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This paper presents a novel approach to construct regularizing operators for severely ill-posed Fredholm integral equations of the first kind by introducing parametrized discretization. The optimal values of discretization and regularization parameters are computed simultaneously by solving a minimization problem formulated based on a regularization parameter search criterion. The effectiveness of the proposed approach is demonstrated through examples of noisy Laplace transform inversions and the deconvolution of nuclear magnetic resonance relaxation data.

We consider {\it local} balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.

Developing an efficient computational scheme for high-dimensional Bayesian variable selection in generalised linear models and survival models has always been a challenging problem due to the absence of closed-form solutions for the marginal likelihood. The RJMCMC approach can be employed to samples model and coefficients jointly, but effective design of the transdimensional jumps of RJMCMC can be challenge, making it hard to implement. Alternatively, the marginal likelihood can be derived using data-augmentation scheme e.g. Polya-gamma data argumentation for logistic regression) or through other estimation methods. However, suitable data-augmentation schemes are not available for every generalised linear and survival models, and using estimations such as Laplace approximation or correlated pseudo-marginal to derive marginal likelihood within a locally informed proposal can be computationally expensive in the "large n, large p" settings. In this paper, three main contributions are presented. Firstly, we present an extended Point-wise implementation of Adaptive Random Neighbourhood Informed proposal (PARNI) to efficiently sample models directly from the marginal posterior distribution in both generalised linear models and survival models. Secondly, in the light of the approximate Laplace approximation, we also describe an efficient and accurate estimation method for the marginal likelihood which involves adaptive parameters. Additionally, we describe a new method to adapt the algorithmic tuning parameters of the PARNI proposal by replacing the Rao-Blackwellised estimates with the combination of a warm-start estimate and an ergodic average. We present numerous numerical results from simulated data and 8 high-dimensional gene fine mapping data-sets to showcase the efficiency of the novel PARNI proposal compared to the baseline add-delete-swap proposal.

In this paper, we present a discontinuity and cusp capturing physics-informed neural network (PINN) to solve Stokes equations with a piecewise-constant viscosity and singular force along an interface. We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface. Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface, we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity, respectively. The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs. These two augmented inputs provide the interface information, so we assume that a level set function is given and its zero level set indicates the position of the interface. The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity, while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation. We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature. Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.

Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method was also extended to find Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix tends to zero if and only if the spectral radius of its standard part less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.

It is disproved the Tokareva's conjecture that any balanced boolean function of appropriate degree is a derivative of some bent function. This result is based on new upper bounds for the numbers of bent and plateaued functions.

The HEat modulated Infinite DImensional Heston (HEIDIH) model and its numerical approximation are introduced and analyzed. This model falls into the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18), introduced for the pricing of forward contracts. The HEIDIH model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process, defined as a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the mild solution to the stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and colored in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Mat\'ern kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.

For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order $O(h^{2})$ in the interior of the interval and a boundary layer where the consistency error does not tend to zero.

We introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function ${l}$, we detail the prior and posterior items, their consequences and exhibit several examples. Taking $\Theta$ as the set of parameters and $Y$ as the set of data (which usually provides {random samples}), a general IFS is a measurable map $\tau:\Theta\times Y \to Y$, which can be interpreted as a family of maps $\tau_\theta:Y\to Y,\,\theta\in\Theta$. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of {information.} We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.

A novel overlapping domain decomposition splitting algorithm based on a Crank-Nisolson method is developed for the stochastic nonlinear Schroedinger equation driven by a multiplicative noise with non-periodic boundary conditions. The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws. Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions, as well as the various initial conditions. In particular, we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in [S. Jiang, L. Wang and J. Hong, Commun. Comput. Phys., 2013] and the finite difference splitting scheme in [J. Cui, J. Hong, Z. Liu and W. Zhou, J. Differ. Equ., 2019]. We observe that our proposed algorithm has excellent computational efficiency and is highly competitive. It provides a useful tool for solving stochastic partial differential equations.