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Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the…
Given pointwise samples of an unknown function belonging to a certain model set, one seeks in Optimal Recovery to recover this function in a way that minimizes the worst-case error of the recovery procedure. While it is often known that…
The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios…
Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other.…
It is proved that one cannot approximate stably the first derivative of a smooth function given noisy values of this function and a bound on this function and its first derivative. Such an approximation is shown to be possible if an a…
We present new algorithms and fast implementations to find efficient approximations for modelling stochastic processes. For many numerical computations it is essential to develop finite approximations for stochastic processes. While the…
We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary…
In many practical decision-making problems it happens that functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. In this paper, a global optimization problem is considered…
In this paper, we study optimization problems of numerical differentiation and summation methods on classes of univariate functions. Sharp estimates (in order) of the optimal recovery error and information complexity are calculated for…
We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both…
Variational representations of divergences and distances between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in…
Finite-difference methods are a class of algorithms designed to solve black-box optimization problems by approximating a gradient of the target function on a set of directions. In black-box optimization, the non-smooth setting is…
The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do not mean the accuracy within the discrete setting, which…
A heuristic formula for 5-point approximation of the first derivative of an unknown function whose values are measured with an error at unequally spaced points is proposed. The derivative at a given point is calculated using the effective…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…