Academic literature on the topic 'Sup-convolution problem'

In this paper, we consider the problem of Pareto optimal allocation in a general framework, involving preference functionals defined on a general real vector space. The optimization problem is equivalent to a modified sup-convolution of the different agents’ preference functionals. The results are then applied to a multi-period setting and some further characterization of Pareto optimality for an allocation is obtained for expected utility for processes.

In this paper, we consider the problem of Pareto optimal allocation in a general framework, involving preference functionals defined on a general real vector space. The optimization problem is equivalent to a modified sup-convolution of the different agents’ preference functionals. The results are then applied to a multi-period setting and some further characterization of Pareto optimality for an allocation is obtained for expected utility for processes.

Cauchy problem of Cahn-Hilliard equation with inertial term in three-dimensional space is considered. Using delicate analysis of its Green function and its convolution with nonlinear term, pointwise decay rate is obtained.

Abstract Objective. Alzheimer’s disease (AD), a common disease of the elderly with unknown etiology, has been adversely affecting many people, especially with the aging of the population and the younger trend of this disease. Current artificial intelligence (AI) methods based on individual information or magnetic resonance imaging (MRI) can solve the problem of diagnostic sensitivity and specificity, but still face the challenges of interpretability and clinical feasibility. In this study, we propose an interpretable multimodal deep reinforcement learning model for inferring pathological features and the diagnosis of AD. Approach. First, for better clinical feasibility, the compressed-sensing MRI image is reconstructed using an interpretable deep reinforcement learning model. Then, the reconstructed MRI is input into the full convolution neural network to generate a pixel-level disease probability risk map (DPM) of the whole brain for AD. The DPM of important brain regions and individual information are then input into the attention-based fully deep neural network to obtain the diagnosis results and analyze the biomarkers. We used 1349 multi-center samples to construct and test the model. Main results. Finally, the model obtained 99.6% ± 0.2%, 97.9% ± 0.2%, and 96.1% ± 0.3% area under curve in ADNI, AIBL and NACC, respectively. The model also provides an effective analysis of multimodal pathology, predicts the imaging biomarkers in MRI and the weight of each individual item of information. In this study, a deep reinforcement learning model was designed, which can not only accurately diagnose AD, but analyze potential biomarkers. Significance. In this study, a deep reinforcement learning model was designed. The model builds a bridge between clinical practice and AI diagnosis and provides a viewpoint for the interpretability of AI technology.

Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L, $$b_T(v, x):=\inf\left\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T], M); \gamma(T)=x\right\}\!,$$ where $M = \mathbb{R}^d$, and T >0. We also consider the stochastic counterpart: \begin{align*} \underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,\beta(t,X))\,dt\right]\!; X\in \mathcal{A}, V\sim\mu,X_T\sim \nu\right\}\!, \end{align*} where $\mathcal{A}$ is the set of stochastic processes satisfying dX = βX (t, X) dt + dWt, for some drift βX (t, X), and where Wt is σ(Xs: 0 ≤ s ≤ t)-Brownian motion. Both cases lead to Lax–Oleinik-type formulas on Wasserstein space that relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard–Buffoni and Fathi–Figalli in the deterministic case, and by Mikami–Thieullen in the stochastic setting. While inf-convolution easily covers cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian L is jointly convex on phase space, Bolza-type dualities – well known in the deterministic case but novel in the stochastic case – transform sup-inf problems to sup–sup settings. We also write Eulerian formulations and point to links with the theory of mean-field games.

The flow analysis and low dimensional representation models are of great significance to the study of the complex flow mechanism. However, the turbulent flow field has complex and unstable spatio-temporal evolution features, and it is difficult to establish the low-dimensional representation model for the flow big data. A low-dimensional representation model of complex flow based on the flow time history deep learning method is proposed and verified. One-dimensional linear convolution, nonlinear fully connected and nonlinear convolution autoencoder methods are established to reduce the dimension of unsteady flow time history data. The decoding mapping from low-dimensional space to time domain is obtained to build representation models for turbulence. The proposed method is verified using flow around the square clyinder with Re=2.2×10⁴. The results show that the flow time history deep learning method can effectively realize the low-dimensional representation of the flow and is suitable for complex turbulent flow problems; the nonlinear one-dimensional convolutional autoencoder is superior to the fully connected and linear convolution methods in representing the complex flow features. The method in this paper is an unsupervised training method, which can be widely used in single-point-based sensor data processing, and is a new method to study the characteristics of turbulence and complex flow problems.

2013/2014
It is well known that optimal risk sharing is an argument that deserves both theoretical and practical interest. It originally appears in the context of reinsurance problems, but now is widely used in a variety of financial and economical applications. The problem concerning the existence of individually rational Pareto optimal allocations, namely optimal solutions, is generally treated in the literature by considering the usual requirement of completeness over decision makers’ preferences. In this thesis we present several conditions for the existence of optimal solutions in a modern preference-based approach provided that agents’ preferences are expressed by not necessarily total preorders and by considering a topological context. We prove the equivalence between optimality and maximality with respect to a coalition preorder traducing the problem of finding optimal solutions to that of guaranteeing the existence of maximal elements for a not necessarily total preorder. In this framework a "folk theorem" is of help since it guarantees the existence of a maximal element for an upper semicontinuous preorder on a compact topological space. We study the functional approaches representing optimal risk sharing identified with the so called multi-objective maximization problem and the supconvolution problem, with the aim of incorporating functional representations of not necessarily total preorders, essentially expressed by order preserving functions and multi-utility representations. We use these two notions in order to guarantee the existence of optimal solutions, and to this aim we appropriately refer to well known results in mathematical utility theory (for example, Rader’s theorem). The case of individual preferences expressed by translation invariant total preorders is also considered, completing fundamental results from the literature also extended to the case of comonotone super-additive and positively homogeneous utility functions. When comonotone allocations are considered, we limit the research of maximal elements with respect to the coalition preorder to the set of comonotone allocations, provided that monotonicity conditions with respect to second order stochastic dominance are imposed to the individual preorders. In all our framework, we deal with risks belonging to some space of nonnegative random variables on a common probability space and, as a natural application of all our considerations, we consider the Choquet Integral when the topology L∞ is considered. Come noto, il problema di risk sharing è un argomento che interessa sia aspetti teorici che applicativi. Originariamente introdotto in contesti di riassicurazione, attualmente è ampiamente utilizzato in una varietà di applicazioni finanziarie ed economiche. Il problema legato all’esistenza di allocazioni Pareto ottimali ed individualmente razionali, definite soluzioni ottime, è generalmente trattato in letteratura considerando l’usuale assioma di completezza sulle preferenze degli agenti. In questa tesi presentiamo diverse condizioni per l'esistenza di soluzioni ottime in un moderno approccio di preferenza caratterizzato dall'espressione delle preferenze individuali per mezzo di preordini non necessariamente totali e considerando un contesto topologico. Viene dimostrata l’equivalenza tra ottimalità e massimalità rispetto ad un preordine di coalizione, traducendo così il problema di trovare soluzioni ottime nel garantire l’esistenza di elementi massimali per un preordine non necessariamente totale. In questo quadro di riferimento, un "folk theorem" è di aiuto in quanto garantisce l’esistenza di un elemento massimale per un preordine superiormente semicontinuo definito su uno spazio topologico compatto. Vengono studiati approcci funzionali legati al problema di risk sharing, identificati con il problema di massimizzazione multi-obiettivo ed il problema di sup-convoluzione, con l’obiettivo di incorporare rappresentazioni funzionali di preordini non necessariamente totali, essenzialmente definite da funzioni order preserving e rappresentazioni di multi-utilità. Queste due notazioni vengono utilizzate in modo da garantire l’esistenza di soluzioni ottime, e a questo scopo ci riferiamo in modo appropriato a ben noti risultati in teoria dell’utilità (ad esempio, il teorema di Rader). Il caso di preferenze individuali espresse da preordini totali invarianti per traslazioni è anche considerato, a completamento di fondamentali risultati presenti in letteratura ed estesi anche al caso di funzioni di utilità che soddisfino alle proprietà di comonotona super-additività e positiva omogeneità. Quando si considerano allocazioni comonotone, ci limitiamo alla ricerca di elementi massimali rispetto al preordine di coalizione nell’insieme delle allocazioni comonotone, purchè vengano imposte condizioni di monotonia sui preordini individuali rispetto alla dominanza stocastica di secondo ordine. In tutto il nostro contesto di riferimento affrontiamo il caso di rischi appartenenti a spazi di variabili aleatorie non-negative definite su un comune spazio di probabilità e come naturale applicazione consideriamo l’integrale di Choquet nel caso venga considerata la topologia L∞.
XXVII Ciclo
1985