Quantitative Continuity and Computable Analysis in Coq

We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the results of this paper extend the library. While full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The paper includes a brief description of the most important concepts of Incone and its sub libraries mf and Metric. The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that the countably infinite product of a space with itself is isomorphic to a space of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also describe many non-computational results that support the correctness of definitions from the library. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers

Formalizing Hyperspaces for Extracting Efficient Exact Real Computation

We propose a framework for certified computation on hyperspaces by formalizing various higher-order data types and operations in a constructive dependent type theory. Our approach builds on our previous work on axiomatization of exact real computation where we formalize nondeterministic first-order partial computations over real and complex numbers. Based on the axiomatization, we first define open, closed, compact and overt subsets in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis. From these proofs we extract programs for testing inclusion, overlapping of sets, et cetera. To improve extracted programs, our framework specializes the Euclidean space ?^m making use of metric properties. To define interesting operations over hyperspaces of Euclidean space, we introduce a nondeterministic version of a continuity principle valid under the standard type-2 realizability interpretation. Instead of choosing one of the usual formulations, we define it in a way similar to an interval extension operator, which often is already available in exact real computation software. We prove that the operations on subsets preserve the encoding, and thereby define a small calculus to built new subsets from given ones, including limits of converging sequences with regards to the Hausdorff metric. From the proofs, we extract programs that generate drawings of subsets of ?^m with any given precision efficiently. As an application we provide a function that constructs fractals, such as the Sierpinski triangle, from iterated function systems using the limit operation, resulting in certified programs that errorlessly draw such fractals up to any desired resolution

Average-Case Polynomial-Time Computability of Hamiltonian Dynamics

We apply average-case complexity theory to physical problems modeled by continuous-time dynamical systems. The computational complexity when simulating such systems for a bounded time-frame mainly stems from trajectories coming close to complex singularities of the system. We show that if for most initial values the trajectories do not come close to singularities the simulation can be done in polynomial time on average. For Hamiltonian systems we relate this to the volume of "almost singularities" in phase space and give some general criteria to show that a Hamiltonian system can be simulated efficiently on average. As an application we show that the planar circular-restricted three-body problem is average-case polynomial-time computable

Continuous and monotone machines

We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq as part of Incone, a Coq library for computable analysis. The correctness proofs use a classical meta-theory with countable choice. Along the way we formally prove some known results such as the existence of a self-modulating modulus of continuity for partial continuous operators on Baire space. To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics

Algebraic Temporal Blocking for Sparse Iterative Solvers on Multi-Core CPUs

Sparse linear iterative solvers are essential for many large-scale simulations. Much of the runtime of these solvers is often spent in the implicit evaluation of matrix polynomials via a sequence of sparse matrix-vector products. A variety of approaches has been proposed to make these polynomial evaluations explicit (i.e., fix the coefficients), e.g., polynomial preconditioners or s-step Krylov methods. Furthermore, it is nowadays a popular practice to approximate triangular solves by a matrix polynomial to increase parallelism. Such algorithms allow to evaluate the polynomial using a so-called matrix power kernel (MPK), which computes the product between a power of a sparse matrix A and a dense vector x, or a related operation. Recently we have shown that using the level-based formulation of sparse matrix-vector multiplications in the Recursive Algebraic Coloring Engine (RACE) framework we can perform temporal cache blocking of MPK to increase its performance. In this work, we demonstrate the application of this cache-blocking optimization in sparse iterative solvers. By integrating the RACE library into the Trilinos framework, we demonstrate the speedups achieved in preconditioned) s-step GMRES, polynomial preconditioners, and algebraic multigrid (AMG). For MPK-dominated algorithms we achieve speedups of up to 3x on modern multi-core compute nodes. For algorithms with moderate contributions from subspace orthogonalization, the gain reduces significantly, which is often caused by the insufficient quality of the orthogonalization routines. Finally, we showcase the application of RACE-accelerated solvers in a real-world wind turbine simulation (Nalu-Wind) and highlight the new opportunities and perspectives opened up by RACE as a cache-blocking technique for MPK-enabled sparse solvers.Comment: 25 pages, 11 figures, 3 table

GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems

While many of the architectural details of future exascale-class high performance computer systems are still a matter of intense research, there appears to be a general consensus that they will be strongly heterogeneous, featuring "standard" as well as "accelerated" resources. Today, such resources are available as multicore processors, graphics processing units (GPUs), and other accelerators such as the Intel Xeon Phi. Any software infrastructure that claims usefulness for such environments must be able to meet their inherent challenges: massive multi-level parallelism, topology, asynchronicity, and abstraction. The "General, Hybrid, and Optimized Sparse Toolkit" (GHOST) is a collection of building blocks that targets algorithms dealing with sparse matrix representations on current and future large-scale systems. It implements the "MPI+X" paradigm, has a pure C interface, and provides hybrid-parallel numerical kernels, intelligent resource management, and truly heterogeneous parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We describe the details of its design with respect to the challenges posed by modern heterogeneous supercomputers and recent algorithmic developments. Implementation details which are indispensable for achieving high efficiency are pointed out and their necessity is justified by performance measurements or predictions based on performance models. The library code and several applications are available as open source. We also provide instructions on how to make use of GHOST in existing software packages, together with a case study which demonstrates the applicability and performance of GHOST as a component within a larger software stack.Comment: 32 pages, 11 figure

Computable analysis for verified exact real computation

We use ideas from computable analysis to formalize exact real number computation in the Coq proof assistant. Our formalization is built on top of the Incone library, a Coq library for computable analysis. We use the theoretical framework that computable analysis provides to systematically generate target specifications for real number algorithms. First we give very simple algorithms that fulfill these specifications based on rational approximations. To provide more efficient algorithms, we develop alternate representations that utilize an existing formalization of floating-point algorithms and interval arithmetic in combination with methods used by software packages for exact real arithmetic that focus on execution speed. We also define a general framework to define real number algorithms independently of their concrete encoding and to prove them correct. Algorithms verified in our framework can be extracted to Haskell programs for efficient computation. The performance of the extracted code is comparable to programs produced using non-verified software packages. This is without the need to optimize the extracted code by hand. As an example, we formalize an algorithm for the square root function based on the Heron method. The algorithm is parametric in the implementation of the real number datatype, not referring to any details of its implementation. Thus the same verified algorithm can be used with different real number representations. Since Boolean valued comparisons of real numbers are not decidable, our algorithms use basic operations that take values in the Kleeneans and Sierpinski space. We develop some of the theory of these spaces. To capture the semantics of non-sequential operations, such as the “parallel or”, we use multivalued functions

Integrin control of the transforming growth factor-β pathway in glioblastoma

Transforming growth factor-β is a central mediator of the malignant phenotype of glioblastoma, the most common and malignant form of intrinsic brain tumours. Transforming growth factor-β promotes invasiveness and angiogenesis, maintains cancer cell stemness and induces profound immunosuppression in the host. Integrins regulate cellular adhesion and transmit signals important for cell survival, proliferation, differentiation and motility, and may be involved in the activation of transforming growth factor-β. We report that αvβ3, αvβ5 and αvβ8 integrins are broadly expressed not only in glioblastoma blood vessels but also in tumour cells. Exposure to αv, β3 or β5 neutralizing antibodies, RNA interference-mediated integrin gene silencing or pharmacological integrin inhibition using the cyclic RGD peptide EMD 121974 (cilengitide) results in reduced phosphorylation of Smad2 in most glioma cell lines, including glioma-initiating cell lines and reduced transforming growth factor-β-mediated reporter gene activity, coinciding with reduced transforming growth factor-β protein levels in the supernatant. Time course experiments indicated that the loss of transforming growth factor-β bioactivity due to integrin inhibition likely results from two distinct mechanisms: an early effect on activation of preformed inactive protein, and second, major effect on transforming growth factor-β gene transcription as confirmed by decreased activity of the transforming growth factor-β gene promoter and decreased transforming growth factor-β1 and transforming growth factor-β2 messenger RNA expression levels. In vivo, EMD 121974 (cilengitide), which is currently in late clinical development as an antiangiogenic agent in newly diagnosed glioblastoma, was a weak antagonist of pSmad2 phosphorylation. These results validate integrin inhibition as a promising strategy not only to inhibit angiogenesis, but also to block transforming growth factor-β-controlled features of malignancy including invasiveness, stemness and immunosuppression in human glioblastom

ESSEX: Equipping Sparse Solvers for Exascale

The ESSEX project investigates computational issues arising at exascale for large-scale sparse eigenvalue problems and develops programming concepts and numerical methods for their solution. The project pursues a coherent co-design of all software layers where a holistic performance engineering process guides code development across the classic boundaries of application, numerical method and basic kernel library. Within ESSEX the numerical methods cover both widely applicable solvers such as classic Krylov, Jacobi-Davidson or recent FEAST methods as well as domain specific iterative schemes relevant for the ESSEX quantum physics application. This report introduces the project structure and presents selected results which demonstrate the potential impact of ESSEX for efficient sparse solvers on highly scalable heterogeneous supercomputers

Worldline Monte Carlo for fermion models at large N_f

Strongly-coupled fermionic systems can support a variety of low-energy phenomena, giving rise to collective condensation, symmetry breaking and a rich phase structure. We explore the potential of worldline Monte Carlo methods for analyzing the effective action of fermionic systems at large flavor number N_f, using the Gross-Neveu model as an example. Since the worldline Monte Carlo approach does not require a discretized spacetime, fermion doubling problems are absent, and chiral symmetry can manifestly be maintained. As a particular advantage, fluctuations in general inhomogeneous condensates can conveniently be dealt with analytically or numerically, while the renormalization can always be uniquely performed analytically. We also critically examine the limitations of a straightforward implementation of the algorithms, identifying potential convergence problems in the presence of fermionic zero modes as well as in the high-density region.Comment: 40 pages, 13 figure