the performance of our matrix-free GPU code by comparing it to a sequential CPU implementation and to a matrix-explicit GPU code which uses existing libraries. The absolute performance of the algorithm for different problem sizes is quantified in terms of floating point throughput and global memory bandwidth.Comment: 18 pages, 7 figure

Deep surrogate accelerated delayed-acceptance HMC for Bayesian inference of spatio-temporal heat fluxes in rotating disc systems

We study the Bayesian inverse problem of inferring the Biot number, a spatio-temporal heat-flux parameter in a PDE model. This is an ill-posed problem where standard optimisation yields unphysical inferences. We introduce a training scheme that uses temperature data to adaptively train a neural-network surrogate to simulate the parametric forward model. This approach approximates forward and inverse solution together, by simultaneously identifying an approximate posterior distribution over the Biot number, and weighting the forward training loss according to this approximation. Utilising random Chebyshev series, we outline how to approximate an arbitrary Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to efficiently sample from the corresponding posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss function approaches zero. Furthermore, we describe how this surrogate-accelerated HMC approach can be combined with a traditional PDE solver in a delayed-acceptance scheme to a-priori control the posterior accuracy, thus overcoming a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high-dimensional parameter spaces, whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results compare the accuracy and efficiency of the adaptive and general training regimes, as well as various Markov chain Monte Carlo proposals strategies

Deep surrogate accelerated delayed-acceptance HMC for Bayesian inference of spatio-temporal heat fluxes in rotating disc systems

We study the Bayesian inverse problem of inferring the Biot number, a spatio-temporal heat-flux parameter in a PDE model. This is an ill-posed problem where standard optimisation yields unphysical inferences. We introduce a training scheme that uses temperature data to adaptively train a neural-network surrogate to simulate the parametric forward model. This approach approximates forward and inverse solution together, by simultaneously identifying an approximate posterior distribution over the Biot number, and weighting the forward training loss according to this approximation. Utilising random Chebyshev series, we outline how to approximate an arbitrary Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to efficiently sample from the corresponding posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss function approaches zero. Furthermore, we describe how this surrogate-accelerated HMC approach can be combined with a traditional PDE solver in a delayed-acceptance scheme to a-priori control the posterior accuracy, thus overcoming a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high-dimensional parameter spaces, whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results compare the accuracy and efficiency of the adaptive and general training regimes, as well as various Markov chain Monte Carlo proposals strategies