Academic literature on the topic 'Hessian metric'

We present a unified framework for the study of Wishart matrices WpnΣ, which generalize the chi-squared distribution to matrix-variate settings and model the covariance structure of multivariate Gaussian data. After recalling their defining properties—additivity under independent summation W1+W2∼Wpn1+n2Σ, equivariance under linear maps AWAT∼WqnAΣAT, and their role as sample covariance matrices—we embed the positive-definite cone Sp+ within Monge-Ampère geometry. Here, Sp+ acquires a Hessian manifold structure with affine-invariant metric and volume form ω=detΣ−p+12dΣ, under which the Wishart density acts as a soliton of natural geometric flows. We then show that the collection of Wishart distributions forms a symmetric monoidal category W, where objects are WpnΣ and whose morphisms are linear maps A:ℝp→ℝq. The tensor product encodes block-diagonal coupling, with braiding given by block permutation, and axioms enforcing Monge-Ampère functoriality, additivity, and convex duality via the Legendre transform. Applications to quantum error correction are discussed: Wishart laws model correlated noise, Wasserstein geodesics optimize error-mitigation cost, tensor structure captures independent error channels, and Legendre duality underpins entropy-driven decoding.

Abstract During the last few years of work with D.B., it became possible to test subjects with visual defects in a constant environment (space kindly made available by the National Hospital, Queen Square, London) and with a fixed procedure, so that a spectrum of capacities could be measured for comparative purposes. D.B. was tested on some of these procedures on three occasions, in September 1979, April 1982, and lastly in August 1983. The room was light-proofed, and surfaces that might be reflective were draped with black velvet. The front wall was covered with dark green hessian material, providing the tangent screen of 100° × 100°, with a subject sitting 1 metre from the fixation point in the centre of the screen. Eight Type Q3D projectors were positioned 2.44 metres from the screen, behind the subject, with two directed to each of the four quadrants of the screen. All stimuli for testing were presented by a projecting tachistoscope (Forth Instruments), consisting of a Kodak Carousel projector fitted with a beam interrupter in the focal plane, driven by an electronically timed solenoid. A zoom lens was fitted to adjust the size of each stimulus set precisely. The projector was situated 2.44 metres from the screen, on an adjustable elevated rack.

Abstract The Kalgoorlie gold camp in the Yilgarn craton of Western Australia comprises the supergiant Golden Mile and the smaller Mt. Charlotte, Mt. Percy, and Hidden Secret deposits. Since the camp’s discovery in 1893, ~1,950 metric tons (t) of Au have been produced from a total estimated endowment of ~2,300 t. The camp is located within Neoarchean rocks of the Kalgoorlie terrane, within the Eastern Goldfields superterrane of the eastern Yilgarn craton. Gold mineralization is distributed along an 8- × 2-km, NNW-trending corridor, which corresponds to the Boulder Lefroy-Golden Mile fault system. The host stratigraphic sequence, dated at ca. 2710 to 2660 Ma, comprises lower ultramafic and mafic lava flow rocks, and upper felsic to intermediate volcaniclastic, epiclastic, and lava flow rocks intruded by highly differentiated dolerite sills such as the ca. 2685 Ma Golden Mile Dolerite. Multiple sets of NNW-trending, steeply dipping porphyry dikes intruded this sequence from ca. 2675 to 2640 Ma. From ca. 2685 to 2640 Ma, rocks of the Kalgoorlie gold camp were subjected to multiple deformation increments and metamorphism. Early D1 deformation from ca. 2685 to 2675 Ma generated the Golden Mile fault and F1 folds. Prolonged sinistral transpression from ca. 2675 to 2655 Ma produced overprinting, NNW-trending sets of D2-D3 folds and faults. The last deformation stage (D4; < ca. 2650 Ma) is recorded by N- to NNE-trending, dextral faults which offset earlier structures. The main mineralization type in the Golden Mile comprises Fimiston lodes: steeply dipping, WNW- to NNW-striking, gold- and telluride-bearing carbonate-quartz veins with banded, colloform, and crustiform textures surrounded by sericite-carbonate-quartz-pyrite-telluride alteration zones. These lodes were emplaced during the earlier stages of regional sinistral transpression (D2) as Riedel shear-type structures. During a later stage of regional sinistral transpression (D3), exceptionally high grade Oroya-type mineralization developed as shallowly plunging ore shoots with “Green Leader” quartz-sericite-carbonate-pyrite-telluride alteration typified by vanadium-bearing muscovite. In the Hidden Secret orebody, ~3 km north-northwest of the Golden Mile, lode mineralization is a silver-rich variety characterized by increased abundance of hessite and petzite and decreased abundance of calaverite. At the adjacent Mt. Charlotte deposit, the gold-, silver-, and telluride-bearing lodes become subordinate to the Mt. Charlotte-type stockwork veins. The stockwork veins occur as planar, 2- to 50-cm thick, auriferous quartz-carbonate-sulfide veins that define steeply NW- to SE-dipping and shallowly N-dipping sets broadly coeval with D4 deformation. Despite extensive research, there is no consensus on critical features of ore formation in the camp. Models suggest either (1) distinct periods of mineralization over a protracted, ca. 2.68 to 2.64 Ga orogenic history; or (2) broadly synchronous formation of the different types of mineralization at ca. 2.64 Ga. The nature of fluids, metal sources, and mineralizing processes remain debated, with both metamorphic and magmatic models proposed. There is strong evidence for multiple gold mineralization events over the course of the ca. 2.68 to 2.64 orogenic window, differing in genesis and contributions from either magmatic or metamorphic ore-forming processes. However, reconciling these models with field relationships and available geochemical and geochronological constraints remains difficult and is the subject of ongoing research.

Abstract The Recursive Quadratic Programming (RQP) method has been shown to be one of the most effective and efficient algorithms for solving engineering optimization problems. The RQP method uses variable metric updates to build approximations of the Hessian of the Lagrangian. If the approximation of the Hessian of the Lagrangian converges to the true Hessian of the Lagrangian, then the RQP method converges quadratically. The convergence of the Hessian approximation is affected by the choice of the variable metric update. Most of the research that has been performed with the RQP method uses the Broyden Fletcher Shanno (BFS) or Symmetric Rank One (SR1) variable metric update. The SR1 update has been shown to yield better estimates of the Hessian of the Lagrangian than those found when the BFS update is used, though there are cases where the SR1 update becomes unstable. This paper describes a hybrid variable metric update that is shown to yield good approximations of the Hessian of the Lagrangian. The hybrid update combines the best features of the SRI and BFS updates and is more stable than the SR1 update. Testing of the method shows that the efficiency of the RQP method is not affected by the new update, but more accurate Hessian approximations are produced. This should increase the accuracy of the solutions and provide more reliable information for post optimality analyses, such as parameter sensitivity studies.

Abstract Naturally fractured reservoirs (NFRs), such as fractured carbonate reservoirs, are commonly located worldwide and have the potential to be good sources of long-term storage of carbon dioxide (CO2). The numerical reservoir simulation models are an excellent source for evaluating the likelihood and comprehending the physics underlying behind the interaction of CO2 and brine in subsurface formations. For various reasons, including the rock's highly fractured and heterogeneous nature, the rapid spread of the CO2 plume in the fractured network, and the high capillary contrast between matrix and fractures, simulating fluid flow behavior in NFR reservoirs during CO2 injection is computationally expensive and cumbersome. This paper presents a deep-learning approach to capture the spatial and temporal dynamics of CO2 saturation plumes during the injection and monitoring periods of Geological Carbon Sequestration (GCS) sequestration in NFRs. To achieve our purpose, we have first built a base case physics-based numerical simulation model to simulate the process of CO2 injection in naturally fractured deep saline aquifers. A standalone package was coded to couple the discrete fracture network in a fully compositional numerical simulation model. Then the base case reservoir model was sampled using the Latin-Hypercube approach to account for a wide range of petrophysical, geological, reservoir, and decision parameters. These samples generated a massive physics-informed database of around 900 cases that provides a sufficient training dataset for the DL model. The performance of the DL model was improved by applying multiple filters, including the Median, Sato, Hessian, Sobel, and Meijering filters. The average absolute percentage error (AAPE), root mean square error (RMSE), Structural similarity index metric (SSIM), peak signal-to-noise ratio (PSNR), and coefficient of determination (R2) were used as error metrics to examine the performance of the surrogate DL models. The developed workflow showed superior performance by giving AAPE less than 5% and R2 more than 0.94 between ground truth and predicted values. The proposed DL-based surrogate model can be used as a quick assessment tool to evaluate the long-term feasibility of CO2 movement in a fracture carbonate medium.