Definition 1. A classic solution (sometimes called strong solution) to the problem is a function u ∈ C2([a, b]) such that it vanishes at the endpoints and pointwise it satisfies the equation −u′′(x)+ u(x) = f(x). Let’s change approach. Suppose that we formally multiply the differential equation by an arbitrary function φ ∈ C1([a, b]) with φ(a) = φ(b) = 0 and we integrate over the interval [a, b...

In this paper we consider the European continuous installment call option. Then  its linear complementarity formulation is given. Writing the resulted problem in variational form, we prove the existence and uniqueness of its weak solution. Finally finite element method is applied to price the European continuous installment call option.

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in Rd defines a d-dimensional space-time manifold in the space-time continuum Rd+1. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new wellposedness and stabili...

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in Rd defines a d-dimensional spacetime manifold in the space-time continuum Rd+1. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stabili...

This thesis is concerned with state estimation in partially observed diffusion processes with discrete time observations. This problem can be solved exactly in a Bayesian framework, up to a set of generally intractable stochastic partial differential equations. Numerous approximate inference methods exist to tackle the problem in a practical way. This thesis introduces a novel deterministic app...

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method. The microtubule is modeled as an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken into account as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling between the ...

Sparse Gaussian process methods that use inducing variables require the selection of the inducing inputs and the kernel hyperparameters. We introduce a variational formulation for sparse approximations that jointly infers the inducing inputs and the kernel hyperparameters by maximizing a lower bound of the true log marginal likelihood. The key property of this formulation is that the inducing i...

In this work we present a variational formulation for a multilayer perceptron neural network. With this formulation any learning task for the neural network is defined in terms of finding a function that is an extremal for some functional. Thus the multilayer perceptron provides a direct method for solving general variational problems. The application of this numerical method is investigated th...

A nonlocal variational formulation for interpolating a sparsely sampled image is introduced in this paper. The proposed variational formulation, originally motivated by image inpainting problems, encourages the transfer of information between similar image patches, following the paradigm of exemplar-based methods. Contrary to the classical inpainting problem, no complete patches are available f...